Online math open (OMO) 2012 2017 fall, EN with solutions

The Online Math Open Fall Contest
September 24-October 1, 2012


Contest Information
Format
The test will start Monday September 24 and end Monday October 1. You will have until 7pm EST on
October 1 to submit your answers. The test consists of 30 short answer questions, each of which has
a nonnegative integer answer. The problem difficulties range from those of AMC problems to those of
Olympiad problems. Problems are ordered in roughly increasing order of difficulty.

Team Guidelines
Students may compete in teams of up to four people. Participating students must not have graduated from
high school. International students may participate. No student can be a part of more than one team. The
members of each team do not get individual accounts; they will all share the team account.
Each team will submit its final answers through its team account. Though teams can save drafts for their
answers, the current interface does not allow for much flexibility in communication between team members.
We recommend using Google Docs and Spreadsheets to discuss problems and compare answers, especially if
teammates cannot communicate in person. Teams may spend as much time as they like on the test before
the deadline.

Aids
Drawing aids such as graph paper, ruler, and compass are permitted. However, electronic drawing aids are
not allowed. This is includes (but is not limited to) Geogebra and graphing calculators. Published print
and electronic resources are not permitted. (This is a change from last year’s rules.)
Four-function calculators are permitted on the Online Math Open. That is, calculators which perform only
the four basic arithmetic operations (+-*/) may be used. Any other computational aids such as scientific
and graphing calculators, computer programs and applications such as Mathematica, and online databases
is prohibited. All problems on the Online Math Open are solvable without a calculator. Four-function
calculators are permitted only to help participants reduce computation errors.

Clarifications
Clarifications will be posted as they are answered. For the Fall 2012-2013 Contest, they will be posted at here.
If you have a question about a problem, please email with “Clarification”
in the subject. We have the right to deny clarification requests that we feel we cannot answer.

Scoring
Each problem will be worth one point. Ties will be broken based on the highest problem number that a
team answered correctly. If there are still ties, those will be broken by the second highest problem solved,
and so on.

Results
After the contest is over, we will release the answers to the problems within the next day. If you have a
protest about an answer, you may send an email to (Include “Protest”
in the subject). Solutions and results will be released in the following weeks.


September/October 2012

Fall OMO 2012-2013

Page 3

1. Calvin was asked to evaluate 37 + 31 × a for some number a. Unfortunately, his paper was tilted 45
degrees, so he mistook multiplication for addition (and vice versa) and evaluated 37 × 31 + a instead.
Fortunately, Calvin still arrived at the correct answer while still following the order of operations. For
what value of a could this have happened?
2. Petya gave Vasya a number puzzle. Petya chose a digit X and said, “I am thinking of a number that
is divisible by 11. The hundreds digit is X and the tens digit is 3. Find the units digit.” Vasya was
excited because he knew how to solve this problem, but then realized that the problem Petya gave did
not have an answer. What digit X did Petya chose?

3. Darwin takes an 11 × 11 grid of lattice points and connects every pair of points that are 1 unit apart,
creating a 10 × 10 grid of unit squares. If he never retraced any segment, what is the total length of
all segments that he drew?
4. Let lcm(a, b) denote the least common multiple of a and b. Find the sum of all positive integers x such
that x ≤ 100 and lcm(16, x) = 16x.
5. Two circles have radius 5 and 26. The smaller circle passes through center of the larger one. What
is the difference between the lengths of the longest and shortest chords of the larger circle that are
tangent to the smaller circle?
6. An elephant writes a sequence of numbers on a board starting with 1. Each minute, it doubles the
sum of all the numbers on the board so far, and without erasing anything, writes the result on the
board. It stops after writing a number greater than one billion. How many distinct prime factors does
the largest number on the board have?
7. Two distinct points A and B are chosen at random from 15 points equally spaced around a circle
centered at O such that each pair of points A and B has the same probability of being chosen. The
probability that the perpendicular bisectors of OA and OB intersect strictly inside the circle can be
expressed in the form m
n , where m, n are relatively prime positive integers. Find m + n.
8. In triangle ABC let D be the foot of the altitude from A. Suppose that AD = 4, BD = 3, CD = 2,
and AB is extended past B to a point E such that BE = 5. Determine the value of CE 2 .
9. Define a sequence of integers by T1 = 2 and for n ≥ 2, Tn = 2Tn−1 . Find the remainder when
T1 + T2 + · · · + T256 is divided by 255.
10. There are 29 unit squares in the diagram below. A frog starts in one of the five (unit) squares on the
top row. Each second, it hops either to the square directly below its current square (if that square
exists), or to the square down one unit and left one unit of its current square (if that square exists),
until it reaches the bottom. Before it reaches the bottom, it must make a hop every second. How many
distinct paths (from the top row to the bottom row) can the frog take?


September/October 2012

Fall OMO 2012-2013

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11. Let ABCD be a rectangle. Circles with diameters AB and CD meet at points P and Q inside the
rectangle such that P is closer to segment BC than Q. Let M and N be the midpoints of segments
AB and CD. If ∠M P N = 40◦ , find the degree measure of ∠BP C.
12. Let a1 , a2 , . . . be a sequence defined by a1 = 1 and for n ≥ 1, an+1 =

a2n − 2an + 3 + 1. Find a513 .

13. A number is called 6-composite if it has exactly 6 composite factors. What is the 6th smallest 6composite number? (A number is composite if it has a factor not equal to 1 or itself. In particular, 1
is not composite.)
14. When Applejack begins to buck trees, she starts off with 100 energy. Every minute, she may either
choose to buck n trees and lose 1 energy, where n is her current energy, or rest (i.e. buck 0 trees) and
gain 1 energy. What is the maximum number of trees she can buck after 60 minutes have passed?
15. How many sequences of nonnegative integers a1 , a2 , . . . , an (n ≥ 1) are there such that a1 · an > 0,
n−1

a1 + a2 + · · · + an = 10, and

(ai + ai+1 ) > 0?
i=1

16. Let ABC be a triangle with AB = 4024, AC = 4024, and BC = 2012. The reflection of line AC over
line AB meets the circumcircle of ABC at a point D = A. Find the length of segment CD.
17. Find the number of integers a with 1 ≤ a ≤ 2012 for which there exist nonnegative integers x, y, z
satisfying the equation
x2 (x2 + 2z) − y 2 (y 2 + 2z) = a.
18. There are 32 people at a conference. Initially nobody at the conference knows the name of anyone else.

The conference holds several 16-person meetings in succession, in which each person at the meeting
learns (or relearns) the name of the other fifteen people. What is the minimum number of meetings
needed until every person knows everyone elses name?
19. In trapezoid ABCD, AB < CD, AB ⊥ BC, AB CD, and the diagonals AC, BD are perpendicular
at point P . There is a point Q on ray CA past A such that QD ⊥ DC. If
AP
QP
+
=
AP
QP
AP
BP
−
can be expressed in the form
then
AP
BP
m + n.

m
n

51
14

4

− 2,

for relatively prime positive integers m, n. Compute

20. The numbers 1, 2, . . . , 2012 are written on a blackboard. Each minute, a student goes up to the board,
chooses two numbers x and y, erases them, and writes the number 2x + 2y on the board. This
continues until only one number N remains. Find the remainder when the maximum possible value of
N is divided by 1000.
21. A game is played with 16 cards laid out in a row. Each card has a black side and a red side, and
initially the face-up sides of the cards alternate black and red with the leftmost card black-side-up. A
move consists of taking a consecutive sequence of cards (possibly only containing 1 card) with leftmost


September/October 2012

Fall OMO 2012-2013

Page 5

card black-side-up and the rest of the cards red-side-up, and flipping all of these cards over. The game
ends when a move can no longer be made. What is the maximum possible number of moves that can
be made before the game ends?
22. Let c1 , c2 , . . . , c6030 be 6030 real numbers. Suppose that for any 6030 real numbers a1 , a2 , . . . , a6030 ,
there exist 6030 real numbers {b1 , b2 , . . . , b6030 } such that
n

an =

bgcd(k,n)
k=1

and

bn =

cd an/d
d|n

for n = 1, 2, . . . , 6030. Find c6030 .
23. For reals x ≥ 3, let f (x) denote the function
f (x) =

√
−x + x 4x − 3
.
2

Let a1 , a2 , . . ., be the sequence satisfying a1 > 3, a2013 = 2013, and for n = 1, 2, . . . , 2012, an+1 = f (an ).
Determine the value of
2012
a3i+1
a1 +
.
a2 + ai ai+1 + a2i+1
i=1 i
24. In scalene ABC, I is the incenter, Ia is the A-excenter, D is the midpoint of arc BC of the circumcircle
of ABC, and M is the midpoint of side BC. Extend ray IM past M to point P such that IM = M P .
Let Q be the intersection of DP and M Ia , and R be the point on the line M Ia such that AR DP .
QM
m
a
Given that AI
AI = 9, the ratio RIa can be expressed in the form n for two relatively prime positive

integers m, n. Compute m + n.
25. Suppose 2012 reals are selected independently and at random from the unit interval [0, 1], and then
1
written in nondecreasing order as x1 ≤ x2 ≤ · · · ≤ x2012 . If the probability that xi+1 − xi ≤ 2011
for
m
i = 1, 2, . . . , 2011 can be expressed in the form n for relatively prime positive integers m, n, find the
remainder when m + n is divided by 1000.
26. Find the smallest positive integer k such that
x + kb
12

≡

x
12

(mod b)

for all positive integers b and x. (Note: For integers a, b, c we say a ≡ b (mod c) if and only if a − b is
divisible by c.)
27. Let ABC be a triangle with circumcircle ω. Let the bisector of ∠ABC meet segment AC at D and
circle ω at M = B. The circumcircle of BDC meets line AB at E = B, and CE meets ω at P = C.
The bisector of ∠P M C meets segment AC at Q = C. Given that P Q = M C, determine the degree
measure of ∠ABC.


September/October 2012

28. Find the remainder when

Fall OMO 2012-2013

216

16
2k
(3 · 214 + 1)k (k − 1)2 −1
k

k=1

is divided by 2

16

Page 6

+ 1. (Note: It is well-known that 216 + 1 = 65537 is prime.)

29. In the Cartesian plane, let Si,j = {(x, y) | i ≤ x ≤ j}. For i = 0, 1, . . . , 2012, color Si,i+1 pink if i is
even and gray if i is odd. For a convex polygon P in the plane, let d(P ) denote its pink density, i.e.
the fraction of its total area that is pink. Call a polygon P pinxtreme if it lies completely in the region
S0,2013 and has at least one vertex on each of the lines x = 0 and x = 2013. Given that the minimum
value of d(P ) over all non-degenerate convex pinxtreme polygons P in the plane can be expressed in
the form

√
(1+ p)2
q2

for positive integers p, q, find p + q.

30. Let P (x) denote the polynomial
9

1209

xk + 2

3
k=0

146409

xk +
k=10

xk .
k=1210

Find the smallest positive integer n for which there exist polynomials f, g with integer coefficients
satisfying xn − 1 = (x16 + 1)P (x)f (x) + 11 · g(x).


September/October 2012

Fall OMO 2012-2013

Acknowledgments

Contest Directors
Ray Li, James Tao, Victor Wang

Head Problem Writers
Ray Li, Victor Wang

Problem Contributors
Ray Li, James Tao, Anderson Wang, Victor Wang, David Yang, Alex Zhu

Proofreaders and Test Solvers
Mitchell Lee, James Tao, Anderson Wang, David Yang, Alex Zhu

Website Manager
Ray Li

Page 7


The Online Math Open
January 16-23, 2012


Contest Information
Format
The test will start Monday January 16 and end Monday January 23. The test consists of 50 short answer
questions, each of which has a nonnegative integer answer. The problem difficulties range from those of AMC
problems to those of Olympiad problems. Problems are ordered in roughly increasing order of difficulty.

Team Guidelines
Students may compete in teams of up to four people. Participating students must not have graduated from

high school. International students may participate. No student can be a part of more than one team. The
members of each team do not get individual accounts; they will all share the team account.
The team will submit their final answers through their account. Though teams can save drafts for their
answers, the current interface does not allow for much flexibility in communication between team members.
We recommend using Google Docs and Spreadsheets to discuss problems and compare answers, especially if
teammates cannot communicate in person. Teams may spend as much time as they like on the test before
the deadline.

Aids
Drawing aids such as graph paper, ruler, and compass are permitted. However, electronic drawing aids are
not allowed. This is includes (but is not limited to) Geogebra and graphing calculators. Published print and
electronic resources are permitted.
Four-function calculators are permitted on the Online Math Open. That is, calculators which perform only
the four basic arithmetic operations (+-*/) may be used. No other computational aids such as scientific
and graphing calculators, computer programs and applications such as Mathematica, and online databases
are permitted. All problems on the Online Math Open are solvable without a calculator. Four-function
calculators are permitted only to help participants reduce computation errors.

Scoring
Each problem will be worth one point. Ties will be broken based on the highest problem number that a
team answered correctly. If there are still ties, those will be broken by the second highest problem solved,
and so on.

Results
After the contest is over, we will release the answers to the problems within the next day. If you have a
protest about an answer, you may send an email to (Include ”Protest” in
the subject). Solutions and results will be released in the following weeks.


January 2012

OMO 2012

Page 3

1. The average of two positive real numbers is equal to their difference. What is the ratio of the larger
number to the smaller one?
2. How many ways are there to arrange the letters A, A, A, H, H in a row so that the sequence HA appears
at least once?
3. A lucky number is a number whose digits are only 4 or 7. What is the 17th smallest lucky number?
4. How many positive even numbers have an even number of digits and are less than 10000?
5. Congruent circles Γ1 and Γ2 have radius 2012, and the center of Γ1 lies on Γ2 . Suppose that Γ1 and
Γ2 intersect at A and B. The line through A perpendicular to AB meets Γ1 and Γ2 again at C and
D, respectively. Find the length of CD.
6. Alice’s favorite number has the following properties:
• It has 8 distinct digits.
• The digits are decreasing when read from left to right.
• It is divisible by 180.
What is Alice’s favorite number?
7. A board 64 inches long and 4 inches high is inclined so that the long side of the board makes a 30
degree angle with the ground.
√ The distance from the highest point on the board to the ground can be
expressed in the form a + b c where a, b, c are positive integers and c is not divisible by the square of
any prime. What is a + b + c?
8. An 8 × 8 × 8 cube is painted red on 3 faces and blue on 3 faces such that no corner is surrounded by
three faces of the same color. The cube is then cut into 512 unit cubes. How many of these cubes
contain both red and blue paint on at least one of their faces?
9. At a certain grocery store, cookies may be bought in boxes of 10 or 21. What is the minimum positive
number of cookies that must be bought so that the cookies may be split evenly among 13 people?
10. A drawer has 5 pairs of socks. Three socks are chosen at random. If the probability that there is a

pair among the three is m
n , where m and n are relatively prime positive integers, what is m + n?
11. If

1
1
1
1
1
1
+ 2+ 3+ 4+
+ ··· =
,
5
x 2x
4x
8x
16x
64
and x can be expressed in the form m
n , where m, n are relatively prime positive integers, find m + n.

12. A cross-pentomino is a shape that consists of a unit square and four other unit squares each sharing
a different edge with the first square. If a cross-pentomino is inscribed in a circle of radius R, what is
100R2 ?


January 2012

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13. A circle ω has center O and radius r. A chord BC of ω also has length r, and the tangents to ω at
B and C meet at A. Ray AO meets ω at D past O, and ray OA meets the circle centered at A with
radius AB at E past A. Compute the degree measure of ∠DBE.
14. Al told Bob that he was thinking of 2011 distinct positive integers. He also told Bob the sum of those
integers. From this information, Bob was able to determine all 2011 integers. How many possible sums
could Al have told Bob?
15. Five bricklayers working together finish a job in 3 hours. Working alone, each bricklayer takes at most
36 hours to finish the job. What is the smallest number of minutes it could take the fastest bricklayer
to complete the job alone?
16. Let A1 B1 C1 D1 A2 B2 C2 D2 be a unit cube, with A1 B1 C1 D1 and A2 B2 C2 D2 opposite square faces, and
let M be the center of face A2 B2 C2 D2 . Rectangular pyramid M A1 B1 C√
1 D1 is cut out of the cube. If
the surface area of the remaining solid can be expressed in the form a + b, where a and b are positive
integers and b is not divisible by the square of any prime, find a + b.
17. Each pair of vertices of a regular 10-sided polygon is connected by a line segment. How many unordered
pairs of distinct parallel line segments can be chosen from these segments?
18. The sum of the squares of three positive numbers is 160. One of the numbers is equal to the sum of
the other two. The difference between the smaller two numbers is 4. What is the difference between
the cubes of the smaller two numbers?
19. There are 20 geese numbered 1 through 20 standing in a line. The even numbered geese are standing
at the front in the order 2, 4, . . . , 20, where 2 is at the front of the line. Then the odd numbered geese
are standing behind them in the order, 1, 3, 5, . . . , 19, where 19 is at the end of the line. The geese
want to rearrange themselves in order, so that they are ordered 1, 2, . . . , 20 (1 is at the front), and they
do this by successively swapping two adjacent geese. What is the minimum number of swaps required
to achieve this formation?
20. Let ABC be a right triangle with a right angle at C. Two lines, one parallel to AC and the other
parallel to BC, intersect on the hypotenuse AB. The lines cut the triangle into two triangles and a

rectangle. The two triangles have areas 512 and 32. What is the area of the rectangle?
21. If

2012

20112011

= xx

for some positive integer x, how many positive integer factors does x have?
22. Find the largest prime number p such that when 2012! is written in base p, it has at least p trailing
zeroes.
23. Let ABC be an equilateral triangle with side length 1. This triangle is rotated by some angle about
its center to form triangle DEF. The intersection of ABC and DEF is an equilateral hexagon with
an area that is 54 the area of ABC. The side length of this hexagon can be expressed in the form m
n
where m and n are relatively prime positive integers. What is m + n?
24. Find the number of ordered pairs of positive integers (a, b) with a + b prime, 1 ≤ a, b ≤ 100, and
is an integer.

ab+1
a+b


January 2012

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25. Let a, b, c be the roots of the cubic x3 + 3x2 + 5x + 7. Given that P is a cubic polynomial such that
P (a) = b + c, P (b) = c + a, P (c) = a + b, and P (a + b + c) = −16, find P (0).
26. Xavier takes a permutation of the numbers 1 through 2011 at random, where each permutation has
an equal probability of being selected. He then cuts the permutation into increasing contiguous subsequences, such that each subsequence is as long as possible. Compute the expected number of such
subsequences.
27. Let a and b be real numbers that satisfy
a4 + a2 b2 + b4 = 900,
a2 + ab + b2 = 45.
Find the value of 2ab.
28. A fly is being chased by three spiders on the edges of a regular octahedron. The fly has a speed of 50
meters per second, while each of the spiders has a speed of r meters per second. The spiders choose the
(distinct) starting positions of all the bugs, with the requirement that the fly must begin at a vertex.
Each bug knows the position of each other bug at all times, and the goal of the spiders is for at least
one of them to catch the fly. What is the maximum c so that for any r < c, the fly can always avoid
being caught?
29. How many positive integers a with a ≤ 154 are there such that the coefficient of xa in the expansion of
(1 + x7 + x14 + · · · + x77 )(1 + x11 + x22 + · · · + x77 )
is zero?
30. The Lattice Point Jumping Frog jumps between lattice points in a coordinate plane that are exactly
1 unit apart. The Lattice Point Jumping Frog starts at the origin and makes 8 jumps, ending at
the origin. Additionally, it never lands on a point other than the origin more than once. How many
possible paths could the frog have taken?
31. Let ABC be a triangle inscribed in circle Γ, centered at O with radius 333. Let M be the midpoint
of AB, N be the midpoint of AC, and D be the point where line AO intersects BC. Given that lines
M N and BO concur on Γ and that BC = 665, find the length of segment AD.
32. The sequence {an } satisfies a0 = 201, a1 = 2011, and an = 2an−1 + an−2 for all n ≥ 2. Let
∞

S=
i=1

What is

ai−1
a2i − a2i−1

1
S?

33. You are playing a game in which you have 3 envelopes, each containing a uniformly random amount
of money between 0 and 1000 dollars. (That is, for any real 0 ≤ a < b ≤ 1000, the probability that the
b−a
amount of money in a given envelope is between a and b is 1000
.) At any step, you take an envelope and
look at its contents. You may choose either to keep the envelope, at which point you finish, or discard
it and repeat the process with one less envelope. If you play to optimize your expected winnings, your
expected winnings will be E. What is E , the greatest integer less than or equal to E?


January 2012

OMO 2012

Page 6

34. Let p, q, r be real numbers satisfying
(p + q)(q + r)(r + p)
= 24
pqr
(p − 2q)(q − 2r)(r − 2p)

= 10.
pqr
Given that pq + rq + pr can be expressed in the form
compute m + n.

m
n,

where m, n are relatively prime positive integers,

35. Let s(n) be the number of 1’s in the binary representation of n. Find the number of ordered pairs of
integers (a, b) with 0 ≤ a < 64, 0 ≤ b < 64 and s(a + b) = s(a) + s(b) − 1.
36. Let sn be the number of solutions to a1 + a2 + a3 + a4 + b1 + b2 = n, where a1 , a2 , a3 and a4 are elements
of the set {2, 3, 5, 7} and b1 and b2 are elements of the set {1, 2, 3, 4}. Find the number of n for which
sn is odd.
37. In triangle ABC, AB = 1 and AC = 2. Suppose there exists a point P in the interior of triangle
ABC such that ∠P BC = 70◦ , and that there are points E and D on segments AB and AC, such that
∠BP E = ∠EP A = 75◦ and ∠AP D = ∠DP C = 60◦ . Let BD meet CE at Q, and let AQ meet BC
at F. If M is the midpoint of BC, compute the degree measure of ∠M P F.
38. Let S denote the sum of the 2011th powers of the roots of the polynomial (x − 20 )(x − 21 ) · · · (x −
22010 ) − 1. How many 1’s are in the binary expansion of S?
39. For positive integers n, let ν3 (n) denote the largest integer k such that 3k divides n. Find the number
of subsets S (possibly containing 0 or 1 elements) of {1, 2, . . . , 81} such that for any distinct a, b ∈ S,
ν3 (a − b) is even.
40. Suppose x, y, z, and w are positive reals such that
x2 + y 2 −

wz
xy
= w2 + z 2 +

= 36
2
2
xz + yw = 30.

Find the largest possible value of (xy + wz)2 .
41. Find the remainder when

63

i=2

i2011 − i
.
i2 − 1

is divided by 2016.
42. In triangle ABC, sin ∠A = 45 and ∠A < 90◦ Let D be a point outside triangle ABC such that
3
∠BAD = ∠DAC and ∠BDC = 90◦ . Suppose that AD = 1 and that BD
CD = 2 . If AB + AC can be
expressed in the form

√
a b
c

where a, b, c are pairwise relatively prime integers, find a + b + c?

43. An integer x is selected at random between 1 and 2011! inclusive. The probability that xx − 1 is

divisible by 2011 can be expressed in the form m
n , where m and n are relatively prime positive integers.
Find m.


January 2012

OMO 2012

Page 7

44. Given a set of points in space, a jump consists of taking two points in the set, P and Q, removing P
from the set, and replacing it with the reflection of P over Q. Find the smallest number n such that for
any set of n lattice points in 10-dimensional-space, it is possible to perform a finite number of jumps
so that some two points coincide.
45. Let K1 , K2 , K3 , K4 , K5 be 5 distinguishable keys, and let D1 , D2 , D3 , D4 , D5 be 5 distinguishable doors.
For 1 ≤ i ≤ 5, key Ki opens doors Di and Di+1 (where D6 = D1 ) and can only be used once. The keys
and doors are placed in some order along a hallway. Key$ha walks into the hallway, picks a key and
opens a door with it, such that she never obtains a key before all the doors in front of it are unlocked.
In how many such ways can the keys and doors be ordered if Key$ha can open all the doors?
46. Let f is a function from the set of positive integers to itself such that f (x) ≤ x2 for all natural numbers
x, and f (f (f (x))f (f (y))) = xy for all natural numbers x and y. Find the number of possible values
of f (30).
√
47. Let ABCD be an isosceles trapezoid with bases AB = 5 and CD = 7 and legs BC = AD = 2 10. A
circle ω with center O passes through A, B, C, and D. Let M be the midpoint of segment CD, and ray
AM meet ω again at E. Let N be the midpoint of BE and P be the intersection of BE with CD. Let
Q be the intersection of ray ON with ray DC. There is a point R on the circumcircle of P N Q such
that ∠P RC = 45◦ . The length of DR can be expressed in the form m
n where m and n are relatively

prime positive integers. What is m + n?
48. Suppose that
982

7i

2

i=1

can be expressed in the form 983q + r, where q and r are integers and 0 ≤ r ≤ 492. Find r.
49. Find the magnitude of the product of all complex numbers c such that the recurrence defined by x1 = 1,
x2 = c2 − 4c + 7, and xn+1 = (c2 − 2c)2 xn xn−1 + 2xn − xn−1 also satisfies x1006 = 2011.
50. In tetrahedron SABC, the circumcircles of faces SAB, SBC, and SCA each have radius 108 The
inscribed sphere of SABC, centered at I, has radius 35. Additionally, SI = 125. Let R is the largest
possible value of the circumradius of face ABC. Give that R can be expressed in the form m
n , where
m and n are relatively prime positive integers, find m + n.


January 2012

OMO 2012

Acknowledgments
Test Directors
Ray Li, Anderson Wang, and Alex Zhu

Head Problem Writers
Ray Li, Anderson Wang, and Alex Zhu

Problem Contributors
Mitchell Lee, Ray Li, Anderson Wang, and Alex Zhu

Proofreaders
Timothy Chu, Mitchell Lee, Victor Wang, David Yang, and George Xing

Website Manager
Mitchell Lee

Page 8


The Online Math Open Fall Contest
October 18 - 29, 2013


Acknowledgements
Contest Directors
• Evan Chen

Head Problem Writers
• Evan Chen
• Michael Kural
• David Stoner

Problem Contributors, Proofreaders, and Test Solvers
• Ray Li
• Calvin Deng
• Mitchell Lee

• James Tao
• Anderson Wang
• Victor Wang
• David Yang
• Alex Zhu

Website Manager
• Douglas Chen

LATEX/Python Geek
• Evan Chen


Contest Information
These rules supersede any rules found elsewhere about the OMO. Please send any further questions directly
to the OMO Team at

Team Registration and Eligibility
Students may compete in teams of up to four people, but no student can belong to more than one team.
Participants must not have graduated from high school (or the equivalent secondary school institution in
other countries). Teams need not remain the same between the Fall and Spring contests, and students are
permitted to participate in one contest but not the other.
Only one member on each team needs to register an account on the website. Please check the
website, for registration instructions.
Note: when we say “up to four”, we really do mean “up to”! Because the time limit is so long, partial teams
are not significantly disadvantaged, and we welcome their participation.

Contest Format and Rules
The 2013 Fall Contest will consist of 30 problems; the answer to each problem will be a nonnegative
integer not exceeding 263 − 2 = 9223372036854775806. The contest window will be October 18 - 29,

2013, from 7PM ET on the start day to 7PM ET on the end day. There is no time limit other than the
contest window.
1. Four-function calculators (calculators which can perform only the four basic arithmetic operations)
are permitted on the Online Math Open. Any other computational aids, including scientific
calculators, graphing calculators, or computer programs is prohibited. All problems on the
Online Math Open are solvable without a calculator. Four-function calculators are permitted only to
help participants reduce computation errors.
2. Drawing aids such as graph paper, ruler, and compass are permitted. However, electronic drawing
aids, such as Geogebra and graphing calculators, are not allowed. Print and electronic
publications are also not allowed.
3. Members of different teams cannot communicate with each other about the contest while the contest
is running.
4. Your score is the number of questions answered correctly; that is, every problem is worth one point.
Ties will be broken based on the ”hardest” problem that a team answered correctly. Remaining ties
will be broken by the second hardest problem solved, and so on. (Problem m is harder than problem
n if fewer teams solve problem m OR if the number of solves is equal and m > n.)
5. Participation in the Online Math Open is free.

Clarifications and Results
Clarifications will be posted as they are answered. For the most recent contests, they will be posted at
If you have a question about problem wording,
please email with “Clarification” in the subject. We have the right to deny
clarification requests that we feel we cannot answer.
After the contest is over, we will release the answers to the problems within the next day. Please do not
discuss the test until answers are released. If you have a protest about an answer, you may send an email
to (Include “Protest” in the subject). Results will be released in the
following weeks. (Results will be counted only for teams that submit answers at least once. Teams that only
register an account will not be listed in the final rankings.)

OMO Fall 2013
October 18 - 29, 2013
1. Determine the value of 142857 + 285714 + 428571 + 571428.
2. The figure below consists of several unit squares, M of which are white and N of which are green.
Compute 100M + N .

3. A palindromic table is a 3 × 3 array of letters such that the words in each row and column read the
same forwards and backwards. An example of such a table is shown below.
O
N
O

M
M
M

O
N
O

How many palindromic tables are there that use only the letters O and M ? (The table may contain
only a single letter.)
4. Suppose a1 , a2 , a3 , . . . is an increasing arithmetic progression of positive integers. Given that a3 = 13,
compute the maximum possible value of
aa1 + aa2 + aa3 + aa4 + aa5 .
5. A wishing well is located at the point (11, 11) in the xy-plane. Rachelle randomly selects an integer
y from the set {0, 1, . . . , 10}. Then she randomly selects, with replacement, two integers a, b from the
set {1, 2, . . . , 10}. The probability the line through (0, y) and (a, b) passes through the well can be
expressed as m
n , where m and n are relatively prime positive integers. Compute m + n.

6. Find the number of integers n with n ≥ 2 such that the remainder when 2013 is divided by n is equal
to the remainder when n is divided by 3.
7. Points M , N , P are selected on sides AB, AC, BC, respectively, of triangle ABC. Find the area of
triangle M N P given that AM = M B = BP = 15 and AN = N C = CP = 25.
8. Suppose that x1 < x2 < · · · < xn is a sequence of positive integers such that xk divides xk+2 for each
k = 1, 2, . . . , n − 2. Given that xn = 1000, what is the largest possible value of n?
9. Let AXY ZB be a regular pentagon with area 5 inscribed in a circle with center O. Let Y denote the
reflection of Y over AB and suppose C is the center of a circle passing through A, Y and B. Compute
the area of triangle ABC.
10. In convex quadrilateral AEBC, ∠BEA = ∠CAE = 90◦ and AB = 15, BC = 14 and CA = 13. Let D
be the foot of the altitude from C to AB. If ray CD meets AE at F , compute AE · AF .
11. Four orange lights are located at the points (2, 0), (4, 0), (6, 0) and (8, 0) in the xy-plane. Four yellow
lights are located at the points (1, 0), (3, 0), (5, 0), (7, 0). Sparky chooses one or more of the lights to
turn on. In how many ways can he do this such that the collection of illuminated lights is symmetric
around some line parallel to the y-axis?
12. Let an denote the remainder when (n + 1)3 is divided by n3 ; in particular, a1 = 0. Compute the
remainder when a1 + a2 + · · · + a2013 is divided by 1000.
1


OMO Fall 2013
October 18 - 29, 2013
13. In the rectangular table shown below, the number 1 is written in the upper-left hand corner, and every
number is the sum of the any numbers directly to its left and above. The table extends infinitely
downwards and to the right.
1 1 1 1 1 ···
1 2 3 4 5 ···
1 3 6 10 15 · · ·
1 4 10 20 35 · · ·
1 5 15 35 70 · · ·

.. .. ..
..
.. . .
.
. . .
.
.
Wanda the Worm, who is on a diet after a feast two years ago, wants to eat n numbers (not necessarily
distinct in value) from the table such that the sum of the numbers is less than one million. However,
she cannot eat two numbers in the same row or column (or both). What is the largest possible value
of n?
14. In the universe of Pi Zone, points are labeled with 2 × 2 arrays of positive reals. One can teleport from
point M to point M if M can be obtained from M by multiplying either a row or column by some
1 2
1 20
1 20
positive real. For example, one can teleport from
to
and then to
.
3 4
3 40
6 80
A tourist attraction is a point where each of the entries of the associated array is either 1, 2, 4, 8 or
16. A company wishes to build a hotel on each of several points so that at least one hotel is accessible
from every tourist attraction by teleporting, possibly multiple times. What is the minimum number
of hotels necessary?
15. Find the positive integer n such that
f (f (· · · f (n) · · · )) = 20142 + 1
2013 f ’s

where f (n) denotes the nth positive integer which is not a perfect square.
16. Al has the cards 1, 2, . . . , 10 in a row in increasing order. He first chooses the cards labeled 1, 2, and
3, and rearranges them among their positions in the row in one of six ways (he can leave the positions
unchanged). He then chooses the cards labeled 2, 3, and 4, and rearranges them among their positions
in the row in one of six ways. (For example, his first move could have made the sequence 3, 2, 1, 4, 5, . . . ,
and his second move could have rearranged that to 2, 4, 1, 3, 5, . . . .) He continues this process until he
has rearranged the cards with labels 8, 9, 10. Determine the number of possible orderings of cards he
can end up with.
17. Let ABXC be a parallelogram. Points K, P, Q lie on BC in this order such that BK = 13 KC and
BP = P Q = QC = 13 BC. Rays XP and XQ meet AB and AC at D and E, respectively. Suppose
that AK ⊥ BC, EK − DK = 9 and BC = 60. Find AB + AC.
18. Given an n × n grid of dots, let f (n) be the largest number of segments between adjacent dots which
can be drawn such that (i) at most one segment is drawn between each pair of dots, and (ii) each dot
has 1 or 3 segments coming from it. (For example, f (4) = 16.) Compute f (2000).
19. Let σ(n) be the number of positive divisors of n, and let rad n be the product of the distinct prime
divisors of n. By convention, rad 1 = 1. Find the greatest integer not exceeding
∞

σ(n)σ(n rad n)
100
n2 σ(rad n)
n=1

1
3

.

20. A positive integer n is called mythical if every divisor of n is two less than a prime. Find the unique

mythical number with the largest number of divisors.
2


OMO Fall 2013
October 18 - 29, 2013
21. Let ABC be a triangle with AB = 5, AC = 8, and BC = 7. Let D be on side AC such that AD = 5
and CD = 3. Let I be the incenter of triangle
ABC and E be the intersection of the perpendicular
√
a b
bisectors of ID and BC. Suppose DE = c where a and c are relatively prime positive integers, and
b is a positive integer not divisible by the square of any prime. Find a + b + c.
22. Find the sum of all integers m with 1 ≤ m ≤ 300 such that for any integer n with n ≥ 2, if 2013m
divides nn − 1 then 2013m also divides n − 1.
23. Let ABCDE be a regular pentagon, and let F be a point on AB with ∠CDF = 55◦ . Suppose F C
and BE meet at G, and select H on the extension of CE past E such that ∠DHE = ∠F DG. Find
the measure of ∠GHD, in degrees.
√
1
24. The real numbers a0 , a1 , . . . , a2013 and b0 , b1 , . . . , b2013 satisfy an = 63
2n + 2 + an−1 and bn =
√
1
2n
+
2
−
b
for

every
integer
n
=
1,
2,
.
.
.
,
2013.
If
a
=
b
and
b
=
a2013 , compute
n−1
0
2013
0
96
2013

(ak bk−1 − ak−1 bk ) .
k=1

25. Let ABCD be a quadrilateral with AD = 20 and BC = 13. The area of

of DBC is 212. Compute the smallest possible perimeter of ABCD.

ABC is 338 and the area

26. Let ABC be a triangle with AB = 13, AC = 25, and tan A = 43 . Denote the reflections of B, C across
AC, AB by D, E, respectively, and let O be the circumcenter of triangle ABC. Let P be a point such
that DP O ∼ P EO, and let X and Y be the midpoints of the major and minor arcs BC of the
circumcircle of triangle ABC. Find P X · P Y .
27. Ben has a big blackboard, initially empty, and Francisco has a fair coin. Francisco flips the coin 2013
times. On the nth flip (where n = 1, 2, . . . , 2013), Ben does the following if the coin flips heads:
(i) If the blackboard is empty, Ben writes n on the blackboard.
(ii) If the blackboard is not empty, let m denote the largest number on the blackboard. If m2 + 2n2
is divisible by 3, Ben erases m from the blackboard; otherwise, he writes the number n.
No action is taken when the coin flips tails. If probability that the blackboard is empty after all 2013
flips is 2k2u+1
, where u, v, and k are nonnegative integers, compute k.
(2v+1)
28. Let n denote the product of the first 2013 primes. Find the sum of all primes p with 20 ≤ p ≤ 150
such that
(i) p+1
2 is even but is not a power of 2, and
(ii) there exist pairwise distinct positive integers a, b, c for which
an (a − b)(a − c) + bn (b − c)(b − a) + cn (c − a)(c − b)
is divisible by p but not p2 .
29. Kevin has 255 cookies, each labeled with a unique nonempty subset of {1, 2, 3, 4, 5, 6, 7, 8}. Each day,
he chooses one cookie uniformly at random out of the cookies not yet eaten. Then, he eats that cookie,
and all remaining cookies that are labeled with a subset of that cookie (for example, if he chooses the
cookie labeled with {1, 2}, he eats that cookie as well as the cookies with {1} and {2}). The expected
value of the number of days that Kevin eats a cookie before all cookies are gone can be expressed in
the form m

n , where m and n are relatively prime positive integers. Find m + n.
30. Let P (t) = t3 + 27t2 + 199t + 432. Suppose a, b, c, and x are distinct positive reals such that
P (−a) = P (−b) = P (−c) = 0, and
a+b+c
=
x
If x =

m
n

b+c+x
+
a

c+a+x
+
b

a+b+x
.
c

for relatively prime positive integers m and n, compute m + n.
3


The Online Math Open Winter Contest
January 4, 2013–January 14, 2013

Acknowledgments
Contest Directors
Ray Li, James Tao, Victor Wang

Head Problem Writers
Evan Chen, Ray Li, Victor Wang

Additional Problem Contributors
James Tao, Anderson Wang, David Yang, Alex Zhu

Proofreaders and Test Solvers
Evan Chen, Calvin Deng, Mitchell Lee, James Tao, Anderson Wang, David Yang, Alex Zhu

Website Manager
Ray Li

LATEX/Document Manager
Evan Chen


Contest Information
Format
The test will start Friday, January 4 and end Monday, January 14. You will have until 7pm EST on January
14 to submit your answers. The test consists of 50 short answer questions, each of which has a nonnegative
integer answer. The problem difficulties range from those of AMC problems to those of Olympiad problems.
Problems are ordered in roughly increasing order of difficulty.

Team Guidelines
Students may compete in teams of up to four people. Participating students must not have graduated from

high school. International students may participate. No student can be a part of more than one team. The
members of each team do not get individual accounts; they will all share the team account.
Each team will submit its final answers through its team account. Though teams can save drafts for their
answers, the current interface does not allow for much flexibility in communication between team members.
We recommend using Google Docs and Spreadsheets to discuss problems and compare answers, especially if
teammates cannot communicate in person. Teams may spend as much time as they like on the test before
the deadline.

Aids
Drawing aids such as graph paper, ruler, and compass are permitted. However, electronic drawing aids are
not allowed. This is includes (but is not limited to) Geogebra and graphing calculators. Published print
and electronic resources are not permitted. (This is a change from last year’s rules.)
Four-function calculators are permitted on the Online Math Open. That is, calculators which perform only
the four basic arithmetic operations (+-*/) may be used. Any other computational aids such as scientific
and graphing calculators, computer programs and applications such as Mathematica, and online databases
are prohibited. All problems on the Online Math Open are solvable without a calculator. Four-function
calculators are permitted only to help participants reduce computation errors.

Clarifications
Clarifications will be posted as they are answered. For the Fall 2012-2013 Contest, they will be posted at here.
If you have a question about a problem, please email with “Clarification”
in the subject. We have the right to deny clarification requests that we feel we cannot answer.

Scoring
Each problem will be worth one point. Ties will be broken based on the “hardest” problem that a team
answered correctly. Remaining ties will be broken by the second hardest problem solved, and so on. Problem
X is defined to be “harder” than Problem Y if and only if
(i) X was solved by less teams than Y , OR
(ii) X and Y were solved by the same number of teams and X appeared later in the test than Y .
Note: This is a change from prior tiebreaking systems.

problems by approximate difficulty.

However, we will still order the

Results
After the contest is over, we will release the answers to the problems within the next day. If you have a
protest about an answer, you may send an email to (Include “Protest”
in the subject). Solutions and results will be released in the following weeks.


January 2012

Winter OMO 2012-2013

Page 1

1. Let x be the answer to this problem. For what real number a is the answer to this problem also a−x?
2. The number 123454321 is written on a blackboard. Evan walks by and erases some (but not all) of the
digits, and notices that the resulting number (when spaces are removed) is divisible by 9. What is the
fewest number of digits he could have erased?
3. Three lines m, n, and lie in a plane such that no two are parallel. Lines m and n meet at an acute
angle of 14◦ , and lines m and meet at an acute angle of 20◦ . Find, in degrees, the sum of all possible
acute angles formed by lines n and .
4. For how many ordered pairs of positive integers (a, b) with a, b < 1000 is it true that a times b is equal
to b2 divided by a? For example, 3 times 9 is equal to 92 divided by 3.

Figure 1: xkcd 759
5. At the Mountain School, Micchell is assigned a submissiveness rating of 3.0 or 4.0 for each class he
takes. His college potential is then defined as the average of his submissiveness ratings over all classes
taken. After taking 40 classes, Micchell has a college potential of 3.975. Unfortunately, he needs a

college potential of at least 3.995 to get into the South Harmon Institute of Technology. Otherwise, he
becomes a rock. Assuming he receives a submissiveness rating of 4.0 in every class he takes from now
on, how many more classes does he need to take in order to get into the South Harmon Institute of
Technology?
6. Circle S1 has radius 5. Circle S2 has radius 7 and has its center lying on S1 . Circle S3 has an integer
radius and has its center lying on S2 . If the center of S1 lies on S3 , how many possible values are there
for the radius of S3 ?
7. Jacob’s analog clock has 12 equally spaced tick marks on the perimeter, but all the digits have been
erased, so he doesn’t know which tick mark corresponds to which hour. Jacob takes an arbitrary tick
mark and measures clockwise to the hour hand and minute hand. He measures that the minute hand
is 300 degrees clockwise of the tick mark, and that the hour hand is 70 degrees clockwise of the same
tick mark. If it is currently morning, how many minutes past midnight is it?
8. How many ways are there to choose (not necessarily distinct) integers a, b, c from the set {1, 2, 3, 4}
c
such that a(b ) is divisible by 4?
9. David has a collection of 40 rocks, 30 stones, 20 minerals and 10 gemstones. An operation consists of
removing three objects, no two of the same type. What is the maximum number of operations he can
possibly perform?
10. At certain store, a package of 3 apples and 12 oranges costs 5 dollars, and a package of 20 apples and
5 oranges costs 13 dollars. Given that apples and oranges can only be bought in these two packages,
what is the minimum nonzero amount of dollars that must be spent to have an equal number of apples
and oranges?
11. Let A, B, and C be distinct points on a line with AB = AC = 1. Square ABDE and equilateral
triangle ACF are drawn on the same side of line BC. What is the degree measure of the acute angle
formed by lines EC and BF ?