Navvye/Olympiad-Problems-CSV huggingface dataset:
2005 Czech-Polish-Slovak Match
Find all integers $n \ge 3$ for which the polynomial

$$W(x) = x^n - 3x^{n-1} + 2x^{n-2} + 6$$

can be written as a product of two non-constant polynomials with integer coefficients.
2005 Czech-Polish-Slovak Match
We distribute $n\ge1$ labelled balls among nine persons $A,B,C, \dots , I$. How many ways are there to do this so that $A$ gets the same number of balls as $B,C,D$ and $E$ together?
2005 Czech-Polish-Slovak Match
Given a convex quadrilateral $ABCD$, find the locus of the points $P$ inside the quadrilateral such that

$$S_{PAB}\cdot S_{PCD} = S_{PBC}\cdot S_{PDA}$$

(where $S_X$ denotes the area of triangle $X$).
2005 Czech-Polish-Slovak Match
Determine all pairs of integers $(x, y)$ satisfying the equation

$$y(x + y) = x^3- 7x^2 + 11x - 3.$$
2004 Czech-Polish-Slovak Match
Show that real numbers, $p, q, r$ satisfy the condition $p^4(q-r)^2 + 2p^2(q+r) + 1 = p^4$ if and only if the quadratic equations $x^2 + px + q = 0$ and $y^2 - py + r = 0$ have real roots (not necessarily distinct) which can be labeled by $x_1,x_2$ and $y_1,y_2$, respectively, in such a way that $x_1y_1 - x_2y_2 = 1$.
2004 Czech-Polish-Slovak Match
Show that for each natural number $k$ there exist only finitely many triples $(p, q, r)$ of distinct primes for which $p$ divides $qr-k$, $q$ divides $pr-k$, and $r$ divides $pq - k$.
2004 Czech-Polish-Slovak Match
A point $P$ in the interior of a cyclic quadrilateral $ABCD$ satisfies $\angle BPC = \angle BAP + \angle PDC$. Denote by $E, F$ and $G$ the feet of the perpendiculars from $P$ to the lines $AB, AD$ and $DC$, respectively. Show that the triangles $FEG$ and $PBC$ are similar.
2004 Czech-Polish-Slovak Match
Solve in real numbers the system of equations: \begin{align*}

\frac{1}{xy}&=\frac{x}{z}+1 \\
\frac{1}{yz}&=\frac{y}{x}+1 \\
\frac{1}{zx}&=\frac{z}{y}+1 \\
\end{align*}
2004 Czech-Polish-Slovak Match
Points $K,L,M$ on the sides $AB,BC,CA$ respectively of a triangle $ABC$ satisfy $\frac{AK}{KB} = \frac{BL}{LC} = \frac{CM}{MA}$. Show that the triangles $ABC$ and $KLM$ have a common orthocenter if and only if $\triangle ABC$ is equilateral.
2004 Czech-Polish-Slovak Match
On the table there are $k \ge 3$ heaps of $1, 2, \dots , k$ stones. In the first step, we choose any three of the heaps, merge them into a single new heap, and remove $1$ stone from this new heap. Thereafter, in the $i$-th step ($i \ge 2$) we merge some three heaps containing more than $i$ stones in total and remove $i$ stones from the new heap. Assume that after a number of steps a single heap of $p$ stones remains on the table. Show that the number $p$ is a perfect square if and only if so are both $2k + 2$ and $3k + 1$. Find the least $k$ with this property.
2003 Czech-Polish-Slovak Match
Given an integer $n \ge 2$, solve in real numbers the system of equations \begin{align*}

\max\{1, x_1\} &= x_2 \\
\max\{2, x_2\} &= 2x_3 \\
&\cdots \\
\max\{n, x_n\} &= nx_1. \\ 

\end{align*}
2003 Czech-Polish-Slovak Match
In an acute-angled triangle $ABC$ the angle at $B$ is greater than $45^\circ$. Points $D,E, F$ are the feet of the altitudes from $A,B,C$ respectively, and $K$ is the point on segment $AF$ such that $\angle DKF = \angle KEF$.

(a) Show that such a point $K$ always exists.

(b) Prove that $KD^2 = FD^2 + AF \cdot BF$.
2003 Czech-Polish-Slovak Match
Numbers $p,q,r$ lies in the interval $(\frac{2}{5},\frac{5}{2})$ nad satisfy $pqr=1$. Prove that there exist two triangles of the same area, one with the sides $a,b,c$ and the other with the sides $pa,qb,rc$.
2003 Czech-Polish-Slovak Match
Point $P$ lies on the median from vertex $C$ of a triangle $ABC$. Line $AP$ meets $BC$ at $X$, and line $BP$ meets $AC$ at $Y$ . Prove that if quadrilateral $ABXY$ is cyclic, then triangle $ABC$ is isosceles.
2003 Czech-Polish-Slovak Match
Find all functions $f : \mathbb{R} \to \mathbb{R}$ that satisfy the condition

$$f(f(x) + y) = 2x + f(f(y) - x)\quad \text{ for all } x, y \in\mathbb{R}.$$
2002 Czech-Polish-Slovak Match
Let $a, b$ be distinct real numbers and $k,m$ be positive integers $k + m = n \ge 3, k \le 2m, m \le 2k$. Consider sequences $x_1,\dots , x_n$ with the following properties:

(i) $k$ terms $x_i$, including $x_1$, are equal to $a$;

(ii) $m$ terms $x_i$, including $x_n$, are equal to $b$;

(iii) no three consecutive terms are equal.

Find all possible values of $x_nx_1x_2 + x_1x_2x_3 + \cdots + x_{n-1}x_nx_1$.
2002 Czech-Polish-Slovak Match
A triangle $ABC$ has sides $BC = a, CA = b, AB = c$ with $a < b < c$ and area $S$. Determine the largest number $u$ and the least number $v$ such that, for every point $P$ inside $\triangle ABC$, the inequality $u \le PD + PE + PF \le v$ holds, where $D,E, F$ are the intersection points of $AP,BP,CP$ with the opposite sides.
2002 Czech-Polish-Slovak Match
Let $S = \{1, 2, \cdots , n\}, n \in N$. Find the number of functions $f : S \to S$ with the property that $x + f(f(f(f(x)))) = n + 1$ for all $x \in S$?
2002 Czech-Polish-Slovak Match
An integer $n > 1$ and a prime $p$ are such that $n$ divides $p-1$, and $p$ divides $n^3 - 1$. Prove that $4p - 3$ is a perfect square.
2002 Czech-Polish-Slovak Match
In an acute-angled triangle $ABC$ with circumcenter $O$, points $P$ and $Q$ are taken on sides $AC$ and $BC$ respectively such that $\frac{AP}{PQ} = \frac{BC}{AB}$ and $\frac{BQ}{PQ} =\frac{AC}{AB}$ . Prove that the points $O, P,Q,C$ lie on a circle.
2002 Czech-Polish-Slovak Match
Let $n \ge 2$ be a fixed even integer. We consider polynomials of the form

$$P(x) = x^n + a_{n-1}x^{n-1} + \cdots + a_1x + 1$$

with real coefficients, having at least one real roots. Find the least possible value of $a^2_1 + a^2_2 + \cdots + a^2_{n-1}$.
2001 Czech-Polish-Slovak Match
Prove that for any positive numbers $a_1,\dots,a_n$ $(n \geq 2)$

$$(a_1^3+1)(a_2^3+1)\cdots(a_n^3+1) \geq (a_1^2 a_2+1)(a_2^2 a_3+1)\cdots (a_n^2 a_1+1)$$
2001 Czech-Polish-Slovak Match
A triangle $ABC$ has acute angles at $A$ and $B$. Isosceles triangles $ACD$ and $BCE$ with bases $AC$ and $BC$ are constructed externally to triangle $ABC$ such that $\angle ADC = \angle ABC$ and $\angle BEC = \angle BAC$. Let $S$ be the circumcenter of $\triangle ABC$. Prove that the length of the polygonal line $DSE$ equals the perimeter of triangle $ABC$ if and only if $\angle ACB$ is right.
2001 Czech-Polish-Slovak Match
Distinct points $A$ and $B$ are given on the plane. Consider all triangles $ABC$ in this plane on whose sides $BC,CA$ points $D,E$ respectively can be taken so that

(i) $\frac{BD}{BC}=\frac{CE}{CA}=\frac{1}{3}$;

(ii) points $A,B,D,E$ lie on a circle in this order.



Find the locus of the intersection points of lines $AD$ and $BE$.
2001 Czech-Polish-Slovak Match
Find all functions $f : \mathbb{R} \to \mathbb{R}$ that satisfy

$$f(x^2 + y) + f(f(x) - y) = 2f(f(x)) + 2y^2\quad\text{ for all }x, y \in \mathbb{R}.$$
2001 Czech-Polish-Slovak Match
Points with integer coordinates in cartesian space are called lattice points. We color $2000$ lattice points blue and $2000$ other lattice points red in such a way that no two blue-red segments have a common interior point (a segment is blue-red if its two endpoints are colored blue and red). Consider the smallest rectangular parallelepiped that covers all the colored points.

(a) Prove that this rectangular parallelepiped covers at least $500,000$ lattice points.

(b) Give an example of a coloring for which the considered rectangular paralellepiped covers at most $8,000,000$ lattice points.
2000 Czech and Slovak Match
a,b,c are +ve real numbers which satisfy 5abc>a^3+b^3+c^3.prove that a,b,c can form a triangle.
2000 Czech and Slovak Match
Let ${ABC}$ be a triangle, ${k}$  its incircle and ${k_a,k_b,k_c}$  three circles orthogonal to ${k}$  passing through ${B}$  and ${C, A}$  and ${C}$ , and ${A}$  and ${B}$  respectively. The circles ${k_a,k_b}$  meet again in ${C'}$ ; in the same way we obtain the points ${B'}$  and ${A'}$ . Prove that the radius of the circumcircle of ${A'B'C'}$  is half the radius of ${k}$ .
2000 Czech and Slovak Match
Let $n$ be a positive integer. Prove that $n$ is a power of two if and only if there exists an integer $m$ such that $2^n-1$ is a divisor of $m^2 +9$.
2000 Czech and Slovak Match
Let $P(x)$ be a polynomial with integer coefficients. Prove that the polynomial $Q(x) = P(x^4)P(x^3)P(x^2)P(x)+1$ has no integer roots.
2000 Czech and Slovak Match
Let $ABCD$ be an isosceles trapezoid with bases $AB$ and $CD$. The incircle of the triangle $BCD$ touches $CD$ at $E$. Point $F$ is chosen on the bisector of the angle $DAC$ such that the lines $EF$ and $CD$ are perpendicular. The circumcircle of the triangle $ACF$ intersects the line $CD$ again at $G$. Prove that the triangle $AFG$ is isosceles.
2000 Czech and Slovak Match
Suppose that every integer has been given one of the colors red, blue, green, yellow. Let $x$ and $y$ be odd integers such that $|x| \ne  |y|$. Show that there are two integers of the same color whose difference has one of the following values: $x,y,x+y,x-y$.
1999 Czech and Slovak Match
Leta,b,c are postive real numbers,proof  that $ \frac{a}{b+2c}+\frac{b}{c+2a}+\frac{c}{a+2b}\geq1$
1999 Czech and Slovak Match
The altitudes through the vertices $A,B,C$ of an acute-angled triangle $ABC$ meet  the opposite sides at $D,E,F,$ respectively. The line through $D$ parallel to $EF$ meets the lines $AC$ and $AB$ at $Q$ and $R$, respectively. The line $EF$ meets $BC$ at $P$. Prove that the circumcircle of the triangle $PQR$ passes through the midpoint of $BC$.
1999 Czech and Slovak Match
Find all natural numbers $k$ for which there exists a set $M$ of ten real numbers such that there are exactly $k$ pairwise non-congruent triangles whose side lengths are three (not necessarily distinct) elements of $M$.
1999 Czech and Slovak Match
Czech and Slovak Match 1999 P4



Quote:

Find all positive integers $k$ for which the following assertion holds:

If $F(x)$ is polynomial with integer coefficients ehich satisfies $0  \leq F(c) \leq k$ for all $c \in \{0,1, \cdots,k+1 \}$, then $$F(0)= F(1) = \cdots =F(k+1).$$
1999 Czech and Slovak Match
Find all functions $f: (1,\infty)\text{to R}$ satisfying

$f(x)-f(y)=(y-x)f(xy)$ for all $x,y>1$.

hintyou may try to find $f(x^5)$ by two ways and then continue the solution.

I have also solved by using this method.By finding $f(x^5)$ in two ways

I found that $f(x)=xf(x^2)$ for all $x>1$.
1999 Czech and Slovak Match
Prove that for any integer $n \ge  3$, the least common multiple of the numbers $1,2, ...  ,n$ is greater than $2^{n-1}$.
1998 Czech and Slovak Match
Let $P$ be an interior point of the parallelogram $ABCD$. Prove that $\angle APB+ \angle CPD = 180^\circ$ if and only if $\angle PDC = \angle PBC$.
1998 Czech and Slovak Match
A polynomial $P(x)$ of degree $n \ge 5$ with integer coefficients has $n$ distinct integer roots, one of which is $0$. Find all integer roots of the polynomial $P(P(x))$.
1998 Czech and Slovak Match
Let $ABCDEF$ be a convex hexagon such that $AB = BC, CD = DE, EF = FA$.

Prove that $\frac{BC}{BE} +\frac{DE}{DA} +\frac{FA}{FC} \ge \frac{3}{2}$  . When does equality occur?
1998 Czech and Slovak Match
Find all functions $f : N\rightarrow N - \{1\}$ satisfying $f (n)+ f (n+1)= f (n+2) +f (n+3) -168$ for all $n \in N$

.
1998 Czech and Slovak Match
In a triangle $ABC$, $T$ is the centroid and $ \angle TAB = \angle ACT$. Find the maximum possible value of $sin \angle CAT +sin \angle CBT$.
1998 Czech and Slovak Match
In a summer camp there are $n$ girls $D_1,D_2, ... ,D_n$ and $2n-1$ boys $C_1,C_2, ...,C_{2n-1}$.

The girl $D_i, i = 1,2,... ,n,$ knows only the boys $C_1,C_2, ... ,C_{2i-1}$.

Let $A(n, r)$ be the number of different ways in which $r$ girls can dance with $r$ boys forming $r$ pairs,

each girl with a boy she knows.

Prove that $A(n, r) = \binom{n}{r} \frac{r!}{(n-r)!}.$
1997 Czech and Slovak Match
Points $K$ and $L$ are chosen on the sides $AB$ and $AC$ of an equilateral triangle $ABC$ such that $BK = AL$. Segments $BL$ and $CK$ intersect at $P$. Determine the ratio $\frac{AK}{KB}$ for which the segments $AP$ and $CK$ are perpendicular.
1997 Czech and Slovak Match
In a community of more than six people each member exchanges letters with exactly three other members of the community. Show that the community can be partitioned into two nonempty groups so that each member exchanges letters with at least two members of the group he belongs to.
1997 Czech and Slovak Match
Find all functions $f : R\rightarrow R$ such that $f ( f (x)+y) = f (x^2 -y)+4 f (x)y$ for all $x,y \in R$

.
1997 Czech and Slovak Match
Is it possible to place $100$ balls in space so that no two of them have a common interior point and each of them touches at least one third of the others?
1997 Czech and Slovak Match
The sum of several integers (not necessarily distinct) equals $1492$. Decide whether the sum of their seventh powers can equal (a) $1996$; (b) $1998$.
1997 Czech and Slovak Match
In a certain language there are only two letters, $A$ and $B$. The words of this language obey the following rules:

(i) The only word of length $1$ is $A$;

(ii) A sequence of letters $X_1X_2...X_{n+1}$, where $X_i\in \{A,B\}$ for each $i$, forms a word of length $n+1$ if and only if it contains at least one letter $A$ and is not of the form $WA$ for a word $W$ of length $n$.

Show that the number of words consisting of $1998 A$’s and $1998 B$’s and not beginning with $AA$ equals $\binom{3995}{1997}-1$
1996 Czech and Slovak Match
Show that an integer $p > 3$ is a prime if and only if for every two nonzero integers $a,b$ exactly one of the numbers

$N_1 = a+b-6ab+\frac{p-1}{6}$ , $N_2 = a+b+6ab+\frac{p-1}{6}$ is a nonzero integer.
1996 Czech and Slovak Match
Let ⋆ be a binary operation on a nonempty set $M$. That is, every pair $(a,b) \in  M$ is assigned an element $a$ ⋆$ b$ in $M$. Suppose that ⋆ has the additional property that $(a $ ⋆ $b) $ ⋆$ b= a$ and $a$ ⋆ $(a$ ⋆$ b)= b$ for all $a,b \in  M$.

(a) Show that $a$ ⋆ $b = b$ ⋆ $a$ for all $a,b \in  M$.

(b) On which finite sets $M$ does such a binary operation exist?
1996 Czech and Slovak Match
The base of a regular quadrilateral pyramid $\pi$ is a square with side length $2a$ and its lateral edge has length a$\sqrt{17}$. Let $M$ be a point inside the pyramid. Consider the five pyramids which are similar to $\pi$ , whose top vertex is at $M$ and whose bases lie in the planes of the faces of $\pi$ . Show that the sum of the surface areas of these five pyramids is greater or equal to one fifth the surface of $\pi$ , and find for which $M$ equality holds.
1996 Czech and Slovak Match
Decide whether there exists a function $f : Z \rightarrow Z$ such that for each $k =0,1, ...,1996$ and for any integer $m$ the equation $f (x)+kx = m$ has at least one integral solution $x$.
1996 Czech and Slovak Match
Two sets of intervals $A ,B$ on the line are given. The set $A$ contains $2m-1$ intervals, every two of which have an interior point in common. Moreover, every interval from $A$ contains at least two disjoint intervals from $B$. Show that there exists an interval in $B$ which belongs to at least $m$ intervals from $A$ .
1996 Czech and Slovak Match
The points $E$ and $D$ lie in the interior of sides $AC$ and $BC$, respectively, of a triangle $ABC$. Let $F$ be the intersection of the lines $AD$  and $BE$.Show that the area of the traingles $ABC$ and $ABF$ satisfies:



$ \frac{S_{ABC}}{S_{ABF}} = \frac{\mid{AC}\mid}{\mid{AE} \mid} + \frac{\mid{BC}\mid}{\mid{BD}\mid} - 1$.
1995 Czech and Slovak Match
Let $ a_1=2, a_2=5$ and $ a_{n+2}=(2-n^2)a_{n+1}+ (2+n^2)a_n$ for $ n\geq 1$. Do there exist $ p,q,r$ so that $ a_pa_q =a_r$?
1995 Czech and Slovak Match
Find all pairs of functions $f ,g : Z \rightarrow Z $ that satisfy $f (g(x)+y) = g( f (y)+x) $ for all integers $ x,y$

and such that $g(x) = g(y)$ only if $x = y$.
1995 Czech and Slovak Match
Consider all triangles $ABC$ in the cartesian plane whose vertices are at lattice points (i.e. with integer coordinates) and which contain exactly one lattice point (to be denoted $P$) in its interior. Let the line $AP$ meet $BC$ at $E$. Determine the maximum possible value of the ratio $\frac{AP}{PE}$.
1995 Czech and Slovak Match
For each real number $p > 1$, find the minimum possible value of the sum $x+y$, where the numbers $x$ and $y$ satisfy the equation $(x+\sqrt{1+x^2})(y+\sqrt{1+y^2}) = p$.
1995 Czech and Slovak Match
The diagonals of a convex quadrilateral $ABCD$ are orthogonal and intersect at point $E$. Prove that the reflections of $E$ in the sides of quadrilateral $ABCD$ lie on a circle.
1995 Czech and Slovak Match
Find all triples $(x; y; p)$ of two non-negative integers $x, y$ and a prime number p such that $ p^x-y^p=1 $
2019 Danube Mathematical Competition
Solve in $ \mathbb{Z}^2 $ the equation: $ x^2\left( 1+x^2 \right) =-1+21^y. $





Lucian Petrescu
2019 Danube Mathematical Competition
Let be a natural number $ n, $ and $ n $ real numbers $ a_1,a_2,\ldots ,a_n. $ Prove that there exists a real number $ a $ such that $ a+a_1,a+a_2,\ldots ,a+a_n $ are all irrational.
2019 Danube Mathematical Competition
Let be a sequence of $ 51 $ natural numbers whose sum is $ 100. $ Show that for any natural number $ 1\le k<100 $ there are some consecutive numbers from this sequence whose sum is $ k $ or $ 100-k. $
2019 Danube Mathematical Competition
Let $ ABCD $ be a cyclic quadrilateral,$ M $ midpoint of $ AC $ and $ N $ midpoint of $ BD. $ If $ \angle AMB =\angle AMD, $ prove that $ \angle ANB =\angle BNC. $
2019 Danube Mathematical Competition
Find all prime $p$ numbers such that $p^3-4p+9$ is perfect square.
2019 Danube Mathematical Competition
Find all nondecreasing functions $ f:\mathbb{R}\longrightarrow\mathbb{R} $ that verify the relation

$$ f\left( f\left( x^2 \right) +y+f(y) \right) =x^2+2f(y) , $$for any real numbers $ x,y. $
2019 Danube Mathematical Competition
We color some unit squares in a $ 99\times 99 $ square grid  with one of $ 5 $ given distinct colors, such that each color appears  the same number of times. On each row and on each column there are no differently colored unit squares. Find the maximum possible number of colored unit squares.
2019 Danube Mathematical Competition
Let $ APD $ be an acute-angled triangle and let $ B,C $ be two points on the segments (excluding their endpoints) $ AP,PD, $ respectively. The diagonals of $ ABCD $ meet at $ Q. $ Denote by $ H_1,H_2 $ the orthocenters of $ APD,BPC, $ respectively. The circumcircles of $ ABQ $ and $ CDQ $ intersect at $ X\neq Q, $ and the circumcircles of $ ADQ,BCQ $ meet at $ Y\neq Q. $ Prove that if the line $ H_1H_2 $ passes through $ X, $ then it also passes through $ Y. $
2018 Danube Mathematical Competition
Find all the pairs $(n, m)$ of positive integers which fulfil simultaneously the conditions:

i) the number $n$ is composite;

ii) if the numbers $d_1, d_2, ..., d_k, k \in N^*$ are all the proper divisors of $n$, then the numbers $d_1 + 1, d_2 + 1, . . . , d_k + 1$ are all the proper divisors of $m$.
2018 Danube Mathematical Competition
Let $ABC$ be a triangle such that in its interior there exists a point $D$ with $\angle DAC = \angle DCA = 30^o$ and $ \angle DBA = 60^o$. Denote $E$ the midpoint of the segment $BC$, and take $F$ on the segment $AC$ so that $AF = 2FC$. Prove that $DE \perp EF$.
2018 Danube Mathematical Competition
Find all the positive integers $n$ with the property:

there exists an integer $k > 2$ and the positive rational numbers $a_1, a_2, ..., a_k$

such that $a_1 + a_2 + .. + a_k = a_1a_2 . . . a_k = n$.
2018 Danube Mathematical Competition
Let $M$ be the set of positive odd integers.

For every positive integer $n$, denote $A(n)$ the number of the subsets of $M$ whose sum of elements equals $n$.

For instance, $A(9) = 2$, because there are exactly two subsets of $M$ with the sum of their elements equal to $9$: $\{9\}$ and $\{1, 3, 5\}$.

a) Prove that $A(n) \le A(n + 1)$ for every integer $n \ge  2$.

b) Find all the integers $n \ge 2$ such that $A(n) = A(n + 1)$
2018 Danube Mathematical Competition
Suppose we have a necklace of $n$ beads.

Each bead is labeled with an integer and the sum of all these labels is $n - 1$.

Prove that we can cut the necklace to form a string,  whose consecutive labels $x_1,x_2,...,x_n$ satisfy

$\sum_{i=1}^{k} x_i \le k - 1$ for any $k = 1,...,n$
2018 Danube Mathematical Competition
Prove that there are infinitely many pairs of positive integers $(m, n)$ such that simultaneously $m$ divides $n^2 + 1$ and $n$ divides $m^2 + 1$.
2018 Danube Mathematical Competition
Let $ABC$ be an acute non isosceles triangle. The angle bisector of angle $A$ meets again the circumcircle of the triangle $ABC$ in $D$. Let $O$ be the circumcenter of the triangle $ABC$. The angle bisectors of $\angle AOB$, and $\angle AOC$ meet the circle $\gamma$ of diameter $AD$ in $P$ and $Q$ respectively. The line $PQ$ meets the perpendicular bisector of $AD$ in $R$. Prove that $AR //  BC$.
2015 Danube Mathematical Competition
Consider a positive integer $n=\overline{a_1a_2...a_k},k\ge 2$.A trunk of $n$ is a number of the form $\overline{a_1a_2...a_t},1\le t\le k-1$.(For example,the number $23$ is a trunk of $2351$.)

By $T(n)$ we denote the sum of all trunk of $n$ and let $S(n)=a_1+a_2+...+a_k$.Prove that $n=S(n)+9\cdot T(n)$.
2015 Danube Mathematical Competition
Consider the set $A=\{1,2,...,120\}$ and $M$ a subset of $A$ such that $|M|=30$.Prove that there are $5$ different subsets of $M$,each of them having two elements,such that the absolute value of the difference of the elements of each subset is the same.
2015 Danube Mathematical Competition
Solve in N $a^2 = 2^b3^c + 1$.
2015 Danube Mathematical Competition
Let $ABCD$ be a rectangle  with $AB\ge BC$  Point $M$ is located on the side $(AD)$, and the perpendicular bisector of  $[MC]$ intersects the line $BC$ at the point $N$. Let ${Q} =MN\cup AB$ . Knowing that $\angle MQA= 2\cdot  \angle BCQ $, show that the quadrilateral $ABCD$ is a square.
2015 Danube Mathematical Competition
A lantern needs exactly $2$ charged batteries in order to work.We have available $n$ charged batteries and $n$ uncharged batteries,$n\ge 4$(all batteries look the same).

A try consists in introducing two batteries in the lantern and verifying if the lantern works.Prove that we can find a pair of charged batteries in at most $n+2$ tries.
2015 Danube Mathematical Competition
Let $ABCD$ be a cyclic quadrangle, let the diagonals $AC$ and $BD$ cross at $O$, and let $I$ and $J$ be the incentres of the triangles $ABC$ and $ABD$, respectively. The line $IJ$ crosses the segments $OA$ and $OB$ at $M$ and $N$, respectively. Prove that the triangle $OMN$ is isosceles.
2015 Danube Mathematical Competition
Show that the edges of a connected simple (no loops and no multiple edges) finite graph can be oriented so that the number of edges leaving each vertex is even if and only if the total number of edges is even
2015 Danube Mathematical Competition
Determine all positive integers $n$ such that all positive integers less than or equal to $n$ and relatively prime to $n$ are pairwise coprime.
2015 Danube Mathematical Competition
Given an integer $n \ge 2$ ,determine the numbers that written in the form $a_1$$a_2$$+$$a_2$$a_3$$+$$...$$a_{k-1}$$a_k$ , where $k$ is an integer greater than or equal to 2, and $a_1$ ,... $a_k$ are positive integers with sum $n$.
2014 Danube Mathematical Competition
Determine the natural number $a =\frac{p+q}{r}+\frac{q+r}{p}+\frac{r+p}{q}$ where $p, q$ and $r$ are prime positive numbers.
2014 Danube Mathematical Competition
We call word  a sequence of letters $\overline {l_1l_2...l_n}, n\ge 1$ .

A word  $\overline {l_1l_2...l_n}, n\ge 1$ is called palindrome  if  $l_k=l_{n-k+1}$ , for any $k, 1 \le k \le n$.

Consider a word  $X=\overline {l_1l_2...l_{2014}}$ in which $ l_k\in\{A,B\}$ , for any $k, 1\le k \le 2014$.

Prove that there are at least $806$ palindrome  words  to ''stick" together to  get word $X$.
2014 Danube Mathematical Competition
Let $ABC$ be a triangle with $\angle A<90^o, AB \ne AC$. Denote  $H$ the orthocenter of triangle $ABC$, $N$ the midpoint of segment $[AH]$, $M$ the midpoint of segment $[BC]$ and $D$ the intersection point of the angle bisector of $\angle BAC$ with the segment $[MN]$. Prove that $<ADH=90^o$