Navvye/Olympiad-Problems-CSV huggingface dataset:
2021 Macedonian Mathematical Olympiad
For a fixed positive integer $n \geq 3$ we are given a $n$ $\times$ $n$ board with all unit squares initially white. We define a floating plus as a $5$-tuple $(M,L,R,A,B)$ of unit squares such that $L$ is in the same row and left of $M$, $R$ is in the same row and right of $M$, $A$ is in the same column and above $M$ and $B$ is in the same column and below $M$. It is possible for $M$ to form a floating plus with unit squares that are not next to it. Find the largest positive integer $k$ (depending on $n$) such that we can color some $k$ squares black in such a way that there is no black colored floating plus.
2021 Macedonian Mathematical Olympiad
Let $(x_{n})_{n=1}^{+\infty}$ be a sequence defined recursively with $x_{n+1} = x_{n}(x_{n}-2)$ and $x_{1} = \frac{7}{2}$. Let $x_{2021} = \frac{a}{b}$, where $a,b \in \mathbb{N}$ are coprime. Show that if $p$ is a prime divisor of $a$, then either $3|p-1$ or $p=3$.
2020 Macedonian Nationаl Olympiad
Let $a, b$ be positive integers and $p, q$ be prime numbers for which $p \nmid q - 1$ and $q \mid a^p - b^p$. Prove that $q \mid a - b$.
2020 Macedonian Nationаl Olympiad
Let $x_1, ..., x_n$ ($n \ge 2$) be real numbers from the interval $[1, 2]$. Prove that



$|x_1 - x_2| + ... + |x_n - x_1| \le \frac{2}{3}(x_1 + ... + x_n)$,



with equality holding if and only if $n$ is even and the $n$-tuple $(x_1, x_2, ..., x_{n - 1}, x_n)$ is equal to $(1, 2, ..., 1, 2)$  or $(2, 1, ..., 2, 1)$.
2020 Macedonian Nationаl Olympiad
Let $ABC$ be a triangle, and $A_1, B_1, C_1$ be points on the sides $BC, CA, AB$, respectively, such that $AA_1, BB_1, CC_1$ are the internal angle bisectors of $\triangle ABC$. The circumcircle $k' = (A_1B_1C_1)$ touches the side $BC$ at $A_1$. Let $B_2$ and $C_2$, respectively, be the second intersection points of $k'$ with lines $AC$ and $AB$. Prove that $|AB| = |AC|$ or $|AC_1| = |AB_2|$.
2020 Macedonian Nationаl Olympiad
Let $S$ be a nonempty finite set, and $\mathcal {F}$ be a collection of subsets of $S$ such that the following conditions are met:

(i) $\mathcal {F}$ $\setminus$ {$S$} $\neq$ $\emptyset$ ;

(ii) if $F_1, F_2 \in \mathcal {F}$, then $F_1 \cap F_2 \in \mathcal {F}$ and $F_1 \cup F_2 \in \mathcal {F}$.



Prove that there exists $a \in S$ which belongs to at most half of the elements of $\mathcal {F}$.
2019 Macedonia National Olympiad
In an acute-angled triangle $ABC$, point $M$ is the midpoint of side $BC$ and the centers of the $M$- excircles of triangles $AMB$ and $AMC$ are $D$ and $E$, respectively. The circumcircle of triangle $ABD$ intersects line $BC$ at points $B$ and $F$. The circumcircle of triangle $ACE$ intersects line $BC$ at points $C$ and $G$. Prove that $BF\hspace{0.25mm} = \hspace{0.25mm} CG$ .
2019 Macedonia National Olympiad
Let $n$ be a positive integer. If $r\hspace{0.25mm} \equiv \hspace{1mm} n\hspace{1mm} (mod\hspace{1mm} 2)$ and $r\hspace{0.10mm} \in \hspace{0.10mm} \{ 0,\hspace{0.10mm} 1 \} $, find the number of integer solutions to the system of equations



$\left\{\begin{array}{l}x+y+z = r \\ \mid x \mid  + \mid y \mid + \mid z \mid = n \end{array}\right.$
2019 Macedonia National Olympiad
Let $ABC$ be a triangle with $AB=AC$, and let $M$ be the midpoint of $BC$. Let $P$ be a point such that $PB<PC$ and $PA$ is parallel to $BC$. Let $X$ and $Y$ be points on the lines $PB$ and $PC$, respectively, so that $B$ lies on the segment $PX$, $C$ lies on the segment $PY$, and $\angle PXM=\angle PYM$. Prove that the quadrilateral $APXY$ is cyclic.
2019 Macedonia National Olympiad
Determine all functions $f: \mathbb {N} \to \mathbb {N}$ such that



$n!\hspace{1mm} +\hspace{1mm} f(m)!\hspace{1mm}  |\hspace{1mm} f(n)!\hspace{1mm} +\hspace{1mm} f(m!)$,



for all $m$, $n$ $\in$ $\mathbb{N}$.
2019 Macedonia National Olympiad
Let $n$ be a given positive integer. Sisyphus performs a sequence of turns on a board consisting of $n + 1$ squares in a row, numbered $0$ to $n$ from left to right. Initially, $n$ stones are put into square $0$, and the other squares are empty. At every turn, Sisyphus chooses any nonempty square, say with $k$ stones, takes one of these stones and moves it to the right by at most $k$ squares (the stone should say within the board). Sisyphus' aim is to move all $n$ stones to square $n$.

Prove that Sisyphus cannot reach the aim in less than

$$ \left \lceil \frac{n}{1} \right \rceil + \left \lceil \frac{n}{2} \right \rceil + \left \lceil \frac{n}{3} \right \rceil + \dots + \left \lceil \frac{n}{n} \right \rceil $$turns. (As usual, $\lceil x \rceil$ stands for the least integer not smaller than $x$. )
2018 Macedonia National Olympiad
Determine all natural numbers $n$ such that $9^n - 7$ can be represented as a product of at least two consecutive natural numbers.
2018 Macedonia National Olympiad
Let $n$ be a natural number and $C$ a non-negative real number. Determine the number of sequences of real numbers $1, x_{2}, ..., x_{n}, 1$ such that the absolute value of the difference between any two adjacent terms is equal to $C$.
2018 Macedonia National Olympiad
Determine all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that:$$f(\max \left\{ x, y \right\} + \min \left\{ f(x), f(y) \right\}) = x+y $$for all real $x,y \in \mathbb{R}$
2018 Macedonia National Olympiad
Let $t_{k} = a_{1}^k + a_{2}^k +...+a_{n}^k$, where $a_{1}$, $a_{2}$, ... $a_{n}$ are positive real numbers and $k \in \mathbb{N}$. Prove that $$\frac{t_{5}^2 t_1^{6}}{15} - \frac{t_{4}^4 t_{2}^2 t_{1}^2}{6} + \frac{t_{2}^3 t_{4}^5}{10} \geq 0 $$
2018 Macedonia National Olympiad
Given is an acute $\triangle ABC$ with orthocenter $H$. The point $H'$ is symmetric to $H$ over the side $AB$. Let $N$ be the intersection point of $HH'$ and $AB$. The circle passing through $A$, $N$ and $H'$ intersects $AC$ for the second time in $M$, and the circle passing through $B$, $N$ and $H'$ intersects $BC$ for the second time in $P$. Prove that $M$, $N$ and $P$ are collinear.
2017 Macedonia National Olympiad
Find all functions $f:\mathbb{N} \to \mathbb{N}$ such that for each natural integer $n>1$ and for all $x,y \in \mathbb{N}$ the following holds:



$$f(x+y) = f(x) + f(y) + \sum_{k=1}^{n-1} \binom{n}{k}x^{n-k}y^k$$
2017 Macedonia National Olympiad
Find all natural integers $n$ such that $(n^3 + 39n - 2)n! + 17\cdot 21^n + 5$ is a square.
2017 Macedonia National Olympiad
Let $x,y,z \in \mathbb{R}$ such that $xyz = 1$. Prove that:



$$\left(x^4 + \frac{z^2}{y^2}\right)\left(y^4 + \frac{x^2}{z^2}\right)\left(z^4 + \frac{y^2}{x^2}\right) \ge \left(\frac{x^2}{y} + 1 \right)\left(\frac{y^2}{z} + 1 \right)\left(\frac{z^2}{x} + 1 \right).$$
2017 Macedonia National Olympiad
Let $O$ be the circumcenter of the acute triangle $ABC$ ($AB < AC$). Let $A_1$ and $P$ be the feet of the perpendicular lines drawn from $A$ and $O$ to $BC$, respectively. The lines $BO$ and $CO$ intersect $AA_1$ in $D$ and $E$, respectively. Let $F$ be the second intersection point of $\odot ABD$ and $\odot ACE$. Prove that the angle bisector od $\angle FAP$ passes through the incenter of $\triangle ABC$.
2017 Macedonia National Olympiad
Let $n>1 \in \mathbb{N}$ and $a_1, a_2, ..., a_n$ be a sequence of $n$ natural integers. Let:



$$b_1 = \left[\frac{a_2 + \cdots + a_n}{n-1}\right], b_i = \left[\frac{a_1 + \cdots + a_{i-1} + a_{i+1} + \cdots + a_n}{n-1}\right], b_n = \left[\frac{a_1 + \cdots + a_{n-1}}{n-1}\right]$$

Define a mapping $f$ by $f(a_1,a_2, \cdots a_n) = (b_1,b_2,\cdots,b_n)$.



a) Let $g: \mathbb{N} \to \mathbb{N}$ be a function such that $g(1)$ is the number of different elements in $f(a_1,a_2, \cdots a_n)$ and $g(m)$ is the number od different elements in $f^m(a_1,a_2, \cdots a_n) = f(f^{m-1}(a_1,a_2, \cdots a_n)); m>1$. Prove that $\exists k_0 \in \mathbb{N}$ s.t. for $m \ge k_0$ the function $g(m)$ is periodic.



b) Prove that $\sum_{m=1}^k \frac{g(m)}{m(m+1)} < C$ for all $k \in \mathbb{N}$, where $C$ is a function that doesn't depend on $k$.
2016 Macedonia National Olympiad
Solve the equation in the set of natural numbers $1+x^z + y^z = LCM(x^z,y^z)$
2016 Macedonia National Olympiad
A magic square is a square with side 3 consisting of 9 unit squares, such that the numbers written in the unit squares (one number in each square) satisfy the following property: the sum of the numbers in each row is equal to the sum of the numbers in each column and is equal to the sum of all the numbers written in any of the two diagonals.



A rectangle with sides $m\ge3$ and $n\ge3$ consists of $mn$ unit squares. If in each of those unit squares exactly one number is written, such that any square with side $3$ is a magic square, then find the number of most different numbers that can be written in that rectangle.
2016 Macedonia National Olympiad
Solve the equation in the set of natural numbers $xyz+yzt+xzt+xyt = xyzt + 3$
2016 Macedonia National Olympiad
A segment $AB$ is given and it's midpoint $K$. On the perpendicular line to $AB$, passing through $K$ a point $C$, different from $K$ is chosen. Let $N$ be the intersection of $AC$ and the line passing through $B$ and the midpoint of $CK$. Let $U$ be the intersection point of $AB$ and the line passing through $C$ and $L$, the midpoint of $BN$. Prove that the ratio of the areas of the triangles $CNL$ and $BUL$, is independent of the choice of the point $C$.
2016 Macedonia National Olympiad
Let $n\ge3$ and $a_1,a_2,...,a_n \in \mathbb{R^{+}}$, such that $\frac{1}{1+a_1^4} + \frac{1}{1+a_2^4} + ... + \frac{1}{1+a_n^4} = 1$. Prove that: $$a_1a_2...a_n \ge (n-1)^{\frac n4}$$
2015 Macedonia National Olympiad
Let $AH_A, BH_B$ and $CH_C$ be altitudes in $\triangle ABC$. Let $p_A,p_B,p_C$ be the perpendicular lines from vertices $A,B,C$ to $H_BH_C, H_CH_A, H_AH_B$ respectively. Prove that $p_A,p_B,p_C$ are concurrent lines.
2015 Macedonia National Olympiad
Let $a,b,c \in \mathbb{R}^{+}$ such that $abc=1$. Prove that:



$$a^2b + b^2c + c^2a \ge \sqrt{(a+b+c)(ab + bc +ca)}$$
2015 Macedonia National Olympiad
All contestants at one contest are sitting in $n$ columns and are forming a "good" configuration. (We define one configuration as "good" when we don't have 2 friends sitting in the same column). It's impossible for all the students to sit in $n-1$ columns in a "good" configuration. Prove that we can always choose contestants $M_1,M_2,...,M_n$ such that $M_i$ is sitting in the $i-th$ column, for each $i=1,2,...,n$ and $M_i$ is friend of $M_{i+1}$ for each $i=1,2,...,n-1$.
2015 Macedonia National Olympiad
Let $k_1$ and $k_2$ be two circles and let them cut each other at points $A$ and $B$. A line through $B$ is cutting $k_1$ and $k_2$ in $C$ and $D$ respectively, such that $C$ doesn't lie inside of $k_2$ and $D$ doesn't lie inside of $k_1$. Let $M$ be the intersection point of the tangent lines to $k_1$ and $k_2$ that are passing through $C$ and $D$, respectively. Let $P$ be the intersection of the lines $AM$ and $CD$. The tangent line to $k_1$ passing through $B$ intersects $AD$ in point $L$. The tangent line to $k_2$ passing through $B$ intersects $AC$ in point $K$. Let $KP \cap MD \equiv N$ and $LP \cap MC \equiv Q$. Prove that $MNPQ$ is a parallelogram.
2015 Macedonia National Olympiad
Find all natural numbers $m$ having exactly three prime divisors $p,q,r$, such that $$p-1\mid m; \quad qr-1 \mid m; \quad q-1 \nmid m; \quad r-1 \nmid m; \quad 3 \nmid q+r.$$
2014 Macedonia National Olympiad
In a plane, 2014 lines are distributed in 3 groups. in every group all the lines are parallel between themselves. What is the maximum number of triangles that can be formed, such that every side of such triangle lie on one of the lines?
2014 Macedonia National Olympiad
Solve the following equation in $\mathbb{Z}$:



$$3^{2a + 1}b^2 + 1 = 2^c$$
2014 Macedonia National Olympiad
Let $k_1, k_2$ and $k_3$ be three circles with centers $O_1, O_2$ and $O_3$ respectively, such that no center is inside of the other two circles. Circles $k_1$ and $k_2$ intersect at $A$ and $P$, circles $k_1$ and $k_3$ intersect and $C$ and $P$, circles $k_2$ and $k_3$ intersect at $B$ and $P$. Let $X$ be a point on $k_1$ such that the line $XA$ intersects $k_2$ at $Y$ and the line $XC$ intersects $k_3$ at $Z$, such that $Y$ is nor inside $k_1$ nor inside $k_3$ and $Z$ is nor inside $k_1$ nor inside $k_2$.



a) Prove that $\triangle XYZ$ is simular to  $\triangle O_1O_2O_3$

b) Prove that the $P_{\triangle XYZ} \le 4P_{\triangle O_1O_2O_3}$. Is it possible to reach equation?

*Note: $P$ denotes the area of a triangle*
2014 Macedonia National Olympiad
Let $a,b,c$ be real numbers such that $a+b+c = 4$ and $a,b,c > 1$. Prove that:



$$\frac 1{a-1} + \frac 1{b-1} + \frac 1{c-1} \ge \frac 8{a+b} + \frac 8{b+c} + \frac 8{c+a}$$
2014 Macedonia National Olympiad
From an equilateral triangle with side 2014 we cut off another equilateral triangle with side 214, such that both triangles have one common vertex and two of the side of the smaller triangles lie on two of the side of the bigger triangle. Is it possible to cover this figure with figures in the picture without overlapping (rotation is allowed) if all figures are made of equilateral triangles with side 1? Explain the answer!



[asy]

import olympiad;

unitsize(20);

pair A,B,C,D,E,F,G,H;

A=(0,0);

B=(1,0);

C=rotate(60)*B;

D=rotate(60)*C;

E=rotate(60)*D;

F=rotate(60)*E;

G=rotate(60)*F;

draw(A--B); draw(A--C); draw(A--D); draw(A--E); draw(A--F); draw(A--G);

draw(B--C--D--E--F--G--B);

A=(2,0);

B=A+(1,0);

C=A+rotate(60)*(B-A);

D=A+rotate(60)*(C-A);

E=A+rotate(120)*(D-A);

F=A+rotate(60)*(E-A);

G=2*F-E;

H=2*C-D;

draw(A--D--C--A--B--C--H--B--G--F--E--A--F--B);

A=(4,0);

B=A+(1,0);

C=A+rotate(-60)*(B-A);

D=B+rotate(60)*(C-B);

E=B+rotate(60)*(D-B);

F=B+rotate(60)*(E-B);

G=E+rotate(60)*(D-E);

H=E+rotate(60)*(G-E);

draw(A--B--C--A);

draw(C--D--B);

draw(D--E--B);

draw(B--F--E);

draw(E--G--D);

draw(E--H--G);

A=(8.5,0.5);

B=A+(1,0);

C=A+rotate(60)*(B-A);

D=A+rotate(60)*(C-A);

E=A+rotate(60)*(D-A);

F=A+rotate(60)*(E-A);

G=A+rotate(60)*(F-A);

H=G+rotate(60)*(F-G);

draw(A--B); draw(A--C); draw(A--D); draw(A--E); draw(A--F); draw(A--G);

draw(B--C); draw(D--E--F--G--B); draw(G--H--F);[/asy]
2013 Macedonia National Olympiad
Let $ p,q,r $ be prime numbers. Solve the equation $ p^{2q}+q^{2p}=r $
2013 Macedonia National Olympiad
$ 2^n $ coins are given to a couple of kids. Interchange of the coins occurs when some of the kids has at least half of all the coins. Then from the coins of one of those kids to the all other kids are given that much coins as the kid already had. In case when all the coins are at one kid there is no possibility for interchange. What is the greatest possible number of consecutive interchanges? ($ n $ is natural number)
2013 Macedonia National Olympiad
Acute angle triangle is given such that $ BC $ is the longest side. Let $ E $ and $ G $ be the intersection points from the altitude from $ A $ to $ BC $ with the circumscribed circle of  triangle $ ABC $ and $ BC $ respectively. Let the center $ O $ of this circle is positioned on the perpendicular line from $ A $ to $ BE $. Let $ EM $ be perpendicular to $ AC $ and $ EF $ be perpendicular to $ AB $. Prove that the area of $ FBEG $ is greater than the area of $ MFE $.
2013 Macedonia National Olympiad
Let $ a,b,c $ be positive real numbers such that $ a^4+b^4+c^4=3 $. Prove that $$ \frac{9}{a^2+b^4+c^6}+\frac{9}{a^4+b^6+c^2}+\frac{9}{a^6+b^2+c^4}\leq\ a^6+b^6+c^6+6 $$
2013 Macedonia National Olympiad
An arbitrary triangle ABC is given. There are 2 lines, p and q, that are not parallel to each other and they are not perpendicular to the sides of the triangle. The perpendicular lines through points A, B and C to line p we denote with $ p_a, p_b, p_c $ and the perpendicular lines to line q we denote with $ q_a, q_b, q_c $. Let the intersection points of the lines $ p_a, q_a, p_b, q_b, p_c $ and $ q_c $ with $ q_b, p_b, q_c, p_c, q_a $ and $ p_a $ are $ K, L, P, Q, N $ and $ M $. Prove that $ KL, MN $ and $ PQ $ intersect in one point.
2012 Macedonia National Olympiad
Solve the equation $~$ $x^4+2y^4+4z^4+8t^4=16xyzt$ $~$ in the set of integer numbers.
2012 Macedonia National Olympiad
If $~$ $a,\, b,\, c,\, d$ $~$ are positive real numbers such that $~$ $abcd=1$ $~$ then prove that the following inequality holds



$$ \frac{1}{bc+cd+da-1} + \frac{1}{ab+cd+da-1} + \frac{1}{ab+bc+da-1} + \frac{1}{ab+bc+cd-1}\; \le\; 2\, . $$



When does inequality hold?
2012 Macedonia National Olympiad
Find all functions $f : \mathbb{R} \to \mathbb{Z}$ which satisfy the conditions:



$f(x+y) < f(x) + f(y)$



$f(f(x)) = \lfloor {x} \rfloor + 2$
2012 Macedonia National Olympiad
A fixed circle $k$ and collinear points $E,F$ and $G$ are given such that the points $E$ and $G$ lie outside the circle $k$ and $F$ lies inside the circle $k$. Prove that, if $ABCD$ is an arbitrary quadrilateral inscribed in the circle $k$ such that the points $E,F$ and $G$ lie on lines $AB,AD$ and $DC$ respectively, then the side $BC$ passes through a fixed point collinear with $E,F$ and $G$, independent of the quadrilateral $ABCD$.
2012 Macedonia National Olympiad
A hexagonal table is given, as the one on the drawing, which has $~$ $2012$ $~$ columns. There are $~$ $2012$ $~$ hexagons in each of the odd columns, and there are $~$ $2013$ $~$ hexagons in each of the even columns.  The number $~$ $i$ $~$ is written in each hexagon from the $~$ $i$-th column. Changing the numbers in the table is allowed in the following way: We arbitrarily select three adjacent hexagons, we rotate the numbers, and if the rotation is clockwise then the three numbers decrease by one, and if we rotate them counterclockwise the three numbers increase by one (see the drawing below). What's the maximum number of zeros that can be obtained in the table by using the above-defined steps.
2011 Macedonia National Olympiad
Let $~$ $ a,\,b,\,c,\,d\, >\, 0$ $~$ and $~$ $a+b+c+d\, =\, 1\, .$ $~$ Prove the inequality



$$ \frac{1}{4a+3b+c}+\frac{1}{3a+b+4d}+\frac{1}{a+4c+3d}+\frac{1}{4b+3c+d}\; \ge\; 2\, . $$
2011 Macedonia National Olympiad
Acute-angled $~$ $\triangle{ABC}$ $~$ is given. A line $~$ $l$ $~$ parallel to side $~$ $AB$ $~$ passing through vertex $~$ $C$ $~$ is drawn. Let the angle bisectors of $~$ $\angle{BAC}$ $~$ and $~$ $\angle{ABC}$ $~$ intersect the sides $~$ $BC$ and $~$ $AC$ at points $~$ $D$ $~$ and $~$ $F$, and line $~$ $l$ $~$ at points $~$ $E$ $~$ and $~$ $G$ $~$ respectively. Prove that if $~$ $\overline{DE}=\overline{GF}$ $~$ then $~$ $\overline{AC}=\overline{BC}\, .$
2011 Macedonia National Olympiad
Find all natural numbers $n$ for which each natural number written with $~$ $n-1$ $~$ 'ones' and one 'seven' is prime.
2011 Macedonia National Olympiad
Find all functions $~$ $f: \mathbb{R} \to \mathbb{R}$ $~$ which satisfy the equation



$$ f(x+yf(x))\, =\, f(f(x)) + xf(y)\, . $$
2011 Macedonia National Olympiad
A table of the type $~$ $ (n_1, n_2, ... , n_m) ,\ n_1 \ge n_2 \ge ... \ge n_m $ $~$ is defined in the following way: $~$ $n_1$ $~$ squares are ordered horizontally one next to another, then $~$ $n_2$ $~$ squares are ordered horizontally beneath the already ordered $~$ $n_1$ $~$ squares. The procedure continues until a net composed of $~$ $n_1$ $~$ squares in the first row, $~$ $n_2$ $~$ in the second, $~$ $n_i$ $~$ in the $~$ $i$-th row is obtained, such that there are totally $~$ $n=n_1+n_2+...+n_m$ $~$ squares in the net. The ordered rows form a straight line on the left, as shown in the example. The obtained table is filled with the numbers from $~$ $1$ $~$ till $~$ $n$ $~$ in a way that the numbers in each row and column become greater from left to right and from top to bottom, respectively. An example of a table of the type $~$ $(5,4,2,1)$ $~$ and one possible way of filling it is attached to the post. Find the number of ways the table of type $~$ $(4,3,2)$ $~$ can be filled.
2010 Macedonia National Olympiad
Solve the equation



$$ x^3+2y^3-4x-5y+z^2=2012, $$



in the set of integers.
2010 Macedonia National Olympiad
Let $a,b,c$ be positive real numbers for which $a+b+c=3$. Prove the inequality

$$\frac{a^3+2}{b+2}+\frac{b^3+2}{c+2}+\frac{c^3+2}{a+2}\ge3$$
2010 Macedonia National Olympiad
A total of $2010$ coins are distributed in $5$ boxes. At the beginning the quantities of coins in the boxes are consecutive natural numbers. Martha should choose and take one of the boxes, but before that she can do the following transformation finitely many times: from a box with at least 4 coins she can transfer one coin to each of the other boxes.

What is the maximum number of coins that Martha can take away?
2010 Macedonia National Olympiad
The point $O$ is the centre of the circumscribed circle of the acute-angled triangle $ABC$. The line $AO$ cuts the side $BC$ in point $N$, and the line $BO$ cuts the side $AC$ at point $M$. Prove that if $CM=CN$, then $AC=BC$.
2010 Macedonia National Olympiad
Let the boxes in picture $1$ be marked as in picture $2$ below (from top to bottom in layers). In one move it is allowed to switch the empty box with another box adjacent to it (two boxes are adjacent if they share a common side). Can the arrangement of the numbers in picture $3$ be obtained after finitely many moves?
2009 Macedonia National Olympiad
Find all natural numbers $x,y,z$ such that $a+2^x3^y=z^2$.
2009 Macedonia National Olympiad
Let $O$ be the centre of the incircle of $\triangle ABC$. Points $K,L$ are the intersection points of the circles circumscribed about triangles $BOC,AOC$ respectively with the bisectors of the angles at $A,B$ respectively $(K,L\not= O)$. Also $P$ is the midpoint of segment $KL$, $M$ is the reflection of $O$ with respect to $P$ and $N$ is the reflection of $O$ with respect to line $KL$. Prove that the points $K,L,M$ and $N$ lie on the same circle.
2009 Macedonia National Olympiad
The Macedonian Mathematical Olympiad is held in two rooms numbered $1$ and $2$. At the beginning all of the competitors enter room No. $1$. The final arrangement of the competitors to the rooms is obtained in the following way: a list with the names of a few of the competitors is read aloud; after a name is read, the corresponding competitor and all of his/her acquaintances from the rest of the competitors change the room in which they currently are. Hence, to each list of names corresponds one final arrangement of the competitors to the rooms. Show that the total number of possible final arrangements is not equal to $2009$ (acquaintance between competitors is a symmetrical relation).
2009 Macedonia National Olympiad
Let $a,b,c$ be positive real numbers for which $ab+bc+ca=\frac{1}{3}$. Prove the inequality

$$ \frac{a}{a^2-bc+1}+\frac{b}{b^2-ca+1}+\frac{c}{c^2-ab+1}\ge\frac{1}{a+b+c}$$
2009 Macedonia National Olympiad
Solve the following equation in the set of integer numbers:



$$  x^{2010}-2006=4y^{2009}+4y^{2008}+2007y. $$
2008 Macedonia National Olympiad
Find all injective functions $ f : \mathbb{N} \to \mathbb{N}$ which satisfy

$$ f(f(n)) \le\frac{n + f(n)}{2}$$

for each $ n \in \mathbb{N}$.
2008 Macedonia National Olympiad
Positive numbers $ a$, $ b$, $ c$ are such that $ \left(a + b\right)\left(b + c\right)\left(c + a\right) = 8$. Prove the inequality

$$ \frac {a + b + c}{3}\ge\sqrt [27]{\frac {a^3 + b^3 + c^3}{3}}

$$
2008 Macedonia National Olympiad
An acute triangle $ ABC$ with $ AB \neq AC$ is given. Let $ V$ and $ D$ be the feet of the altitude and angle bisector from $ A$, and let $ E$ and $ F$ be the intersection points of the circumcircle of $ \triangle AVD$ with sides $ AC$ and $ AB$, respectively. Prove that $ AD$, $ BE$ and $ CF$ have a common point.
2008 Macedonia National Olympiad
We call an integer $ n > 1$ good if, for any natural numbers $ 1 \le b_1, b_2, \ldots , b_{n-1} \le n - 1$ and any $ i \in \{0, 1, \ldots , n - 1\}$, there is a subset $ I$ of $ \{1, \ldots , n - 1\}$ such that $ \sum_{k\in I} b_k \equiv i \pmod n$. (The sum over the empty set is zero.) Find all good numbers.
2007 Macedonia National Olympiad
Let $a, b, c$ be positive real numbers. Prove that $$1+\frac{3}{ab+bc+ca}\geq\frac{6}{a+b+c}.$$