Tài liệu Tuyển Tập Các Đề Thi Của Các Nước Trên Thế Giới P2 pptx

TRƯỜNG ................
KHOA………
…………..o0o…………..












Tuyển tập các đề thi của các
nước trên thế giới P2 - Cao
Minh Quang







☺ The best problems from around the world Cao Minh Quang

301
13th Mexican 1999

A1. 1999 cards are lying on a table. Each card has a red side and a black side and can be
either side up. Two players play alternately. Each player can remove any number of cards
showing the same color from the table or turn over any number of cards of the same color.
The winner is the player who removes the last card. Does the first or second player have a
winning strategy?
A2. Show that there is no arithmetic progression of 1999 distinct positive primes all less than
12345.
A3. P is a point inside the triangle ABC. D, E, F are the midpoints of AP, BP, CP. The lines
BF, CE meet at L; the lines CD, AF meet at M; and the lines AE, BD meet at N. Show that
area DNELFM = (1/3) area ABC. Show that DL, EM, FN are concurrent.
B1. 10 squares of a chessboard are chosen arbitrarily and the center of each chosen square is
marked. The side of a square of the board is 1. Show that either two of the marked points are a
distance ≤ √2 apart or that one of the marked points is a distance 1/2 from the edge of the
board.
B2. ABCD has AB parallel to CD. The exterior bisectors of
∠
B and
∠
C meet at P, and the
exterior bisectors of
∠
A and
∠
D meet at Q. Show that PQ is half the perimeter of ABCD.
B3. A polygon has each side integral and each pair of adjacent sides perpendicular (it is not
necessarily convex). Show that if it can be covered by non-overlapping 2 x 1 dominos, then at
least one of its sides has even length.

☺ The best problems from around the world Cao Minh Quang

302
14th Mexican 2000

A1. A, B, C, D are circles such that A and B touch externally at P, B and C touch externally

at Q, C and D touch externally at R, and D and A touch externally at S. A does not intersect
C, and B does not intersect D. Show that PQRS is cyclic. If A and C have radius 2, B and D
have radius 3, and the distance between the centers of A and C is 6, find area PQRS.
A2. A triangle is constructed like that below, but with 1, 2, 3, ... , 2000 as the first row. Each
number is the sum of the two numbers immediately above. Find the number at the bottom of
the triangle.
1 2 3 4 5
3 5 7 9
8 12 16
20 28
48
A3. If A is a set of positive integers, take the set A' to be all elements which can be written as
± a
1
± a
2
... ± a
n
, where a
i
are distinct elements of A. Similarly, form A" from A'. What is the
smallest set A such that A" contains all of 1, 2, 3, ... , 40?
B1. Given positive integers a, b (neither a multiple of 5) we construct a sequence as follows:
a
1
= 5, a
n+1
= a a
n
+ b. What is the largest number of primes that can be obtained before the

first composite member of the sequence?
B2. Given an n x n board with squares colored alternately black and white like a chessboard.
An allowed move is to take a rectangle of squares (with one side greater than one square, and
both sides odd or both sides even) and change the color of each square in the rectangle. For
which n is it possible to end up with all the squares the same color by a sequence of allowed
moves?
B3. ABC is a triangle with
∠
B > 90
o
. H is a point on the side AC such that AH = BH and
BH is perpendicular to BC. D, E are the midpoints of AB, BC. The line through H parallel to
AB meets DE at F. Show that
∠
BCF =
∠
ACD.












☺ The best problems from around the world Cao Minh Quang

303
15th Mexican 2001

A1. Find all 7-digit numbers which are multiples of 21 and which have each digit 3 or 7.
A2. Given some colored balls (at least three different colors) and at least three boxes. The
balls are put into the boxes so that no box is empty and we cannot find three balls of different
colors which are in three different boxes. Show that there is a box such that all the balls in all
the other boxes have the same color.
A3. ABCD is a cyclic quadrilateral. M is the midpoint of CD. The diagonals meet at P. The
circle through P which touches CD at M meets AC again at R and BD again at Q. The point S
on BD is such that BS = DQ. The line through S parallel to AB meets AC at T. Show that AT
= RC.
B1. For positive integers n, m define f(n,m) as follows. Write a list of 2001 numbers a
i
,
where a
1
= m, and a
k+1
is the residue of a
k
2
mod n (for k = 1, 2, ... , 2000). Then put f(n,m) = a
1

- a
2
+ a
3

- a
4
+ a
5
- ... + a
2001
. For which n ≥ 5 can we find m such that 2 ≤ m ≤ n/2 and f(m,n)
> 0?
B2. ABC is a triangle with AB < AC and
∠
A = 2
∠
C. D is the point on AC such that CD =
AB. Let L be the line through B parallel to AC. Let L meet the external bisector of
∠
A at M
and the line through C parallel to AB at N. Show that MD = ND.
B3. A collector of rare coins has coins of denominations 1, 2, ... , n (several coins for each
denomination). He wishes to put the coins into 5 boxes so that: (1) in each box there is at
most one coin of each denomination; (2) each box has the same number of coins and the same
denomination total; (3) any two boxes contain all the denominations; (4) no denomination is
in all 5 boxes. For which n is this possible?

☺ The best problems from around the world Cao Minh Quang

304
16th Mexican 2002

A1. The numbers 1 to 1024 are written one per square on a 32 x 32 board, so that the first
row is 1, 2, ... , 32, the second row is 33, 34, ... , 64 and so on. Then the board is divided into
four 16 x 16 boards and the position of these boards is moved round clockwise, so that AB
goes to DA, DC goes to CB
then each of the 16 x 16 boards is divided into four equal 8 x 8 parts and each of these is
moved around in the same way (within the 16 x 16 board). Then each of the 8 x 8 boards is
divided into four 4 x 4 parts and these are moved around, then each 4 x 4 board is divided into
2 x 2 parts which are moved around, and finally the squares of each 2 x 2 part are moved
around. What numbers end up on the main diagonal (from the top left to bottom right)?
A2. ABCD is a parallelogram. K is the circumcircle of ABD. The lines BC and CD meet K
again at E and F. Show that the circumcenter of CEF lies on K.
A3. Does n
2
have more divisors = 1 mod 4 or = 3 mod 4?
B1. A domino has two numbers (which may be equal) between 0 and 6, one at each end. The
domino may be turned around. There is one domino of each type, so 28 in all. We want to
form a chain in the usual way, so that adjacent dominos have the same number at the adjacent
ends. Dominos can be added to the chain at either end. We want to form the chain so that after
each domino has been added the total of all the numbers is odd. For example, we could place
first the domino (3,4), total 3 + 4 = 7. Then (1,3), total 1 + 3 + 3 + 4 = 11, then (4,4), total 11
+ 4 + 4 = 19. What is the largest number of dominos that can be placed in this way? How

many maximum-length chains are there?

B2. A trio is a set of three distinct integers such that two of the numbers are divisors or
multiples of the third. Which trio contained in {1, 2, ... , 2002} has the largest possible sum?
Find all trios with the maximum sum.
B3. ABCD is a quadrilateral with
∠
A =
∠
B = 90
o
. M is the midpoint of AB and
∠
CMD =
90
o
. K is the foot of the perpendicular from M to CD. AK meets BD at P, and BK meets AC
at Q. Show that
∠
AKB = 90
o
and KP/PA + KQ/QB = 1.



☺ The best problems from around the world Cao Minh Quang

305
17th Mexican 2003

A1. Find all positive integers with two or more digits such that if we insert a 0 between the
units and tens digits we get a multiple of the original number.
A2. A, B, C are collinear with B betweeen A and C. K
1
is the circle with diameter AB, and
K
2
is the circle with diameter BC. Another circle touches AC at B and meets K
1
again at P
and K
2
again at Q. The line PQ meets K
1
again at R and K
2
again at S. Show that the lines AR
and CS meet on the perpendicular to AC at B.
A3. At a party there are n women and n men. Each woman likes r of the men, and each man
likes r of then women. For which r and s must there be a man and a woman who like each
other?
B1. The quadrilateral ABCD has AB parallel to CD. P is on the side AB and Q on the side
CD such that AP/PB = DQ/CQ. M is the intersection of AQ and DP, and N is the intersection
of PC and QB. Find MN in terms of AB and CD.
B2. Some cards each have a pair of numbers written on them. There is just one card for each
pair (a,b) with 1 ≤ a < b ≤ 2003. Two players play the following game. Each removes a card
in turn and writes the product ab of its numbers on the blackboard. The first player who
causes the greatest common divisor of the numbers on the blackboard to fall to 1 loses. Which
player has a winning strategy?
B3. Given a positive integer n, an allowed move is to form 2n+1 or 3n+2. The set S

n
is the set
of all numbers that can be obtained by a sequence of allowed moves starting with n. For
example, we can form 5 → 11 → 35 so 5, 11 and 35 belong to S
5
. We call m and n
compatible if S
m
∩ S
n
is non-empty. Which members of {1, 2, 3, ... , 2002} are compatible
with 2003?













☺ The best problems from around the world Cao Minh Quang

306

Polish (1983 – 2003)


















☺ The best problems from around the world Cao Minh Quang

307
34th Polish 1983

A1. The angle bisectors of the angles A, B, C in the triangle ABC meet the circumcircle again
at K, L, M. Show that |AK| + |BL| + |CM| > |AB| + |BC| + |CA|.
A2. For given n, we choose k and m at random subject to 0 ≤ k ≤ m ≤ 2
n
. Let p
n
be the
probability that the binomial coefficient mCk is even. Find lim
n→∞
p
n
.
A3. Q is a point inside the n-gon P
1
P
2
...P
n
which does not lie on any of the diagonals. Show
that if n is even, then Q must lie inside an even number of triangles P
i
P
j
P
k
.
B1. Given a real numbers x

∈
(0,1) and a positive integer N, prove that there exist positive
integers a, b, c, d such that (1) a/b < x < c/d, (2) c/d - a/b < 1/n, and (3) qr - ps = 1.
B2. There is a piece in each square of an m x n rectangle on an infinite chessboard. An
allowed move is to remove two pieces which are adjacent horizontally or vertically and to
place a piece in an empty square adjacent to the two removed and in line with them (as shown
below)
X X . to . . X, or . to X
X .
X .
Show that if mn is a multiple of 3, then it is not possible to end up with only one piece after a
sequence of moves.
B3. Show that if the positive integers a, b, c, d satisfy ab = cd, then we have gcd(a,c) gcd(a,d)
= a gcd(a,b,c,d).

☺ The best problems from around the world Cao Minh Quang

308
35th Polish 1984

A1. X is a set with n > 2 elements. Is there a function f : X → X such that the composition f
n-
1
is constant, but f
n-2
is not constant?
A2. Given n we define a
i,j
as follows. For i, j = 1, 2, ... , n, a
i,j
= 1 for j = i, and 0 for j ≠ i. For
i = 1, 2, ... , n, j = n+1, ... , 2n, a
i,j
= -1/n. Show that for any permutation p of (1, 2, ... , 2n) we
have ∑
i=1
n
|∑
k=1
n
a
i,p(k)
| ≥ n/2.
A3. W is a regular octahedron with center O. P is a plane through the center O. K(O, r

1
) and
K(O, r
2
) are circles center O and radii r
1
, r
2
such that K(O, r
1
)  P∩W  K(O, r
2
). Show that
r
1
/r
2
≤ (√3)/2.
B1. We throw a coin n times and record the results as the sequence α
1
, α
2
, ... , α
n
, using 1 for
head, 2 for tail. Let β
j
= α
1
+ α

2
+ ... + α
j
and let p(n) be the probability that the sequence β
1
,
β
2
, ... , β
n
includes the value n. Find p(n) in terms of p(n-1) and p(n-2).
B2. Six disks with diameter 1 are placed so that they cover the edges of a regular hexagon
with side 1. Show that no vertex of the hexagon is covered by two or more disks.
B3. There are 1025 cities, P
1
, ... , P
1025
and ten airlines A
1
, ... , A
10
, which connect some of the
cities. Given any two cities there is at least one airline which has a direct flight between them.
Show that there is an airline which can offer a round trip with an odd number of flights.

☺ The best problems from around the world Cao Minh Quang

309
36th Polish 1985

A1. Find the largest k such that for every positive integer n we can find at least k numbers in
the set {n+1, n+2, ... , n+16} which are coprime with n(n+17).
A2. Given a square side 1 and 2n positive reals a
1
, b
1
, ... , a
n
, b
n
each ≤ 1 and satisfying ∑ a

i
b
i

≥ 100. Show that the square can be covered with rectangles R
i
with sides length (a
i
, b
i
)
parallel to the square sides.
A3. The function f : R → R satisfies f(3x) = 3f(x) - 4f(x)
3
for all real x and is continuous at x
= 0. Show that |f(x)| ≤ 1 for all x.
B1. P is a point inside the triangle ABC is a triangle. The distance of P from the lines BC,
CA, AB is d
a
, d
b
, d
c
respectively. Show that 2/(1/d
a
+ 1/d
b
+ 1/d
c
) < r < (d

a
+ d
b
+ d
c
)/2, where
r is the inradius.
B2. p(x,y) is a polynomial such that p(cos t, sin t) = 0 for all real t. Show that there is a
polynomial q(x,y) such that p(x,y) = (x
2
+ y
2
- 1) q(x,y).
B3. There is a convex polyhedron with k faces. Show that if k/2 of the faces are such that no
two have a common edge, then the polyhedron cannot have an inscribed sphere.

☺ The best problems from around the world Cao Minh Quang

310
37th Polish 1986

A1. A square side 1 is covered with m
2
rectangles. Show that there is a rectangle with
perimeter at least 4/m.
A2. Find the maximum possible volume of a tetrahedron which has three faces with area 1.
A3. p is a prime and m is a non-negative integer < p-1. Show that ∑
j=1
p
j
m
is divisible by p.
B1. Find all n such that there is a real polynomial f(x) of degree n such that f(x) ≥ f '(x) for all
real x.
B2. There is a chess tournament with 2n players (n > 1). There is at most one match between
each pair of players. If it is not possible to find three players who all play each other, show
that there are at most n
2
matches. Conversely, show that if there are at most n

2
matches, then it
is possible to arrange them so that we cannot find three players who all play each other.
B3. ABC is a triangle. The feet of the perpendiculars from B and C to the angle bisector at A
are K, L respectively. N is the midpoint of BC, and AM is an altitude. Show that K,L,N,M are
concyclic.






















☺ The best problems from around the world Cao Minh Quang

311
38th Polish 1987

A1. There are n ≥ 2 points in a square side 1. Show that one can label the points P
1
, P
2
, ... , P
n

such that ∑
i=1
n
|P
i-1
- P
i
|
2
≤ 4, where we use cyclic subscripts, so that P
0
means P
n
.
A2. A regular n-gon is inscribed in a circle radius 1. Let X be the set of all arcs PQ, where P,
Q are distinct vertices of the n-gon. 5 elements L
1
, L
2
, ... , L

5
of X are chosen at random (so
two or more of the L
i
can be the same). Show that the expected length of L
1
∩ L
2
∩ L
3
∩ L
4
∩
L
5
is independent of n.
A3. w(x) is a polynomial with integral coefficients. Let p
n
be the sum of the digits of the
number w(n). Show that some value must occur infinitely often in the sequence p
1
, p
2
, p
3
, ... .
B1. Let S be the set of all tetrahedra which satisfy (1) the base has area 1, (2) the total face
area is 4, and (3) the angles between the base and the other three faces are all equal. Find the
element of S which has the largest volume.
B2. Find the smallest n such that n

2
-n+11 is the product of four primes (not necessarily
distinct).
B3. A plane is tiled with regular hexagons of side 1. A is a fixed hexagon vertex. Find the
number of paths P such that (1) one endpoint of P is A, (2) the other endpoint of P is a
hexagon vertex, (3) P lies along hexagon edges, (4) P has length 60, and (5) there is no shorter
path along hexagon edges from A to the other endpoint of P.




















☺ The best problems from around the world Cao Minh Quang

312

39th Polish 1988

A1. The real numbers x
1
, x
2
, ... , x
n
belong to the interval (0,1) and satisfy x
1
+ x
2
+ ... + x
n
=
m + r, where m is an integer and r
∈
[0,1). Show that x
1
2
+ x
2
2
+ ... + x
n
2
≤ m + r
2
.
A2. For a permutation P = (p

1
, p
2
, ... , p
n
) of (1, 2, ... , n) define X(P) as the number of j such
that p
i
< p
j
for every i < j. What is the expected value of X(P) if each permutation is equally
likely?
A3. W is a polygon. W has a center of symmetry S such that if P belongs to W, then so does
P', where S is the midpoint of PP'. Show that there is a parallelogram V containing W such
that the midpoint of each side of V lies on the border of W.
B1. d is a positive integer and f : [0,d] → R is a continuous function with f(0) = f(d). Show
that there exists x
∈
[0,d-1] such that f(x) = f(x+1).
B2. The sequence a
1
, a
2
, a
3
, ... is defined by a
1
= a
2
= a

3
= 1, a
n+3
= a
n+2
a
n+1
+ a
n
. Show that for
any positive integer r we can find s such that a
s
is a multiple of r.
B3. Find the largest possible volume for a tetrahedron which lies inside a hemisphere of
radius 1.

☺ The best problems from around the world Cao Minh Quang

313
40th Polish 1989

A1. An even number of politicians are sitting at a round table. After a break, they come back
and sit down again in arbitrary places. Show that there must be two people with the same
number of people sitting between them as before the break.
A2. k
1
, k
2
, k
3
are three circles. k
2
and k
3
touch externally at P, k
3
and k
1
touch externally at Q,

and k
1
and k
2
touch externally at R. The line PQ meets k
1
again at S, the line PR meets k
1

again at T. The line RS meets k
2
again at U, and the line QT meets k
3
again at V. Show that P,
U, V are collinear.
A3. The edges of a cube are labeled from 1 to 12. Show that there must exist at least eight
triples (i, j, k) with 1 ≤ i < j < k ≤ 12 so that the edges i, j, k are consecutive edges of a path.
But show that the labeling can be done so that we cannot find nine such triples.
B1. n, k are positive integers. A
0
is the set {1, 2, ... , n}. A
i
is a randomly chosen subset of A
i-1

(with each subset having equal probability). Show that the expected number of elements of A
k

is n/2
k

.
B2. Three circles of radius a are drawn on the surface of a sphere of radius r. Each pair of
circles touches externally and the three circles all lie in one hemisphere. Find the radius of a
circle on the surface of the sphere which touches all three circles.
B3. Show that for positive reals a, b, c, d we have ((ab + ac + ad + bc + bd + cd)/6)
1/2
≥ ((abc
+ abd + acd + bcd)/4)
1/3
.



















☺ The best problems from around the world Cao Minh Quang

314
41st Polish 1990

A1. Find all real-valued functions f on the reals such that (x-y)f(x+y) - (x+y)f(x-y) = 4xy(x
2
-
y
2
) for all x, y.
A2. For n > 1 and positive reals x
1
, x
2
, ... , x
n
, show that x
1
2
/(x
1
2
+x
2
x
3
) + x
2
2
/(x

2
2
+x
3
x
4
) + ... +
x
n
2
/(x
n
2
+x
1
x
2
) ≤ n-1.
A3. In a tournament there are n players. Each pair of players play each other just once. There
are no draws. Show that either (1) one can divide the players into two groups A and B, such
that every player in A beat every player in B, or (2) we can label the players P
1
, P
2
, ... , P
n
such
that P
i
beat P

i+1
for i = 1, 2, ... n (where we use cyclic subscripts, so that P
n+1
means P
1
).
B1. A triangle with each side length at least 1 lies inside a square side 1. Show that the center
of the square lies inside the triangle.
B2. a
1
, a
2
, a
3
, ... is a sequence of positive integers such that lim
n→∞
n/a
n
= 0. Show that we can
find k such that there are at least 1990 squares between a
1
+ a
2
+ ... + a
k
and a
1
+ a
2
+ ... + a

k+1
.
B3. Show that ∑
k=0
[n/3]
(-1)
k
nC3k is a multiple of 3 for n > 2. (nCm is the binomial
coefficient).

☺ The best problems from around the world Cao Minh Quang

315
42nd Polish 1991

A1. Do there exist tetrahedra T
1
, T
2
such that (1) vol T
1
> vol T
2
, and (2) every face of T
2
has
larger area than any face of T
1
?
A2. Let F(n) be the number of paths P
0
, P
1
, ... , P
n
of length n that go from P
0
= (0,0) to a
lattice point P

n
on the line y = 0, such that each P
i
is a lattice point and for each i < n, P
i
and
P
i+1
are adjacent lattice points a distance 1 apart. Show that F(n) = (2n)Cn.
A3. N is a number of the form ∑
k=1
60
a
k
k
kk
, where each a
k
= 1 or -1. Show that N cannot be a
5th power.
B1. Let V be the set of all vectors (x,y) with integral coordinates. Find all real-valued
functions f on V such that (a) f(v
) = 1 for all v of length 1; (b) f(v + w) = f(v) + f(w) for all
perpendicular v
, w
∈
V. (The vector (0,0) is considered to be perpendicular to any vector.)
B2. k
1
, k

2
are circles with different radii and centers K
1
, K
2
. Neither lies inside the other, and
they do not touch or intersect. One pair of common tangents meet at A on K
1
K
2
, the other pair
meet at B on K
1
K
2
. P is any point on k
1
. Show that there is a diameter of K
2
with one endpoint
on the line PA and the other on the line PB.
B3. The real numbers x, y, z satisfy x
2
+ y
2
+ z
2
= 2. Show that x + y + z ≤ 2 + xyz. When do
we have equality?

☺ The best problems from around the world Cao Minh Quang

316
43rd Polish 1992

A1. Segments AC and BD meet at P, and |PA| = |PD|, |PB| = |PC|. O is the circumcenter of the
triangle PAB. Show that OP and CD are perpendicular.
A2. Find all functions f : Q
+
→ Q
+

, where Q
+
is the positive rationals, such that f(x+1) = f(x)
+ 1 and f(x
3
) = f(x)
3
for all x.
A3. Show that for real numbers x
1
, x
2
, ... , x
n
we have ∑
i=1
m
(∑
j=1
n
x
i
x
j
/(i+j) ) ≥ 0. When do we
have equality?
B1. The functions f
0
, f
1

, f
2
, ... are defined on the reals by f
0
(x) = 8 for all x, f
n+1
(x) = √(x
2
+
6f
n
(x)). For all n solve the equation f
n
(x) = 2x.
B2. The base of a regular pyramid is a regular 2n-gon A
1
A
2
...A
2n
. A sphere passes through the
apex S of the pyramid and cuts the edge SA
i
at B
i
(for i = 1, 2, ... , 2n). Show that ∑ SB
2i-1
= ∑
SB
2i

.
B3. Show that k
3
! is divisible by (k!)
k2+k+1
.























☺ The best problems from around the world Cao Minh Quang

317
44th Polish 1993


A1. Find all rational solutions to:
t
2
- w
2
+ z
2
= 2xy
t
2
- y
2
+ w
2
= 2xz
t
2
- w
2
+ x
2
= 2yz.
A2. A circle center O is inscribed in the quadrilateral ABCD. AB is parallel to and longer
than CD and has midpoint M. The line OM meets CD at F. CD touches the circle at E. Show
that DE = CF iff AB = 2CD.

A3. g(k) is the greatest odd divisor of k. Put f(k) = k/2 + k/g(k) for k even, and 2
(k+1)/2
for k
odd. Define the sequence x
1
, x
2
, x
3
, ... by x
1
= 1, x
n+1
= f(x
n
). Find n such that x
n
= 800.
B1. P is a convex polyhedron with all faces triangular. The vertices of P are each colored with
one of three colors. Show that the number of faces with three vertices of different colors is
even.
B2. Find all real-valued functions f on the reals such that f(-x) = -f(x), f(x+1) = f(x) + 1 for
all x, and f(1/x) = f(x)/x
2
for x ≠ 0.
B3. Is the volume of a tetrahedron determined by the areas of its faces and its circumradius?

☺ The best problems from around the world Cao Minh Quang

318
45th Polish 1994

A1. Find all triples (x,y,z) of positive rationals such that x + y + z, 1/x + 1/y + 1/z and xyz are
all integers.
A2. L, L' are parallel lines. C is a circle that does not intersect L. A is a variable point on L.
The two tangents to C from A meet L' in two points with midpoint M. Show that the line AM
passes through a fixed point (as A varies).
A3. k is a fixed positive integer. Let a
n
be the number of maps f from the subsets of {1, 2, ... ,
n} to {1, 2, ... , k} such that for all subsets A, B of {1, 2, ... , n} we have f(A ∩ B) = min(f(A),
f(B)). Find lim

n→∞
a
n
1/n
.
B1. m, n are relatively prime. We have three jugs which contain m, n and m+n liters. Initially
the largest jug is full of water. Show that for any k in {1, 2, ... , m+n} we can get exactly k
liters into one of the jugs.
B2. A parallelepiped has vertices A
1
, A
2
, ... , A
8
and center O. Show that 4 ∑ |OA
i
|
2
≤
(∑|OA
i
|)
2
.
B3. The distinct reals x
1
, x
2
, ... , x
n

(n > 3) satisfy ∑ x
i
= 0, &sum x
i
2
= 1. Show that four of
the numbers a, b, c, d must satisfy a + b + c + nabc ≤ ∑ x
i
3
≤ a + b + d + nabd.

☺ The best problems from around the world Cao Minh Quang

319
46th Polish 1995

A1. How many subsets of {1, 2, ... , 2n} do not contain two numbers with sum 2n+1?
A2. The diagonals of a convex pentagon divide it into a small pentagon and ten triangles.
What is the largest number of the triangles that can have the same area?
A3. p ≥ 5 is prime. The sequence a
0
, a
1
, a
2
, ... is defined by a
0
= 1, a
1
= 1, ... , a
p-1
= p-1 and a
n

= a
n-1
+ a
n-p
for n ≥ p. Find a
p

3
mod p.
B1. The positive reals x
1
, x
2
, ... , x
n
have harmonic mean 1. Find the smallest possible value
of x
1
+ x
2
2
/2 + x
3
3
/3 + ... + x
n
n
/n.
B2. An urn contains n balls labeled 1, 2, ... , n. We draw the balls out one by one (without
replacing them) until we obtain a ball whose number is divisible by k. Find all k such that the
expected number of balls removed is k.
B3. PA, PB, PC are three rays in space. Show that there is just one pair of points B', C' with
B' on the ray PB and C' on the ray PC such that PC' + B'C' = PA + AB' and PB' + B'C' = PA +
AC'.

☺ The best problems from around the world Cao Minh Quang

320
47th Polish 1996

A1. Find all pairs (n,r) with n a positive integer and r a real such that 2x
2
+2x+1 divides
(x+1)
n
- r.

A2. P is a point inside the triangle ABC such that
∠
PBC =
∠
PCA <
∠
PAB. The line PB
meets the circumcircle of ABC again at E. The line CE meets the circumcircle of APE again
at F. Show that area APEF/area ABP does not depend on P.
A3. a
i
, x
i
are positive reals such that a
1
+ a
2
+ ... + a
n
= x
1
+ x
2
+ ... + x
n
= 1. Show that 2 ∑
i<j

x
i

x
j
≤ (n-2)/(n-1) + ∑ a
i
x
i
2
/(1-a
i
). When do we have equality?
B1. ABCD is a tetrahedron with
∠
BAC =
∠
ACD and
∠
ABD =
∠
BDC. Show that AB =
CD.
B2. Let p(k) be the smallest prime not dividing k. Put q(k) = 1 if p(k) = 2, or the product of
all primes < p(k) if p(k) > 2. Define the sequence x
0
, x
1
, x
2
, ... by x
0
= 1, x

n+1
= xnp(x
n
)/q(x
n
).
Find all n such that x
n
= 111111.
B3. Let S be the set of permutations a
1
a
2
...a
n
of 123...n such that a
i
≥ i. An element of S is
chosen at random. Find all n such that the probability that the chosen permutation satisfies a
i

≤ i+1 exceeds 1/3.

☺ The best problems from around the world Cao Minh Quang

321
48th Polish 1997

A1. The positive integers x
1
, x
2
, ... , x
7
satisfy x
6
= 144, x
n+3
= x
n+2

(x
n+1
+x
n
) for n = 1, 2, 3, 4.
Find x
7
.
A2. Find all real solutions to 3(x
2
+ y
2
+ z
2
) = 1, x
2
y
2
+ y
2
z
2
+ z
2
x
2
= xyz(x + y + z)
3
.
A3. ABCD is a tetrahedron. DE, DF, DG are medians of triangles DBC, DCA, DAB. The

angles between DE and BC, between DF and CA, and between DG and AB are equal. Show
that area DBC ≤ area DCA + area DAB.
B1. The sequence a
1
, a
2
, a
3
, ... is defined by a
1
= 0, a
n
= a
[n/2]
+ (-1)
n(n+1)/2
. Show that for any
positive integer k we can find n in the range 2
k
≤ n < 2
k+1
such that a
n
= 0.
B2. ABCDE is a convex pentagon such that DC = DE and
∠
C =
∠
E = 90
o

. F is a point on
the side AB such that AF/BF = AE/BC. Show that
∠
FCE =
∠
FDE and
∠
FEC =
∠
BDC.
B3. Given any n points on a unit circle show that at most n
2
/3 of the segments joining two
points have length > √2.

☺ The best problems from around the world Cao Minh Quang

322
49th Polish 1998

A1. Find all solutions in positive integers to a + b + c = xyz, x + y + z = abc.
A2. Fn is the Fibonacci sequence F
0
= F
1
= 1, F
n+2
= F
n+1
+ F
n
. Find all pairs m > k ≥ 0 such
that the sequence x
0
, x
1
, x
2

, ... defined by x
0
= F
k
/F
m
and x
n+1
= (2x
n
- 1)/(1 - x
n
) for x
n
≠ 1, or 1
if x
n
= 1, contains the number 1.
A3. PABCDE is a pyramid with ABCDE a convex pentagon. A plane meets the edges PA,
PB, PC, PD, PE in points A', B', C', D', E' distinct from A, B, C, D, E and P. For each of the
quadrilaterals ABB'A', BCC'B, CDD'C', DEE'D', EAA'E' take the intersection of the
diagonals. Show that the five intersections are coplanar.
B1. Define the sequence a
1
, a
2
, a
3
, ... by a
1

= 1, a
n
= a
n-1
+ a
[n/2]
. Does the sequence contain
infinitely many multiples of 7?
B2. The points D, E on the side AB of the triangle ABC are such that (AD/DB)(AE/EB) =
(AC/CB)
2
. Show that
∠
ACD =
∠
BCE.
B3. S is a board containing all unit squares in the xy plane whose vertices have integer
coordinates and which lie entirely inside the circle x
2
+ y
2
= 1998
2
. +1 is written in each
square of S. An allowed move is to change the sign of every square in S in a given row,
column or diagonal. Can we end up with all -1s by a sequence of allowed moves?

☺ The best problems from around the world Cao Minh Quang

323
50th Polish 1999

A1. D is a point on the side BC of the triangle ABC such that AD > BC. E is a point on the
side AC such that AE/EC = BD/(AD-BC). Show that AD > BE.
A2. Given 101 distinct non-negative integers less than 5050 show that one choose four a, b,
c, d such that a + b - c - d is a multiple of 5050.
A3. Show that one can find 50 distinct positive integers such that the sum of each number
and its digits is the same.
B1. For which n do the equations have a solution in integers:
x
1

2
+ x
2
2
+ 50 = 16x
1
+ 12x
2

x
2
2
+ x
3
2
+ 50 = 16x
2
+ 12x
3

...
x
n-1
2
+ x
n
2
+ 50 = 16x
n-1
+ 12x

n

x
n
2
+ x
1
2
+ 50 = 16x
n
+ 12x
1

B2. Show that ∑
1≤i<j≤n
(|a
i
-a
j
| + |b
i
-b
j
|) ≤ ∑
1≤i<j≤n
|a
i
-b
j
| for all integers a

i
, b
i
.
B3. The convex hexagon ABCDEF satisfies
∠
A +
∠
C +
∠
E = 360
o
and AB·CD·EF =
BC·DE·FA. Show that AB·FD·EC = BF·DE·CA.

☺ The best problems from around the world Cao Minh Quang

324
51st Polish 2000

A1. How many solutions in non-negative reals are there to the equations:
x
1
+ x
n
2
= 4x
n

x
2
+ x
1
2
= 4x
1

...
x
n
+ x

n-1
2
= 4x
n-1
?
A2. The triangle ABC has AC = BC. P is a point inside the triangle such that
∠
PAB =
∠
PBC. M is the midpoint of AB. Show that
∠
APM +
∠
BPC = 180
o
.
A3. The sequence a
1
, a
2
, a
3
, ... is defined as follows. a
1
and a
2
are primes. a
n
is the greatest
prime divisor of a

n-1
+ a
n-2
+ 2000. Show that the sequence is bounded.
B1. PA
1
A
2
...A
n
is a pyramid. The base A
1
A
2
...A
n
is a regular n-gon. The apex P is placed so
that the lines PA
i
all make an angle 60
o
with the plane of the base. For which n is it possible to
find B
i
on PA
i
for i = 2, 3, ... , n such that A
1
B
2

+ B
2
B
3
+ B
3
B
4
+ ... + B
n-1
B
n
+ B
n
A
1
< 2A
1
P?
B2. For each n ≥ 2 find the smallest k such that given any subset S of k squares on an n x n
chessboard we can find a subset T of S such that every row and column of the board has an
even number of squares in T.
B3. p(x) is a polynomial of odd degree which satisfies p(x
2
-1) = p(x)
2
- 1 for all x. Show that
p(x) = x.