Navvye/Olympiad-Problems-CSV huggingface dataset:
2016 Estonia Team Selection Test
Prove that for any positive integer $n\ge $, $2 \cdot \sqrt3 \cdot  \sqrt[3]{4} ...\sqrt[n-1]{n} > n$
2016 Estonia Team Selection Test
Let $O$ be the circumcentre of the acute triangle $ABC$. Let $c_1$ and $c_2$ be the circumcircles of triangles $ABO$ and $ACO$. Let $P$ and $Q$ be points on $c_1$ and $c_2$ respectively, such that OP is a diameter of $c_1$ and $OQ$ is a diameter of $c_2$. Let $T$ be the intesection of the tangent to $c_1$ at $P$ and the tangent to $c_2$ at $Q$. Let $D$ be the second intersection of the line $AC$ and the circle $c_1$. Prove that the points $D, O$ and $T$ are collinear
2016 Estonia Team Selection Test
A circle is divided into arcs of equal size by $n$ points ($n \ge 1$). For any positive integer $x$, let $P_n(x)$ denote the number of possibilities for colouring all those points, using colours from $x$ given colours, so that any rotation of the colouring by $ i \cdot \frac{360^o}{n}$ , where i is a positive integer less than $n$, gives a colouring that differs from the original in at least one point. Prove that the function $P_n(x)$ is a polynomial with respect to $x$.
2016 Estonia Team Selection Test
On the sides $AB, BC$ and $CA$ of triangle $ABC$, points $L, M$ and $N$ are chosen, respectively, such that the lines $CL, AM$ and $BN$ intersect at a common point O inside the triangle and the quadrilaterals $ALON, BMOL$ and $CNOM$ have incircles. Prove that

$$\frac{1}{AL\cdot BM} +\frac{1}{BM\cdot CN} +\frac{1}{CN \cdot AL} =\frac{1}{AN\cdot BL} +\frac{1}{BL\cdot CM} +\frac{1}{CM\cdot  AN} $$
2016 Estonia Team Selection Test
Let $x, y$ and $z$ be positive real numbers such that $x + y + z = \frac{1}{x} +  \frac{1}{y} +  \frac{1}{z}$ . Prove that $xy + yz + zx \ge  3$.
2016 Estonia Team Selection Test
Let $n$ be a positive integer such that there exists a positive integer that is less than $\sqrt{n}$ and does not divide $n$. Let $(a_1, . . . , a_n)$ be an arbitrary permutation of $1, . . . , n$. Let $a_{i1} < . . . < a_{ik}$ be its maximal increasing subsequence and let $a_{j1} > . . . > a_{jl}$ be its maximal decreasing subsequence.

Prove that tuples $(a_{i1}, . . . , a_{ik})$ and $(a_{j1}, . . . , a_{jl} )$ altogether contain at least one number that does not divide $n$.
2016 Estonia Team Selection Test
Let $m$ be an integer, $m \ge  2$. Each student in a school is practising $m$ hobbies the most. Among any $m$ students there exist two students who have a common hobby. Find the smallest number of students for which there must exist a hobby which is practised by at least $3$ students .
2016 Estonia Team Selection Test
Find all positive integers $n$ such that $(n^2 + 11n - 4) \cdot  n! + 33 \cdot 13^n + 4$ is a perfect square
2016 Estonia Team Selection Test
The circles $k_1$ and $k_2$ intersect at points $M$ and $N$. The line $\ell$ intersects with the circle $k_1$ at points $A$ and $C$ and with circle $k_2$ at points $B$ and $D$, so that points $A, B, C$ and $D$ are on the line $\ell$  in that order. Let $X$ be a point on line $MN$ such that the point $M$ is between points $X$ and $N$. Lines $AX$ and $BM$ intersect at point $P$ and lines $DX$ and $CM$ intersect at point $Q$. Prove that $PQ \parallel \ell $.
2015 Estonia Team Selection Test
Let $n$ be a natural number, $n \ge  5$, and $a_1, a_2, . . . , a_n$ real numbers such that all possible sums $a_i + a_j$, where $1 \le i < j \le  n$, form $\frac{n(n-1)}{2}$ consecutive members of an arithmetic progression when taken in some order. Prove that $a_1 = a_2 = . . . = a_n$.
2015 Estonia Team Selection Test
A square-shaped pizza with side length $30$ cm is cut into pieces (not necessarily rectangular). All cuts are parallel to the sides, and the total length of the cuts is $240$ cm. Show that there is a piece whose area is at least $36$ cm$^2$
2015 Estonia Team Selection Test
Let $q$ be a fixed positive rational number. Call number $x$ charismatic  if there exist a positive integer $n$ and integers $a_1, a_2, . . . , a_n$ such that $x = (q + 1)^{a_1} \cdot  (q + 2)^{a_2} ...(q + n)^{a_n}$.

a) Prove that $q$ can be chosen in such a way that every positive rational number turns out to be charismatic.

b) Is it true for every $q$ that, for every charismatic number $x$, the number $x + 1$ is charismatic, too?
2015 Estonia Team Selection Test
Altitudes $AD$ and $BE$ of an acute triangle $ABC$ intersect at $H$. Let $C_1 (H,HE)$ and $C_2(B,BE)$ be two circles tangent at $AC$ at point $E$. Let $P\ne E$ be the second point of tangency of the circle $C_1 (H,HE)$ with its tangent line going through point $C$, and $Q\ne E$ be the second point of tangency of the circle $C_2(B,BE)$ with its tangent line going through point $C$. Prove that points $D, P$, and $Q$ are collinear.
2015 Estonia Team Selection Test
Find all functions $f$ from reals to reals which satisfy $f (f(x) + f(y)) = f(x^2) + 2x^2 f(y) + (f(y))^2$ for all real numbers $x$ and $y$.
2015 Estonia Team Selection Test
In any rectangular game board with black and white squares, call a row $X$ a mix of rows $Y$ and $Z$ whenever each cell in row $X$ has the same colour as either the cell of the same column in row $Y$ or the cell of the same column in row $Z$. Let a natural number $m \ge 3$ be given. In some rectangular board, black and white squares lie in such a way that all the following conditions hold.

1) Among every three rows of the board, one is a mix of two others.

2) For every two rows of the board, their corresponding cells in at least one column have different colours.

3) For every two rows of the board, their corresponding cells in at least one column have equal colours.

4) It is impossible to add a new row with each cell either black or white to the board in a way leaving both conditions 1) and 2) still in force

Find all possibilities of what can be the number of rows of the board.
2015 Estonia Team Selection Test
Prove that for every prime number $p$ and positive integer $a$, there exists a natural number $n$ such that $p^n$ contains $a$ consecutive equal digits.
2015 Estonia Team Selection Test
Find all positive integers $n$ for which it is possible to partition a regular  $n$-gon into triangles with diagonals not intersecting inside the $n$-gon such that at every vertex of the $n$-gon an odd number of triangles meet.
2015 Estonia Team Selection Test
The orthocenter of an acute triangle $ABC$ is $H$. Let $K$ and $P$ be the midpoints of lines $BC$ and $AH$, respectively. The angle bisector drawn from the vertex $A$ of the triangle $ABC$ intersects with line $KP$ at $D$. Prove that $HD\perp AD$.
2015 Estonia Team Selection Test
Let $n$ be an integer and $a, b$ real numbers such that $n > 1$ and $a > b > 0$. Prove that $$(a^n - b^n) \left ( \frac{1}{b^{n- 1}} - \frac{1}{a^{n -1}}\right) > 4n(n -1)(\sqrt{a} - \sqrt{b})^2$$
2015 Estonia Team Selection Test
Let $M$ be the midpoint of the side $AB$ of a triangle $ABC$. A circle through point $C$ that has a point of tangency to the line $AB$ at point $A$ and a circle through point $C$ that has a point of tangency to the line $AB$ at point $B$

intersect the second time at point $N$. Prove that $|CM|^2 + |CN|^2 - |MN|^2 = |CA|^2 + |CB|^2 - |AB|^2$.
2015 Estonia Team Selection Test
Call an $n$-tuple $(a_1, . . . , a_n)$ occasionally periodic  if there exist a nonnegative integer $i$ and a positive integer $p$ satisfying $i + 2p \le n$ and $a_{i+j} = a_{i+p+j}$ for every $j = 1, 2, . . . , p$. Let $k$ be a positive integer. Find the least positive integer $n$ for which there exists an $n$-tuple $(a_1, . . . , a_n)$ with elements from set $\{1, 2, . . . , k\}$, which is not occasionally periodic but whose arbitrary extension $(a_1, . . . , a_n, a_{n+1})$ is occasionally periodic for any $a_{n+1} \in  \{1, 2, . . . , k\}$.
2014 Estonia Team Selection Test
In Wonderland, the government of each country consists of exactly $a$ men and $b$ women, where $a$ and $b$ are fixed natural numbers and $b > 1$. For improving of relationships between countries, all possible working groups consisting of exactly one government member from each country, at least $n$ among whom are women, are formed (where $n$ is a fixed non-negative integer). The same person may belong to many working groups. Find all possibilities how many countries can be in Wonderland, given that the number of all working groups is prime.
2014 Estonia Team Selection Test
Let $a, b$ and $c$ be positive real numbers for which $a + b + c = 1$. Prove that $$\frac{a^2}{b^3 + c^4 + 1}+\frac{b^2}{c^3 + a^4 + 1}+\frac{c^2}{a^3 + b^4 + 1} > \frac{1}{5}$$
2014 Estonia Team Selection Test
Three line segments, all of length $1$, form a connected figure in the plane. Any two different line segments can intersect only at their endpoints. Find the maximum area of the convex hull of the figure.
2014 Estonia Team Selection Test
In an acute triangle the feet of altitudes drawn from vertices $A$ and $B$ are $D$ and $E$, respectively. Let $M$ be the midpoint of side $AB$. Line $CM$ intersects the circumcircle of $CDE$ again in point $P$ and the circumcircle of $CAB$ again in point $Q$. Prove that $|MP| = |MQ|$.
2014 Estonia Team Selection Test
In Wonderland there are at least $5$ towns. Some towns are connected directly by roads or railways. Every town is connected to at least one other town and for any four towns there exists some direct connection between at least three pairs of towns among those four. When entering the public transportation network of this land, the traveller must insert one gold coin into a machine, which lets him use a direct connection to go to the next town. But if the traveller continues travelling from some town with the same method of transportation that took him there, and he has paid a gold coin to get to this town, then going to the next town does not cost anything, but instead the traveller gains the coin he last used back. In other cases he must pay just like when starting travelling. Prove that it is possible to get from any town to any other town by using at most $2$ gold coins.
2014 Estonia Team Selection Test
Find all natural numbers $n$ such that the equation $x^2 + y^2 + z^2 = nxyz$ has solutions in positive integers
2013 Estonia Team Selection Test
Find all prime numbers $p$ for which one can find a positive integer $m$ and nonnegative integers $a_0,a_1,...,a_m$ less than $p$ such that

$a_0+a_1p+...+a_{m-1}p^{m-1}+a_{m}p^{m} = 2013$

$a_0+a_1+...+a_{m-1}+a_{m} = 11$
2013 Estonia Team Selection Test
For which positive integers $n \ge 3$ is it possible to mark $n$ points of a plane in such a way that, starting from one marked point and moving on each step to the marked point which is the second closest to the current point, one can walk through all the marked points and return to the initial one? For each point, the second closest marked point must be uniquely determined.
2013 Estonia Team Selection Test
Let $x_1,..., x_n$ be non-negative real numbers, not all of which are zeros.

(i) Prove that

$$1 \le \frac{\left(x_1+\frac{x_2}{2}+\frac{x_3}{3}+...+\frac{x_n}{n}\right)(x_1+2x_2+3x_3+...+nx_n)}{(x_1+x_2+x_3+...+x_n)^2} \le \frac{(n+1)^2}{4n}$$(ii) Show that, for each $n > 1$, both inequalities can hold as equalities.
2013 Estonia Team Selection Test
Let $D$ be the point different from $B$ on the hypotenuse $AB$ of a right triangle $ABC$ such that $|CB| = |CD|$. Let $O$ be the circumcenter of triangle $ACD$. Rays $OD$ and $CB$ intersect at point $P$, and the line through point $O$ perpendicular to side AB and ray $CD$ intersect at point $Q$. Points $A, C, P, Q$ are concyclic. Does this imply that $ACPQ$ is a square?
2013 Estonia Team Selection Test
Call a tuple $(b_m, b_{m+1},..., b_n)$ of integers perfect if both following conditions are fulfilled:

1. There exists an integer $a > 1$ such that $b_k = a^k + 1$ for all $k = m, m + 1,..., n$

2. For all $k = m, m + 1,..., n,$ there exists a prime number $q$ and a non-negative integer $t$ such that $b_k = q^t$.

Prove that if $n - m$ is large enough then there is no perfect tuples, and find all perfect tuples with the maximal number of components.
2013 Estonia Team Selection Test
A class consists of $7$ boys and $13$ girls. During the first three months of the school year, each boy has communicated with each girl at least once. Prove that there exist two boys and two girls such that both boys communicated with both girls first time in the same month.
2012 Estonia Team Selection Test
Prove that for any positive integer $k$ there exist $k$ pairwise distinct integers for which the sum of their squares equals the sum of their cubes.
2012 Estonia Team Selection Test
For a given positive integer $n$ one has to choose positive integers $a_0, a_1,...$ so that the following conditions hold:

(1) $a_i = a_{i+n}$ for any $i$,

(2) $a_i$ is not divisible by $n$ for any $i$,

(3) $a_{i+a_i}$ is divisible by $a_i$ for any $i$.

For which positive integers $n > 1$ is this possible only if the numbers $a_0, a_1, ...$ are all equal?
2012 Estonia Team Selection Test
In a cyclic quadrilateral $ABCD$ we have $|AD| > |BC|$ and the vertices $C$ and $D$ lie on the shorter arc $AB$ of the circumcircle. Rays $AD$ and $BC$ intersect at point $K$, diagonals $AC$ and $BD$ intersect at point $P$. Line $KP$ intersects the side $AB$ at point $L$. Prove that $\angle ALK$ is acute.
2012 Estonia Team Selection Test
Let $ABC$ be a triangle where $|AB| = |AC|$. Points $P$ and $Q$ are different from the vertices of the triangle and lie on the sides $AB$ and $AC$, respectively. Prove that the circumcircle of the triangle $APQ$ passes through the circumcenter of $ABC$ if and only if $|AP| = |CQ|$.
2012 Estonia Team Selection Test
Let $x, y, z$ be positive real numbers whose sum is $2012$. Find the maximum value of $$ \frac{(x^2 + y^2 + z^2)(x^3 + y^3 + z^3)}{(x^4 + y^4 + z^4)}$$
2012 Estonia Team Selection Test
Let $m$ be a positive integer, and consider a $m\times m$ checkerboard consisting of unit squares. At the centre of some of these unit squares there is an ant. At time $0$, each ant starts moving with speed $1$ parallel to some edge of the checkerboard. When two ants moving in the opposite directions meet, they both turn $90^{\circ}$ clockwise and continue moving with speed $1$. When more than $2$ ants meet, or when two ants moving in perpendicular directions meet, the ants continue moving in the same direction as before they met. When an ant reaches one of the edges of the checkerboard, it falls off and will not re-appear.



Considering all possible starting positions, determine the latest possible moment at which the last ant falls off the checkerboard, or prove that such a moment does not necessarily exist.



Proposed by Toomas Krips, Estonia
2011 Estonia Team Selection Test
Two circles lie completely outside each other.Let $A$ be the point of intersection of internal common tangents of the circles and let $K$ be the projection of this point onto one of their external common tangents.The tangents,different from the common tangent,to the circles through point $K$ meet the circles at $M_1$ and $M_2$.Prove that the line $AK$ bisects angle $M_1 KM_2$.
2011 Estonia Team Selection Test
Let $n$ be a positive integer. Prove that for each factor $m$ of the number $1+2+\cdots+n$ such that $m\ge n$, the set $\{1,2,\ldots,n\}$ can be partitioned into disjoint subsets, the sum of the elements of each being equal to $m$.

Edit:Typographical error fixed.
2011 Estonia Team Selection Test
Does there exist an operation $*$ on the set of all integers such that the following conditions hold simultaneously:

$(1)$ for all integers $x,y,z$, $(x*y)*z=x*(y*z)$;

$(2)$ for all integers $x$ and $y$, $x*x*y=y*x*x=y$?
2011 Estonia Team Selection Test
Let $a,b,c$ be positive real numbers such that $2a^2 +b^2=9c^2$.Prove that $\displaystyle \frac{2c}{a}+\frac cb \ge\sqrt 3$.
2011 Estonia Team Selection Test
Prove that if $n$ and $k$ are positive integers such that $1<k<n-1$,Then the binomial coefficient $\binom nk$ is divisible by at least two different primes.
2011 Estonia Team Selection Test
On a square board with $m$ rows and $n$ columns, where $m\le n$, some squares are colored black in such a way that no two rows are alike. Find tha biggest integer $k$ such that, for every possible coloring to start with, one can always color $k$ columns entirely red in such a way that still no two rows are alike.
2010 Estonia Team Selection Test
For arbitrary positive integers $a, b$, denote $a @ b =\frac{a-b}{gcd(a,b)}$

Let $n$ be a positive integer. Prove that the following conditions are equivalent:

(i) $gcd(n, n @  m) = 1$ for every positive integer $m < n$,

(ii) $n = p^k$ where $p$ is a prime number and $k$ is a non-negative integer.
2010 Estonia Team Selection Test
Let $n$ be a positive integer. Find the largest integer $N$ for which there exists a set of $n$ weights such that it is possible to determine the mass of all bodies with masses of $1, 2, ..., N$ using a balance scale .

(i.e. to determine whether a body with unknown mass has a mass $1, 2, ..., N$, and which namely).
2010 Estonia Team Selection Test
Let the angles of a triangle be $\alpha, \beta$, and $\gamma$, the perimeter $2p$ and the radius of the circumcircle $R$. Prove the inequality $\cot^2 \alpha + \cot^2 \beta + \cot^2 \gamma \ge  3 \left(\frac{9R^2}{p^2}-1\right)$. When is the equality achieved?
2010 Estonia Team Selection Test
In an acute triangle $ABC$ the angle $C$ is greater than the angle $A$. Let $AE$ be a diameter of the circumcircle of the triangle. Let the intersection point of the ray $AC$ and the tangent of the circumcircle through the vertex $B$ be $K$. The perpendicular to $AE$ through $K$ intersects the circumcircle of the triangle $BCK$ for the second time at point $D$. Prove that $CE$ bisects the angle $BCD$.
2010 Estonia Team Selection Test
Let $P(x, y)$ be a non-constant homogeneous polynomial with real coefficients such that $P(\sin t, \cos t) = 1$ for every real number $t$. Prove that there exists a positive integer $k$ such that $P(x, y) = (x^2 + y^2)^k$.
2010 Estonia Team Selection Test
Every unit square of a $n \times n$ board is colored either red or blue so that among all 2 $\times 2$ squares on this board all possible colorings of $2 \times 2$ squares with these two colors are represented (colorings obtained from each other by rotation and reflection are considered different).

a) Find the least possible value of $n$.

b) For the least possible value of $n$ find the least possible number of red unit squares
2009 Estonia Team Selection Test
For arbitrary pairwise distinct positive real numbers $a, b, c$, prove the inequality

$$\frac{(a^2- b^2)^3 + (b^2-c^2)^3+(c^2-a^2)^3}{(a- b)^3 + (b-c)^3+(c-a)^3}> 8abc$$
2009 Estonia Team Selection Test
Call a finite set of positive integers independent  if its elements are pairwise coprime, and nice  if the arithmetic mean of the elements of every non-empty subset of it is an integer.

a) Prove that for any positive integer $n$ there is an $n$-element set of positive integers which is both independent and nice.

b) Is there an infinite set of positive integers whose every independent subset is nice and which has an $n$-element independent subset for every positive integer $n$?
2009 Estonia Team Selection Test
Find all natural numbers $n$ for which there exists a convex polyhedron satisfying the following conditions:

(i) Each face is a regular polygon.

(ii) Among the faces, there are polygons with at most two different numbers of edges.

(iii) There are two faces with common edge that are both $n$-gons.
2009 Estonia Team Selection Test
Points $A', B', C'$ are chosen on the sides $BC, CA, AB$ of triangle $ABC$, respectively, so that $\frac{|BA'|}{|A'C|}=\frac{|CB'|}{|B'A|}=\frac{|AC'|}{|C'B|}$. The line which is parallel to line $B'C'$ and goes  through point $A$ intersects the lines $AC$ and $AB$ at $P$ and $Q$, respectively. Prove that   $\frac{|PQ|}{|B'C'|} \ge 2$
2009 Estonia Team Selection Test
A strip consists of $n$ squares which are numerated in their order by integers $1,2,3,..., n$. In the beginning, one square is empty while each remaining square contains one piece. Whenever a square contains a piece and its some neighbouring square contains another piece while the square immediately following the neighbouring square is empty, one may raise the first piece over the second one to the empty square, removing the second piece from the strip.

Find all possibilites which square can be initially empty, if it is possible to reach a state where the strip contains only one piece and

a) $n = 2008$,

b) $n = 2009$.
2009 Estonia Team Selection Test
For any positive integer $n$, let $c(n)$ be the largest divisor of $n$ not greater than $\sqrt{n}$ and let $s(n)$ be the least integer $x$ such that $n < x$ and the product $nx$ is divisible by an integer $y$ where $n < y < x$. Prove that, for every $n$, $s(n) = (c(n) + 1) \cdot \left( \frac{n}{c(n)}+1\right)$
2008 Estonia Team Selection Test
There are $2008$ participants in a programming competition. In every round, all programmers are divided into two equal-sized teams. Find the minimal number of rounds after which there can be a situation in which every two programmers have been in different teams at least once.
2008 Estonia Team Selection Test
Let $ABCD$ be a cyclic quadrangle whose midpoints of diagonals $AC$ and $BD$ are $F$ and $G$, respectively.

a) Prove the following implication: if the bisectors of angles at $B$ and $D$ of the quadrangle intersect at diagonal $AC$ then $\frac14 \cdot |AC| \cdot |BD| = | AG| \cdot |BF| \cdot |CG| \cdot |DF|$.

b) Does the converse implication also always hold?
2008 Estonia Team Selection Test
Let $ n$ be a positive integer, and let $ x$ and $ y$ be a positive real number such that $ x^n + y^n = 1.$ Prove that

$$ \left(\sum^n_{k = 1} \frac {1 + x^{2k}}{1 + x^{4k}} \right) \cdot \left( \sum^n_{k = 1} \frac {1 + y^{2k}}{1 + y^{4k}} \right) < \frac {1}{(1 - x) \cdot (1 - y)}.

$$



Author: Juhan Aru, Estonia
2008 Estonia Team Selection Test
Sequence $(G_n)$ is defined by $G_0 = 0, G_1 = 1$ and $G_n = G_{n-1} + G_{n-2} + 1$ for every $n \ge2$. Prove that for every positive integer $m$ there exist two consecutive terms in the sequence that are both divisible by $m$.
2008 Estonia Team Selection Test
Points $A$ and $B$ are fixed on a circle $c_1$. Circle $c_2$, whose centre lies on $c_1$, touches line $AB$ at $B$. Another line through $A$ intersects $c_2$ at points $D$ and $E$, where $D$ lies between $A$ and $E$. Line $BD$ intersects $c_1$ again at $F$. Prove that line $EB$ is tangent to $c_1$ if and only if $D$ is the midpoint of the segment $BF$.
2008 Estonia Team Selection Test
A string of parentheses is any word that can be composed by the following rules.

1) () is a string of parentheses.

2) If $s$ is a string of parentheses then $(s)$ is a string of parentheses.

3) If $s$ and t are strings of parentheses then $st$ is a string of parentheses.

The midcode  of a string of parentheses is the tuple of natural numbers obtained by finding, for all pairs of opening and its corresponding closing parenthesis, the number of characters remaining to the left from the medium position between these parentheses, and writing all these numbers in non-decreasing order. For example, the midcode of $(())$ is $(2,2)$ and the midcode of ()() is $(1,3)$. Prove that midcodes of arbitrary two different strings of parentheses are different.
2007 Estonia Team Selection Test
On the control board of a nuclear station, there are $n$ electric switches ($n > 0$), all in one row. Each switch has two possible positions: up and down. The switches are connected to each other in such a way that, whenever a switch moves down from its upper position, its right neighbour (if it exists) automatically changes position. At the beginning, all switches are down. The operator of the board first changes the position of the leftmost switch once, then the position of the second leftmost switch twice etc., until eventually he changes the position of the rightmost switch n times. How many switches are up after all these operations?