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13-th Nordic Mathematical Contest
April 12, 1999
1. The function f is defined for nonnegative integers and satisfies
f (f (n + 11)) if n ≤ 1999,
n−5
if n > 1999.

f (n) =

Find all solutions of the equation f (n) = 1999.
2. A convex heptagon with all different sides is inscribed in a circle. At most,
how many angles equal to 120◦ can this heptagon have?
3. Nonnegative integers a and b are given. A soldier is walking on the infinite
lattice Z × Z as follows. In each step, from a point (x, y) he is only allowed
to go to one of the points (x ± a, y ± b) and (x ± b, y ± a). Find all values
of a and b for which the soldier can visit every point of the lattice during
his infinite walk.
4. Let a1 , a2 , . . . , an be positive numbers (n ≥ 1). Show that
n

1
1
+ ···+
a1
an

≥

1
1
+ ··· +
1 + a1
1 + an

n+

1
1
+ ···+
a1
an

.

When does equality hold?

1

The IMO Compendium Group,

D. Djuki´c, V. Jankovi´c, I. Mati´c, N. Petrovi´c
www.imo.org.yu


14-th Nordic Mathematical Contest
March 30, 2000
1. In how many ways can the number 2000 be written as a sum of three
positive, not necesarily different integers? (The order of summands is
irrelevant.)
2. The persons P1 , P2 , . . . , Pn sit around a table in this order, and each one
has a number of coins. Initially, P1 has one coin more than P2 , P2 has one
coin more than P3 , etc. Now P1 gives one coin to P2 , who in turn gives
two coins to P3 , etc., up to Pn who gives n coins to P1 ; then P1 continues
by giving n + 1 coins to P2 , etc. The transactions go on until someone has
not enough coins to give away one coin more than he just received. After
this process ends, it turns out that there are two neighbors at the table
one of whom has five times as many coins as the other. Find the number
of persons and the number of coins circulating around the table.
3. In the triangle ABC, the bisectors of angles B and C meet the opposite
sides at D and E, respectively. The bisectors intersect at point O such
that OD = OE. Prove that either △ABC is isosceles or ∠A = 60◦ .
4. A real function defined for 0 ≤ x ≤ 1 satisfies f (0) = 0, f (1) = 1, and
f (x) − f (y)
1
≤
≤2
2
f (y) − f (z)
whenever 0 ≤ x < y < z ≤ 1 and z − y = y − x. Show that

1
7

≤f

1
3

≤ 74 .

1

The IMO Compendium Group,
D. Djuki´c, V. Jankovi´c, I. Mati´c, N. Petrovi´c
www.imo.org.yu


15-th Nordic Mathematical Contest
March 29, 2001
1. Let A be a finite set of unit squares in the coordinate plane, each of which
has vertices at integer points. Show that there exists a subset B of A
consisting of at least 1/4 of the squares in A such that no two distinct
squares in B have a common vertex.
2. A function f : R → R is bounded and satisfies
f

x+

1
3

+f

x+

1
2

= (x) + f

x+

5
6

for all real x. Show that f is periodic.
3. Find the number of real roots of the equation
x8 − x7 + 2x6 − 2x5 + 3x4 − 3x3 + 4x2 − 4x +

5
= 0.
2

4. Each of the diagonals AD, BE and CF of a convex hexagon ABCDEF
divides its area into two equal parts. Prove that these three diagonals pass
through the same point.

1

The IMO Compendium Group,

D. Djuki´c, V. Jankovi´c, I. Mati´c, N. Petrovi´c
www.imo.org.yu


16-th Nordic Mathematical Contest
April 4, 2002
1. A trapezoid ABCD with AB CD and AD < CD is inscribed in a circle
c. Let DP be a chord parallel to AC. The tangent to c at D meets the line
AB at E, and the lines P B and DC meet at Q. Prove that EQ = AC.
2. Let N balls, numbered 1 through N , be distributed over two urns. A
ball is taken from one urn and moved to the other. It turns out that the
arithmetic means of the numbers on the balls in each urn increased by the
same number x. What is the greatest possible value of x?
3. Let a1 , a2 , . . . , an , b1 , b2 , . . . , bn be real numbers with a1 , . . . , an distinct.
Show that if the product (ai + b1 )(ai + b2 ) · · · (ai + bn) takes the same value
for every i = 1, 2, . . . , n, then the product (a1 + bj )(a2 + bj ) · · · (an + bj )
also takes the same value for every j = 1, 2, . . . , n.
4. Eva, Per, and Anna randomly select different nine-digit integers made of
digits 1, 2, . . . , 9 and check if they are divisible by 11. Anna claims that
the probability that the number is divisible by 11 is exactly 1/11; Eva
believes that this probability is less than 1/11, while Per thinks that it is
greater than 1/11. Who of them is right?

1

The IMO Compendium Group,
D. Djuki´c, V. Jankovi´c, I. Mati´c, N. Petrovi´c
www.imo.org.yu

17-th Nordic Mathematical Contest
April 3, 2003
1. The squares of a rectangular chessboard with 10 rows and 14 columns are
colored alternatingly black and white in the usual manner. Some stones
are placed the board (possibly more than one on the same square) so that
there are an odd number of stones in each row and each column. Show
that the total number of stones on black squares is even.
2. Find all triples (x, y, z) of integers satisfying the equation
x3 + y 3 + z 3 − 3xyz = 2003.
3. An interior point D of an equilateral triangle ABC is taken so that
∠ADC = 150◦ . Prove that the triangle whose sides are congruent to
AD, BD and CD is right-angled.
4. Find all functions f from R \ {0} to itself satisfying
f (x) + f (y) = f (xyf (x + y))
for all x, y = 0 with x + y = 0.

1

The IMO Compendium Group,
D. Djuki´c, V. Jankovi´c, I. Mati´c, N. Petrovi´c
www.imo.org.yu


18-th Nordic Mathematical Contest
April 1, 2004
1. Twenty-seven balls labelled from 1 to 27 are distributed in three bowls:
red, blue, and yellow. What are the possible values of the number of balls
in the red bowl if the average labels in the red, blue and yellow bowl are
15, 3, and 18, respectively?
2. The Fibonacci sequence is defined by f1 = 0, f2 = 1, and fn+2 = fn+1 +fn

for n ≥ 1. Prove that there is a strictly increasing arithmetic progression
whose no term is in the Fibonacci sequence.
3. Given a finite sequence x1,1 , x2,1 , . . . , xn,1 of integers (n ≥ 2), not all equal,
define the sequences x1,k , . . . , xn,k by
xi,k+1 =

1
(xi,k + xi+1,k ),
2

where xn+1,k = x1,k .

Show that if n is odd, then not all xj,k are integers. Is this also true for
even n?
4. Let a, b, c be the sides and R be the circumradius of a triangle. Prove that
1
1
1
1
+
+
≥ 2.
ab bc ca
R

1

The IMO Compendium Group,
D. Djuki´c, V. Jankovi´c, I. Mati´c, N. Petrovi´c
www.imo.org.yu

19-th Nordic Mathematical Contest
April 5, 2005
1. Find all positive integers k such that the product of the decimal digits of
k equals 25
8 k − 211.
2. If a, b, c are positive numbers, prove the inequality
2a2
2b2
2c2
+
+
≥ a + b + c.
b+c c+a a+b
3. There are 2005 boys and girls sitting at a round table. No more than 668
of them are boys. A girl G is said to be in a strong position if, counting
from G to either direction at any length (G herself included), the number
of girls is always strictly larger than the number of boys. Prove that there
always exists a girl in a strong position.
4. Circle C1 touches circle C2 internally at A. A line through A intersects C1
again at B and C2 again at C. The tangent to C1 at B intersects C2 at D
and E. The tangents to C1 through C touch C1 at F and G. Prove that
points D, E, F, G are concyclic.

1

The IMO Compendium Group,
D. Djuki´c, V. Jankovi´c, I. Mati´c, N. Petrovi´c
www.imo.org.yu

20th Nordic Mathematical Contest
Thursday March 30, 2006
English version
Time allowed: 4 hours. Each problem is worth 5 points.
Problem 1. Let B and C be points on two fixed rays emanating from a
point A such that AB + AC is constant.
Prove that there exists a point D = A such that the circumcircles of the
triangels ABC pass through D for every choice of B and C.
Problem 2. The real numbers x, y and z are not all equal and fulfil
x+

1
1
1
=y+ =z+ =k
y
z
x

Determine all possible values of k.
Problem 3. A sequence of positive integers {an } is given by
a0 = m

and

an+1 = a5n + 487 for all n ≥ 0

Determine all values of m for which the sequence contains as many square

numbers as possible.
Problem 4. The squares of a 100 × 100 chessboard are painted with 100
different colours. Each square has only one colour and every colour is used
exactly 100 times.
Show that there exists a row or a column on the chessboard in which at least
10 colours are used.
Only writing and drawing sets are allowed


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▼❛r❝❤ ✷✾✱ ✷✵✵✼
❊♥❣❧✐s❤ ✈❡rs✐♦♥

❚✐♠❡ ❛❧❧♦✇❡❞✿ ✹ ❤♦✉rs✳ ❊❛❝❤ ♣r♦❜❧❡♠ ✐s ✇♦rt❤ ✺ ♣♦✐♥ts✳ ❖♥❧② ✇r✐t✐♥❣ ❛♥❞ ❞r❛✇✐♥❣
s❡ts ❛r❡ ❛❧❧♦✇❡❞✳

Pr♦❜❧❡♠ ✶
❋✐♥❞

♦♥❡

s♦❧✉t✐♦♥ ✐♥ ♣♦s✐t✐✈❡ ✐♥t❡❣❡rs t♦ t❤❡ ❡q✉❛t✐♦♥

x2 − 2x − 2007y 2 = 0 .

Pr♦❜❧❡♠ ✷
❆ tr✐❛♥❣❧❡✱ ❛ ❧✐♥❡ ❛♥❞ t❤r❡❡ r❡❝t❛♥❣❧❡s✱ ✇✐t❤ ♦♥❡ s✐❞❡ ♣❛r❛❧❧❡❧ t♦ t❤❡ ❣✐✈❡♥ ❧✐♥❡✱ ❛r❡
❣✐✈❡♥ ✐♥ s✉❝❤ ❛ ✇❛② t❤❛t t❤❡ r❡❝t❛♥❣❧❡s ❝♦♠♣❧❡t❡❧② ❝♦✈❡r t❤❡ s✐❞❡s ♦❢ t❤❡ tr✐❛♥❣❧❡✳
Pr♦✈❡ t❤❛t t❤❡ r❡❝t❛♥❣❧❡s ♠✉st ❝♦♠♣❧❡t❡❧② ❝♦✈❡r t❤❡ ✐♥t❡r✐♦r ♦❢ t❤❡ tr✐❛♥❣❧❡✳

Pr♦❜❧❡♠ ✸
❚❤❡ ♥✉♠❜❡r

102007

✐s ✇r✐tt❡♥ ♦♥ ❛ ❜❧❛❝❦❜♦❛r❞✳ ❆♥♥❡ ❛♥❞ ❇❡r✐t ♣❧❛② ❛ ❣❛♠❡ ✇❤❡r❡

t❤❡ ♣❧❛②❡r ✐♥ t✉r♥ ♠❛❦❡s ♦♥❡ ♦❢ t✇♦ ♦♣❡r❛t✐♦♥s✿
✭✐✮ r❡♣❧❛❝❡ ❛ ♥✉♠❜❡r
t❤❛♥ ✶ s✉❝❤ t❤❛t

x ♦♥ t❤❡ ❜❧❛❝❦❜♦❛r❞ ❜② t✇♦ ✐♥t❡❣❡r ♥✉♠❜❡rs a ❛♥❞ b ❣r❡❛t❡r
x = ab ;

✭✐✐✮ ❡r❛s❡ ♦♥❡ ♦r ❜♦t❤ ♦❢ t✇♦ ❡q✉❛❧ ♥✉♠❜❡rs ♦♥ t❤❡ ❜❧❛❝❦❜♦❛r❞✳
❚❤❡ ♣❧❛②❡r ✇❤♦ ✐s ♥♦t ❛❜❧❡ t♦ ♠❛❦❡ ❤❡r t✉r♥ ❧♦s❡s t❤❡ ❣❛♠❡✳ ❲❤♦ ✇✐❧❧ ✇✐♥ t❤❡
❣❛♠❡ ✐❢ ❆♥♥❡ ❜❡❣✐♥s ❛♥❞ ❜♦t❤ ♣❧❛②❡rs ❛❝t ✐♥ ❛♥ ♦♣t✐♠❛❧ ✇❛②❄

Pr♦❜❧❡♠ ✹
❆ ❧✐♥❡ t❤r♦✉❣❤ ❛ ♣♦✐♥t
t❤❛t

B

❧✐❡s ❜❡t✇❡❡♥

A

A

✐♥t❡rs❡❝ts ❛ ❝✐r❝❧❡ ✐♥ t✇♦ ♣♦✐♥ts✱

❛♥❞

C✳

❋r♦♠ t❤❡ ♣♦✐♥t

❝✐r❝❧❡✱ ♠❡❡t✐♥❣ t❤❡ ❝✐r❝❧❡ ❛t ♣♦✐♥ts

ST

❛♥❞

AC ✳

❙❤♦✇ t❤❛t

S ❛♥❞ T ✳ ▲❡t
AP/P C = 2 · AB/BC ✳

A
P

B

❛♥❞

C✱

✐♥ s✉❝❤ ❛ ✇❛②

❞r❛✇ t❤❡ t✇♦ t❛♥❣❡♥ts t♦ t❤❡
❜❡ t❤❡ ✐♥t❡rs❡❝t✐♦♥ ♦❢ t❤❡ ❧✐♥❡s


The 22nd Nordic Mathematical Contest
31 March 2008
English version
Time allowed is 4 hours. Each problem is worth 5 points. The only permitted
aids are writing and drawing materials.
Problem 1
Determine all real numbers A, B and C such that there exists a real function
f that satisfies
f (x + f (y)) = Ax + By + C
for all real x and y.
Problem 2
Assume that n ≥ 3 people with different names sit around a round table. We
call any unordered pair of them, say M and N , dominating, if
(i) M and N do not sit on adjacent seats, and
(ii) on one (or both) of the arcs connecting M and N along the table edge,
all people have names that come alphabetically after the names of M
and N .
Determine the minimal number of dominating pairs.
Problem 3
Let ABC be a triangle and let D and E be points on BC and CA, respectively,
such that AD and BE are angle bisectors of ABC . Let F and G be points
on the circumcircle of ABC such that AF and DE are parallel and FG and
BC are parallel. Show that
AB + AC

AG
=
.
BG
AB + BC
Problem 4
The difference between the cubes of two consecutive positive integers is a
square n2 , where n is a positive integer. Show that n is the sum of two
squares.


The 23rd Nordic Mathematical Contest
Thursday April 2, 2009
English version

Time allowed is 4 hours. Each problem is worth 5 points. The only permitted aids
are writing and drawing tools.
Problem 1
A point P is chosen in an arbitrary triangle. Three lines are drawn through P which
are parallel to the sides of the triangle. The lines divide the triangle into three smaller
triangles and three parallelograms. Let f be the ratio between the total area of the
three smaller triangles and the area of the given triangle. Show that f ≥ 31 and
determine those points P for which f = 13 .
Problem 2
On a faded piece of paper it is possible, with some effort, to discern the following:
(x2 + x + a)(x15 − . . .) = x17 + x13 + x5 − 90x4 + x − 90.
Some parts have got lost, partly the constant term of the first factor of the left side,
partly the main part of the other factor. It would be possible to restore the polynomial
forming the other factor, but we restrict ourselves to asking the question: What is
the value of the constant term a? We assume that all polynomials in the statement

above have only integer coefficients.
Problem 3
The integers 1, 2, 3, 4 and 5 are written on a blackboard. It is allowed to wipe out
two integers a and b and replace them with a + b and ab. Is it possible, by repeating
this procedure, to reach a situation where three of the five integers on the blackboard
are 2009?
Problem 4
There are 32 competitors in a tournament. No two of them are equal in playing
strength, and in a one against one match the better one always wins. Show that the
gold, silver, and bronze medal winners can be found in 39 matches.


Version: English

24th Nordic Mathematical Contest
13th of April, 2010

1. A function f : Z+ → Z+ , where Z+ is the set of positive integers, is non-decreasing and
satisfies f (mn) = f (m)f (n) for all relatively prime positive integers m and n. Prove that
f (8)f (13) ≥ (f (10))2 .
2. Three circles ΓA , ΓB and ΓC share a common point of intersection O. The other common
point of ΓA and ΓB is C, that of ΓA and ΓC is B, and that of ΓC and ΓB is A. The line AO
intersects the circle ΓA in the point X = O. Similarly, the line BO intersects the circle ΓB
in the point Y = O, and the line CO intersects the circle ΓC in the point Z = O. Show
that
|AY | |BZ| |CX|
= 1.
|AZ| |BX| |CY |
3. Laura has 2010 lamps connected with 2010 buttons in front of her. For each button, she
wants to know the corresponding lamp. In order to do this, she observes which lamps

are lit when Richard presses a selection of buttons. (Not pressing anything is also a
possible selection.) Richard always presses the buttons simultaneously, so the lamps are
lit simultaneously, too.
a) If Richard chooses the buttons to be pressed, what is the maximum number of different
combinations of buttons he can press until Laura can assign the buttons to the lamps
correctly?
b) Supposing that Laura will choose the combinations of buttons to be pressed, what
is the minimum number of attempts she has to do until she is able to associate the
buttons with the lamps in a correct way?
4. A positive integer is called simple if its ordinary decimal representation consists entirely of
zeroes and ones. Find the least positive integer k such that each positive integer n can be
written as n = a1 ± a2 ± a3 ± . . . ± ak where a1 , . . . , ak are simple.

Time allowed is 4 hours.
Each problem is worth 5 points.
Only writing and drawing tools are permitted.


The 25th Nordic Mathematical Contest
Monday 4 April 2011
English version

The time allowed is 4 hours. Each problem is worth 5 points. The only aids permitted
are writing and drawing tools.

Problem 1
When a0 , a1 , . . . , a1000 denote digits, can the sum of the 1001-digit numbers
a0 a1 . . . a1000 and a1000 a999 . . . a0 have odd digits only?
Problem 2
In a triangle ABC assume AB = AC, and let D and E be points on the extension of

segment BA beyond A and on the segment BC, respectively, such that the lines CD
4h
and AE are parallel. Prove CD ≥
CE, where h is the height from A in triangle
BC
ABC. When does equality hold?
Problem 3
Find all functions f such that
f (f (x) + y) = f (x2 − y) + 4yf (x)
for all real numbers x and y.
Problem 4
1
, where a and b are
ab
1
relatively prime positive integers such that a < b ≤ n and a + b > n, equals .
2
(Integers a and b are called relatively prime if the greatest common divisor of a and
b is 1.)

Show that for any integer n ≥ 2 the sum of the fractions


The 26th Nordic Mathematical Contest
Tuesday, 27 March 2012
English Version
The time allowed is 4 hours. Each problem is worth 5 points. The only
permitted aids are writing and drawing tools.

Problem 1. The real numbers a, b, c are such that a2 + b2 = 2c2 , and also such

that a = b, c = −a, c = −b. Show that
(a + b + 2c)(2a2 − b2 − c2 )
(a − b)(a + c)(b + c)
is an integer.
Problem 2. Given a triangle ABC, let P lie on the circumcircle of the triangle
and be the midpoint of the arc BC which does not contain A. Draw a straight
line l through P so that l is parallel to AB. Denote by k the circle which passes
through B, and is tangent to l at the point P . Let Q be the second point of
intersection of k and the line AB (if there is no second point of intersection,
choose Q = B). Prove that AQ = AC.
Problem 3. Find the smallest positive integer n, such that there exist n integers
x1 , x2 , . . . , xn (not necessarily different), with 1 ≤ xk ≤ n, 1 ≤ k ≤ n, and such
that
n(n + 1)
, and x1 x2 · · · xn = n!,
x1 + x2 + · · · + xn =
2
but {x1 , x2 , . . . , xn } = {1, 2, . . . , n}.
Problem 4. The number 1 is written on the blackboard. After that a sequence
of numbers is created as follows: at each step each number a on the blackboard
is replaced by the numbers a − 1 and a + 1; if the number 0 occurs, it is erased
immediately; if a number occurs more than once, all its occurrences are left on
the blackboard. Thus the blackboard will show 1 after 0 steps; 2 after 1 step;
1, 3 after 2 steps; 2, 2, 4 after 3 steps, and so on. How many numbers will there
be on the blackboard after n steps?

1


The 27th Nordic Mathematical Contest

Monday, 8 April 2013
English Version
The time allowed is 4 hours. Each problem is worth 5 points.
The only permitted aids are writing and drawing tools.
Problem 1. Let (an )n≥1 be a sequence with a1 = 1 and
an+1 = an +

√

an +

1
2

for all n ≥ 1, where x denotes the greatest integer less than or equal to x. Find
all n ≤ 2013 such that an is a perfect square.
Problem 2. In a football tournament there are n teams, with n ≥ 4, and each
pair of teams meets exactly once. Suppose that, at the end of the tournament,
the final scores form an arithmetic sequence where each team scores 1 more point
than the following team on the scoreboard. Determine the maximum possible score
of the lowest scoring team, assuming usual scoring for football games (where the
winner of a game gets 3 points, the loser 0 points, and if there is a tie both teams
get 1 point).
Problem 3. Define a sequence (nk )k≥0 by n0 = n1 = 1, and n2k = nk + nk−1 and
n2k+1 = nk for k ≥ 1. Let further qk = nk /nk−1 for each k ≥ 1. Show that every
positive rational number is present exactly once in the sequence (qk )k≥1 .
Problem 4. Let ABC be an acute angled triangle, and H a point in its interior.
Let the reflections of H through the sides AB and AC be called Hc and Hb ,
respectively, and let the reflections of H through the midpoints of these same sides
be called Hc and Hb , respectively. Show that the four points Hb , Hb , Hc , and Hc

are concyclic if and only if at least two of them coincide or H lies on the altitude
from A in triangle ABC.


The 28th Nordic Mathematical Contest
Monday, 31 March 2014
English Version
The time allowed is 4 hours. Each problem is worth 5 points.
The only permitted aids are writing and drawing tools.

Problem 1
Find all functions f : N → N (where N is the set of the natural numbers and is
assumed to contain 0), such that
f (x2 ) − f (y 2 ) = f (x + y)f (x − y),
for all x, y ∈ N with x ≥ y.
Problem 2
Given an equilateral triangle, find all points inside the triangle such that the
distance from the point to one of the sides is equal to the geometric mean of the
distances from the point to the other two sides of the triangle.
√
[The geometric mean of two numbers x and y equals xy.]
Problem 3
Find all nonnegative integers a, b, c, such that
√
√
√
√
a + b + c = 2014.

Problem 4

A game is played on an n × n chessboard. At the beginning there are 99 stones
on each square. Two players A and B take turns, where in each turn the player
chooses either a row or a column and removes one stone from each square in the
chosen row or column. They are only allowed to choose a row or a column, if it
has least one stone on each square. The first player who cannot move, looses the
game. Player A takes the first turn. Determine all n for which player A has a
winning strategy.


The 29th Nordic Mathematical Contest
Tuesday, March 24, 2015

Problem 1.
Let ABC be a triangle and Γ the circle with diameter AB. The bisectors of ∠BAC and
∠ABC intersect Γ (also) at D and E, respectively. The incircle of ABC meets BC and
AC at F and G, respectively. Prove that D, E, F and G are collinear.
Problem 2.
Find the primes p, q, r, given that one of the numbers pqr and p + q + r is 101 times the
other.
Problem 3.
Let n > 1 and p(x) = xn + an−1 xn−1 + · · · + a0 be a polynomial with n real roots (counted
with multiplicity). Let the polynomial q be defined by
2015

p(x + j).

q(x) =
j=1

We know that p(2015) = 2015. Prove that q has at least 1970 different roots r1 , . . . , r1970

such that |rj | < 2015 for all j = 1, . . . , 1970.
Problem 4.
An encyclopedia consists of 2000 numbered volumes. The volumes are stacked in order
with number 1 on top and 2000 in the bottom. One may perform two operations with the
stack:
(i) For n even, one may take the top n volumes and put them in the bottom of the stack
without changing the order.
(ii) For n odd, one may take the top n volumes, turn the order around and put them on
top of the stack again.
How many different permutations of the volumes can be obtained by using these two
operations repeatedly?

Time allowed: 4 hours.
Each problem is worth 7 points.
Only writing and drawing tools are allowed.


The 30th Nordic Mathematical Contest
Tuesday, April 5, 2016
English version
Time allowed: 4 hours. Each problem is worth 7 points.
Only writing and drawing tools are allowed.

Problem 1
Determine all sequences of non-negative integers a1 , . . . , a2016 all less than or equal to 2016
satisfying i + j | iai + jaj for all i, j ∈ {1, 2, . . . , 2016}.
Problem 2
Let ABCD be a cyclic quadrilateral satisfying AB = AD and AB + BC = CD.
Determine ∠CDA.
Problem 3

Find all a ∈ R for which there exists a function f : R → R, such that
(i) f (f (x)) = f (x) + x, for all x ∈ R,
(ii) f (f (x) − x) = f (x) + ax, for all x ∈ R.
Problem 4
King George has decided to connect the 1680 islands in his kingdom by bridges. Unfortunately the rebel movement will destroy two bridges after all the bridges have been built,
but not two bridges from the same island.
What is the minimal number of bridges the King has to build in order to make sure that
it is still possible to travel by bridges between any two of the 1680 islands after the rebel
movement has destroyed two bridges?


The 31st Nordic Mathematical Contest
Monday, 3 April 2017
English version
Time allowed: 4 hours. Each problem is worth 7 points.
Only writing and drawing tools are allowed.

Problem 1 Let n be a positive integer. Show that there exist positive integers a and b
such that:
a2 + a + 1
= n2 + n + 1.
b2 + b + 1

Problem 2 Let a, b, α, β be real numbers such that 0 ≤ a, b ≤ 1, and 0 ≤ α, β ≤ π2 . Show
that if
√
ab cos(α − β) ≤ (1 − a2 )(1 − b2 ),
then
a cos α + b sin β ≤ 1 + ab sin(β − α).

Problem 3 Let M and N be the midpoints of the sides AC and AB, respectively, of
an acute triangle ABC, AB ̸= AC. Let ωB be the circle centered at M passing through
B, and let ωC be the circle centered at N passing through C. Let the point D be such
that ABCD is an isosceles trapezoid with AD parallel to BC. Assume that ωB and ωC
intersect in two distinct points P and Q. Show that D lies on the line P Q.

Problem 4 Find all integers n and m, n > m > 2, and such that a regular n-sided
polygon can be inscribed in a regular m-sided polygon so that all the vertices of the n-gon
lie on the sides of the m-gon.


❚❤❡ ✸✷♥❞ ◆♦r❞✐❝ ▼❛t❤❡♠❛t✐❝❛❧ ❈♦♥t❡st
▼♦♥❞❛②✱ ✾ ❆♣r✐❧ ✷✵✶✽
❊♥❣❧✐s❤ ✈❡rs✐♦♥

❚✐♠❡ ❛❧❧♦✇❡❞✿ ✹ ❤♦✉rs✳ ❊❛❝❤ ♣r♦❜❧❡♠ ✐s ✇♦rt❤ ✼ ♣♦✐♥ts✳
❖♥❧② ✇r✐t✐♥❣ ❛♥❞ ❞r❛✇✐♥❣ t♦♦❧s ❛r❡ ❛❧❧♦✇❡❞✳

Pr♦❜❧❡♠ ✶ ▲❡t k ❜❡ ❛ ♣♦s✐t✐✈❡ ✐♥t❡❣❡r ❛♥❞ P ❛ ♣♦✐♥t ✐♥ t❤❡ ♣❧❛♥❡✳ ❲❡ ✇✐s❤
t♦ ❞r❛✇ ❧✐♥❡s✱ ♥♦♥❡ ♣❛ss✐♥❣ t❤r♦✉❣❤ P ✱ ✐♥ s✉❝❤ ❛ ✇❛② t❤❛t ❛♥② r❛② st❛rt✐♥❣
❢r♦♠ P ✐♥t❡rs❡❝ts ❛t ❧❡❛st k ♦❢ t❤❡s❡ ❧✐♥❡s✳ ❉❡t❡r♠✐♥❡ t❤❡ s♠❛❧❧❡st ♥✉♠❜❡r ♦❢
❧✐♥❡s ♥❡❡❞❡❞✳
Pr♦❜❧❡♠ ✷ ❆ s❡q✉❡♥❝❡ ♦❢ ♣r✐♠❡s p1 , p2 , . . . ✐s ❣✐✈❡♥ ❜② t✇♦ ✐♥✐t✐❛❧ ♣r✐♠❡s p1
❛♥❞ p2 ✱ ❛♥❞ pn+2 ❜❡✐♥❣ t❤❡ ❣r❡❛t❡st ♣r✐♠❡ ❞✐✈✐s♦r ♦❢ pn + pn+1 + 2018 ❢♦r ❛❧❧
n ≥ 1✳ Pr♦✈❡ t❤❛t t❤❡ s❡q✉❡♥❝❡ ♦♥❧② ❝♦♥t❛✐♥s ✜♥✐t❡❧② ♠❛♥② ♣r✐♠❡s ❢♦r ❛❧❧
♣♦ss✐❜❧❡ ✈❛❧✉❡s ♦❢ p1 ❛♥❞ p2 ✳
Pr♦❜❧❡♠ ✸ ▲❡t ABC ❜❡ ❛ tr✐❛♥❣❧❡ ✇✐t❤ AB < AC ✳ ▲❡t D ❛♥❞ E ❜❡ ♦♥ t❤❡
❧✐♥❡s CA ❛♥❞ BA✱ r❡s♣❡❝t✐✈❡❧②✱ s✉❝❤ t❤❛t CD = AB ✱ BE = AC ✱ ❛♥❞ A✱ D
❛♥❞ E ❧✐❡ ♦♥ t❤❡ s❛♠❡ s✐❞❡ ♦❢ BC ✳ ▲❡t I ❜❡ t❤❡ ✐♥❝❡♥tr❡ ♦❢ tr✐❛♥❣❧❡ ABC ✱
❛♥❞ ❧❡t H ❜❡ t❤❡ ♦rt❤♦❝❡♥tr❡ ♦❢ tr✐❛♥❣❧❡ BCI ✳ ❙❤♦✇ t❤❛t D✱ E ✱ ❛♥❞ H ❛r❡

❝♦❧❧✐♥❡❛r✳
Pr♦❜❧❡♠ ✹ ▲❡t f = f (x, y, z) ❜❡ ❛ ♣♦❧②♥♦♠✐❛❧ ✐♥ t❤r❡❡ ✈❛r✐❛❜❧❡s x, y, z s✉❝❤
t❤❛t
f (w, w, w) = 0

❢♦r ❛❧❧ w ∈ R✳ ❙❤♦✇ t❤❛t t❤❡r❡ ❡①✐st t❤r❡❡ ♣♦❧②♥♦♠✐❛❧s A, B, C ✐♥ t❤❡s❡ s❛♠❡
t❤r❡❡ ✈❛r✐❛❜❧❡s s✉❝❤ t❤❛t A + B + C = 0 ❛♥❞
f (x, y, z) = A(x, y, z) · (x − y) + B(x, y, z) · (y − z) + C(x, y, z) · (z − x).

■s t❤❡r❡ ❛♥② ♣♦❧②♥♦♠✐❛❧ f ❢♦r ✇❤✐❝❤ t❤❡s❡ A, B, C ❛r❡ ✉♥✐q✉❡❧② ❞❡t❡r♠✐♥❡❞❄

✶








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❳❩⑦✰➔❙♥❻❨↔➑➜❫⑩❹q❨✂❹❻❸✐❶▼⑤
✄➄❪✺❹■→✰❫⑩❨✮❵✘❬✺♠❙♥■☎
❬ ✡ ➒ ✕ ☛ ✍ á➤➂❲❹❤❨✸❾⑩❱✬❸❖❹❤❪✳➩✂❜✍❞✰➎❽✎♦ ✒❲❿➁♥➙❱❺⑧❑❳✘❦✰❹❤❨✮❳✐❫⑩❨⑨➑➜❳➏→①❳❩❪❽♥■❾①❣✪♥❩❱✮➑➯⑤❤➊✮❹q❪➄❹❤❨✮❳➧❢❻❹q❨✮❳
❿✂♥❩❱✤❼❙➑➃❳❧❬✺♠❙♥■✫
❬ ✡ ➒ ☛ ✕ ê ➱ ✍ á✛➂❂❸❖❹❤❪☞♥■❾⑩❾❯➩➽÷ð❬❭♠✮❫➣➑✝❣✐❹❤❼✮❾Ô❦➢❸❖❹❤❾❴❾⑩❹■❣ó❸❖❪✺❹q❿à❬❭♠✮❳❧♥■⑧❑❹■→①❳❧❫❴❨❙❦✏❼❙③④❬❭❫⑩❹q❨❷ø❽❢
❣✪♠✮❫Ô③Û♠✃❦✏❳❩❬❭❳❩❪✺❿❐❫⑩❨✮❳■✬➑ ✡ ➒ ✕ ☛ ✍ ⑤qÙ✝❨✮❹q❬✺♠✮❳➏❪❄❿✤❳➏❬✺♠✮❹✰❦☞❫➣➑✳⑧✏❱☞③➯❹❤❼✮❨✰❬✺❫❴❨✮❵✝❬❭♠✮❳❇❸s♥q③④❬✺❹q❪❽➑❄❹■✭❸ ✆✏⑤➧➓❲♠✮❳❇❸s♥q③④❬✺❹q❪
♦ ✒✓✮ ❫➣➑✐❦✏❫⑩→✰❫➣➑➃❫⑩⑧✮❾❴❳✪⑧✏✯❱ ✆❧♥➹❬❭❹❤❬②♥❻❾✳❹❻❸✾♦ ✒ ➆✢➂✬❬❭❫⑩❿✤❳■➑➯⑤①➓❲♠✮❳✪➩ ✮ ❿✂♥❩❱❂⑧❙❳✃❦✏❫⑩→✰❫➣❦✰❳■❦✼⑧✰✯❱ ✆❂♥❂❬✺❹q❬❽♥■❾✳❹❻❸
✰ ➩❙➥■✎♦ ✱❙➀ ✰ ➩❑➥❩✲✈ ✱❙➀➋➼■➼❩➼✮❬❭❫⑩❿✤❳➙➑✳♥❻❨❙❦✤➑➃❫⑩❿✤❫❴❾➣♥■❪✺❾⑩❱✪❸❖❹❤❪✘rs✎♦ ✒❷➆✃➩❑t ✮ ✔①✎♦ ✒■➥■♦ ✌✴✳ ✰ ➩❙➥■♦ ✌ ✱❙➀ ✰ r➇✎♦ ✒✳➆✃➩❑t✺➥➙♦ ✌ ✱
❣✪❫⑩❬❭♠➸❳➙Ü✉❼❙♥■❾⑩❫❴❬➜❱✂❹q❨✮❾⑩❱☎❫➭❸✪❬❭♠✮❳❩❪❭❳❧❫➣➑✬❨✮❹✂❪✺❹q❼✮❨❙❦✏❫❴❨✮❵☎❦✏❹■❣✪❨✳❢▼➑➜❹➋❬❭♠✮❳❧❬✺❹q❬❽♥■❾❇❨✰❼✮❿⑨⑧❙❳❩❪✝❹■❸✪❸s♥❤③④❬❭❹❤❪②➑
❹❻✵❸ ✆➋❹✰③➏③④❼✮❪✺❫❴❨✮❵➋❫❴❨➽❨✰❼✮❿✤❳❩❪②♥❻❬✺❹q❪✃♥❻❨❙❦➽❦✰❳❩❨✮❹❤❿❐❫⑩❨❙♥■❬✺❹❤❪✃❹❻✶
❸ ✡ ➒ ✕ ☛ ✍ ❫➣✸➑ ✷❩❳❩❪✺❹☎❹❤❨✮❾❴❱➸❫ ❸✪❬❭♠✮❳❩❪✺❳❐❫➣➑➹❨✸❹
❪✺❹q❼✮❨❙❦✏❫❴❨✮❵❐❦✏❹■❣✪❨➋❫⑩❨❐❬✺♠✮❳☞➔✮❪❭❳❩→✏❫❴❹❤❼❙➑➯⑤❙➓❲♠✮❫➣➑❲♠↔♥❻➔✮➔❙❳❩❨❙➑✘❹❤❨✮❾❴❱✤❫➭❸❯➩✂❜✍❞❧❹❤❪❺✎♦ ✒■⑤
➱ ❈ ❇ ❶ ✮ ⑤✪Ù❺➑➋♥❻⑧❙❹■→①❳①❢
✹✤✻▲ ✺■◆✰✽● ✼▼✾× ✺❀✿✻❁❯❃
◆ ❂❩❍❙❅▲ ❄❆✺❀✿❖✦❍ ✼❇❘☞ù❙❳❩❬✡➻✃r➇❶➐t★✚ ❜ ➝ó❣✪♠✮❳➏❪✺❳★✎♦ ✒✲❇ ❶ ✮ ⑧✮❼✸❬➋✓♦ ✒ ú ❉
➻❂rs❶➐t➄❋
❜ ❊❍✌● ✜ ➱ ✰ ❶✳➥■♦ ✌ ✱❤⑤❙↕ð❸❅❶➋❋
❜ ❊ ✚☞✒ ✜ ✖ ✎➫ ✚②♦ ❢❙➫ ✌✣■❑❏ ❞✰➎❩✽➂ ▲✰❢✮❬❭♠✮❳❩❨✂❣❲❳✬❥❙❨↔❦➢❬✺♠↔♥❻❬
➻❂r➇❶✳t❅❜ ✙ ✚ ➫◆✚②♦ ✚✺ê ✌ ❜ ✚✢✙ ✜ ✒ ➫✎✚➏r➇♦ ✚ ➆✍➂❻t❅❜✕❶✤➆€❖☞r➇❶➐t
☞➱ ▼ ✌ ▼ ▼ ✒
✖
❣✪♠✮❳❩❪❭◗❳ ❖➹rs❶➐t➄❜ ❊ ✚☞✒ ✜ ✖ ➫ ✚ ❫Ô➑➲❬❭♠✮❳☞➑➃❼✮❿➍❹❻❸✾❬✺♠✮❳☞⑧✮❫❴❨❙♥❻❪❭❱✂❦✏❫⑩❵q❫⑩❬②➑④⑤
⑥✝❹■❣☞❢✳❣❲❳✂➑➃❳❩❳⑨❬✺♠❙♥■❘
❬ ✡ ✕ ❚ú✕ ❙ ✍ ❫Ô➑➹❹✰❦✮❦➽❫➭❸❺♥■❨❙❦➸❹q❨✮❾⑩❱★❫➭❸➲➻❂r⑩➩✤➀✇➦✉t❂❜ ➻❂r❴➩❑t❅➀✇➻❂r➇➦①t✝❣✪♠✸❫➣③❽♠➸❫Ô➑
❳■Ü①❼✮❫⑩→q♥■❾⑩❳❩❨✰✚ ❬❄❬❭◗❹ ❖➹r❴➩❲➀✂➦✉t❅❋
❜ ❖☞r⑩➩❙t①✶
➀ ❖☞r➇➦①t❽⑤q➓❲♠✮❫➣➑❅③④❳❩❪❭❬❽♥■❫⑩❨✮❾❴❱➹❹✰③➏③④❼✮❪✺❳➙➑ü❫ ❸✸➩➢❜ ❊ ✚✢✒ ✜ ✖ ❯➩ ✚➯♦ ✚ ❢✰➦➹❜
❊ ✚✢✒ ✜ ✖ ❱➦ ✚②♦ ❣✪♠✸❳❩❪✺❳▼✛➩ ✚➙➎❽❱➦ ✚ ■❲❏ ❞✏➎❩❀➂ ▲✰❢❽✛➩ ✚■➀☞❳➦ ✚✘ò➈♦✰❢❤♥q➑➐❬❭♠✮❫➣➑❷❿➁♥❻➨①❳■➑✮❬✺♠✸❳❲⑧✮❫❴❨❙♥❻❪❭❱✬❪✺❳❩➔✸❪✺❳■➑➃❳❩❨✏❬②♥❻❬❭❫⑩❹❤❨
❸❖❹❤❪☞➩✂➀➪➦➸❜ ❊ ✚✢✒ ✜ ✖ r❴➩ ✚ ➀✕➦ ✚ t➟♦ ✚ ⑤➄➵★❳✂❿✗❼↔➑➜❬☞❬✺♠✮❳➏❨ï➔✸❪✺❹■→①❳✂❬✺♠❙♥■❬❧❹❤❬✺♠✸❳❩❪✺❣✪❫Ô➑➜❳①❨❢ ❖☞r⑩➩✤➀✕➦①t✤ò
❖☞r⑩➩❑t✳❩

➀ ❖☞r➇➦①t❽⑤✸↕➟❬❺➑➜❼✰ý✂③④❳➙➑❲❬✺❹✤➔✮❪❭❹■→①❳➹❬❭♠✮❫➣➑❲❸❖❹❤❪❺➦✼❜➪✓♦ ❬✳♥❻❨❙❦➋❬❭♠✮❳❩❨➋❼↔➑➜❳❂❬❭♠✮❫➣➑✪❪✺❳➙➑➜❼✮❾❴❬❲❸❖❹❤❪✬❳■♥q③Û♠
➦ ❬ ❜➤➂❤⑤
ù❙❳❩❬❺➩➽⑧❑❳☎♥q➑❧♥❻⑧❙❹■→①❳①❢➄➦✡❜➍♦ ❬ ⑤✾↕s✵❸ ❭☞èÝ➝★❹❤❪☞➩ ❬ ❜ ❞✰❢➄❣❲❳⑨❵❤❳❩✝❬ ❖☞r⑩➩✤➀✇➦✉t❧❪
❜ ❖➹r❴➩❑t❅➀➤➂➋❜
❖☞r⑩➩❑t❯➀€❖☞r➇➦①t❽⑤❴❫✘❬✺♠✮❳❩❪❭❣✪❫➣➑➃❳①❢✏❣✐❳✬❵❤❳❩❬❅➩❂➀➅➦➹❜➤r❴➩❧➆➽♦ ❬ t✳➀☎➩ ❬ ú ➱ ⑤✮Ù✬➑✬❖☞r⑩➩✃➆✢♦ ❬ t❅❜❋❖☞r⑩➩❑t➌➆➈➂q❢
❫⑩❨↔❦✏❼❙③④❬❭❫⑩❹❤❨✂❱✰❫⑩❳❩❾Ô❦✮❵➑ ❖☞r⑩➩❂➀➅➦✉t❅❛
➞ ❖➹r❴➩❑t✐❜
ò ❖☞r⑩➩❑t✳❝
➀ ❖➹rs➦✉t②⑤
⑥✝❹■❣☞❢✮❵❤❫❴→①❳❩❨❧➩❧➀✢➦❂❜✕❶▼❢❙❶✓❜ ❊ ✚✢✒ ✜ ✖ ❞❶ ✚➯♦ ✚ ❥❙⑦✰❳■❦❷❢✏❣✐❳☞❵❤❳❩✶
❬ ✡ ✕❢Ú ❡ ❙ ✍ ❹✰❦✮❦➋❳❩⑦✸♥❤③④❬❭❾⑩❱⑨❫⑩❨✂❬✺♠✸❳❂③➏♥❤➑➜❳➙➑
❣✪♠✮❳❩❪❭❳✬➩✡♥■❨❙❦➋➦⑨♥❻❪❭❳❂❿✂♥q❦✏❳✬❸❖❪✺❹q❿➠➑➜➔✸❾⑩❫⑩❬❭❬✺❫❴❨✮❵❧❬✺♠✮❳❀♦ ✚ ❸❖❹❤❪✬❣✪♠✸❫➣③❽♠➋❴❶ ✚✬❜ô❥Ú ➂➹✐ ❫❴❨✏❬❭❹✗❬➃❣❲❹⑨❵❤❪❭❹❤❼✮➔↔➑④⑤
Ù✬➑❲❬❭♠✮❳❩❪✺❳✃♥■❪✺❳◗❖➹rs❶➐t✐➑➃❼❙③Û♠✂❬❭❳❩❪✺❿✂➑➯❢①❬✺♠✮❫Ô➑✪③❩♥❻❨✂⑧❙❳❧❦✏❹q❨✮❳☞❫⑩❨✂❳❩⑦✸♥❤③④❬❭❾⑩❱❐♦✽❣✦❤ ❦✏❫❧❦❷❳➏❪✺❳❩❨✰❬❲❣✥♥➙❱✸➑④⑤

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