Navvye/Olympiad-Problems-CSV huggingface dataset:
2020 Caucasus Mathematical Olympiad
By one magic nut, Wicked Witch can either turn a flea into a beetle or a spider into a bug; while by one magic acorn, she can either turn a flea into a spider or a beetle into a bug. In the evening Wicked Witch had spent 20 magic nuts and 23 magic acorns. By these actions, the number of beetles increased by 5. Determine what was the change in the number of spiders. (Find all possible answers and prove that the other answers are impossible.)
2020 Caucasus Mathematical Olympiad
Let $\omega_1$ and $\omega_2$ be two non-intersecting circles. Let one of its internal tangents touches $\omega_1$ and $\omega_2$ at $A_1$ and $A_2$, respectively, and let one of its external tangents touches $\omega_1$ and $\omega_2$ at $B_1$ and $B_2$, respectively. Prove that if $A_1B_2 = A_2B_1$, then   $A_1B_2 \perp A_2B_1$.
2020 Caucasus Mathematical Olympiad
Let $a_n$ be a sequence given by $a_1 = 18$, and $a_n = a_{n-1}^2+6a_{n-1}$, for $n>1$. Prove that this sequence contains no perfect powers.
2020 Caucasus Mathematical Olympiad
Positive integers $n$, $k>1$ are given. Pasha and Vova play a game on a board $n\times k$.  Pasha begins, and further they alternate the following moves. On each move a player should place a border of length 1 between two adjacent cells. The player loses if after his move there is no way from the bottom left cell to the top right without crossing any order. Determine who of the players has a winning strategy.
2020 Caucasus Mathematical Olympiad
Find the number of pairs of positive integers $a$ and $b$ such that  $a\leq 100\,000$, $b\leq 100\,000$, and

$$

\frac{a^3-b}{a^3+b}=\frac{b^2-a^2}{b^2+a^2}.

$$
2020 Caucasus Mathematical Olympiad
All vertices of a regular 100-gon are colored in 10 colors. Prove that there exist 4 vertices of the given 100-gon which are the vertices of a rectangle and which are colored in at most 2 colors.
2020 Caucasus Mathematical Olympiad
A regular triangle $ABC$ is given. Points $K$ and $N$ lie in the segment $AB$, a point $L$ lies in the segment $AC$, and a point $M$  lies in the segment $BC$ so that $CL=AK$, $CM=BN$, $ML=KN$. Prove that $KL \parallel MN$.
2020 Caucasus Mathematical Olympiad
Let real  $a$, $b$, and $c$ satisfy $$abc+a+b+c=ab+bc+ca+5.$$Find the least possible value of $a^2+b^2+c^2$.
2020 Caucasus Mathematical Olympiad
Let $\omega_1$ and $\omega_2$ be two non-intersecting circles. Let one of its internal tangents touches $\omega_1$ and $\omega_2$ at $A_1$ and $A_2$, respectively, and let one of its external tangents touches $\omega_1$ and $\omega_2$ at $B_1$ and $B_2$, respectively. Prove that if $A_1B_2\perp  A_2B_1$, then   $A_1B_2 = A_2B_1$.
2020 Caucasus Mathematical Olympiad
Peter and Basil play the following game on a horizontal table $1\times{2019}$. Initially Peter chooses $n$ positive integers and writes them on a board. After that Basil puts a coin in one of the cells. Then at each move, Peter announces a number s among the numbers written on the board, and Basil needs to shift the coin by $s$ cells, if it is possible: either to the left, or to the right, by his decision. In case it is not possible to shift the coin by $s$ cells neither to the left, nor to the right, the coin stays in the current cell. Find the least $n$ such that Peter can play so that the coin will visit all the cells, regardless of the way Basil plays.
2020 Caucasus Mathematical Olympiad
Find all functions $f : \mathbb{N}\rightarrow{\mathbb{N}}$ such that for all positive integers $m$ and $n$ the number $f(m)+n-m$ is divisible by $f(n)$.
2020 Caucasus Mathematical Olympiad
Morteza wishes to take two real numbers $S$ and $P$, and then to arrange six pairwise distinct real numbers on a circle so that for each three consecutive numbers at least one of the two following conditions holds:

1) their sum equals $S$

2) their product equals $P$.

Determine if Morteza’s wish could be fulfilled.
2020 Caucasus Mathematical Olympiad
In $\triangle ABC$ with $AB\neq{AC}$ let $M$ be the midpoint of $AB$, let $K$ be the midpoint of the arc $BAC$ in the circumcircle of $\triangle ABC$, and let the perpendicular bisector of $AC$ meet the bisector of $\angle BAC$ at $P$ . Prove that $A, M, K, P$ are concyclic.
2020 Caucasus Mathematical Olympiad
Peter wrote $100$ distinct integers on a board. Basil needs to fill the cells of a table $100\times{100}$ with integers so that the sum in each rectangle $1\times{3}$ (either vertical, or horizontal) is equal to one of the numbers written on the board. Find the greatest $n$ such that, regardless of numbers written by Peter, Basil can fill the table so that it would contain each of numbers $(1,2,...,n)$ at least once (and possibly some other integers).
2019 Caucasus Mathematical Olympiad
In the kindergarten there is a big box with balls of three colors: red, blue and green, 100 balls in total. Once Pasha took out of the box 30 red, 10 blue, and 20 green balls and played with them. Then he lost five balls and returned the others back into the box. The next day, Sasha took out of the box 8 red, 18 blue, and 48 green balls. Is it possible to determine the color of at least one lost ball?
2019 Caucasus Mathematical Olympiad
Determine if there exist five consecutive positive integers such that their LCM is a perfect square.
2019 Caucasus Mathematical Olympiad
Points $A'$ and $B'$ lie inside the parallelogram $ABCD$ and points $C'$ and $D'$ lie outside of it, so that all sides of 8-gon $AA'BB'CC'DD'$ are equal. Prove that $A'$, $B'$, $C'$, $D'$ are concyclic.
2019 Caucasus Mathematical Olympiad
Vova has a square grid $72\times 72$. Unfortunately, $n$ cells are  stained with coffee. Determine if Vova always can cut out a clean square $3\times 3$ without its central cell, if

a) $n=699$;

b) $n=750$.
2019 Caucasus Mathematical Olympiad
Vasya has a numeric expression

$$ \Box \cdot \Box +\Box \cdot \Box $$and 4 cards with numbers that can be put on 4 free places in the expression. Vasya tried to put cards in all possible ways and all the time obtained the same value as a result. Prove that equal numbers are written on three of his cards.
2019 Caucasus Mathematical Olympiad
In a triangle $ABC$ with  $\angle BAC = 90^{\circ}$ let $BL$ be the bisector, $L\in AC$. Let $D$ be a point symmetrical to $A$ with respect to $BL$. Let  $M$ be the circumcenter of $ADC$. Prove that $CM$, $DL$, and $AB$ are concurrent.
2019 Caucasus Mathematical Olympiad
15 boxes are given. They all  are initially empty. By one move it is allowed to choose some boxes and to put in them numbers of apricots which are pairwise distinct powers of 2. Find the least positive integer $k$ such that it is possible

to have equal numbers of apricots in all the boxes after $k$ moves.
2019 Caucasus Mathematical Olympiad
Determine if there exist positive integers $a_1,a_2,...,a_{10}$, $b_1,b_2,...,b_{10}$ satisfying the following property: for each non-empty subset $S$ of $\{1,2,\ldots,10\}$  the sum  $\sum\limits_{i\in S}a_i$  divides $\left( 12+\sum\limits_{i\in S}b_i \right)$.
2019 Caucasus Mathematical Olympiad
Pasha placed numbers from 1 to 100 in the cells of the square $10\times 10$, each number exactly once. After that, Dima considered all sorts of squares, with the sides going along the grid lines, consisting of more than one cell, and painted in green the largest number in each such square (one number could be colored many times). Is it possible that all two-digit numbers are painted green?
2019 Caucasus Mathematical Olympiad
In a triangle $ABC$ let $I$ be the incenter. Prove that the circle passing through $A$ and touching $BI$ at $I$, and the circle passing through $B$ and touching $AI$ at $I$, intersect at a point on the circumcircle of $ABC$.
2019 Caucasus Mathematical Olympiad
Find all positive integers $n\geqslant 2$ such that there exists a permutation $a_1$, $a_2$, $a_3$, \ldots, $a_{2n}$ of the numbers  $1, 2, 3, \ldots, 2n$ satisfying $$a_1\cdot a_2 + a_3\cdot a_4 + \ldots + a_{2n-3} \cdot  a_{2n-2} = a_{2n-1} \cdot a_{2n}.$$
2019 Caucasus Mathematical Olympiad
Dima has 100 rocks with pairwise distinct weights. He also has a strange pan scales: one should put exactly 10 rocks on each side. Call a pair of rocks {\it clear} if Dima can find out which of these two rocks is heavier. Find the least possible number of clear pairs.
2019 Caucasus Mathematical Olympiad
Given a triangle  $ABC$ with $BC=a$, $CA=b$, $AB=c$, $\angle BAC = \alpha$, $\angle CBA = \beta$, $\angle ACB = \gamma$. Prove that $$ a \sin(\beta-\gamma) + b \sin(\gamma-\alpha) +c\sin(\alpha-\beta) = 0.$$
2019 Caucasus Mathematical Olympiad
15 boxes are given. They all  are initially empty. By one move it is allowed to choose some boxes and to put in them numbers of apricots which are pairwise distinct powers of 2. Find the least positive integer $k$ such that it is possible to have equal numbers of apricots in all the boxes after $k$ moves.
2019 Caucasus Mathematical Olympiad
On sides $BC$, $CA$, $AB$ of a triangle $ABC$ points $K$, $L$, $M$ are chosen, respectively, and a point $P$ is inside $ABC$ is chosen so that  $PL\parallel BC$, $PM\parallel CA$, $PK\parallel AB$. Determine if it is possible that each of  three trapezoids $AMPL$, $BKPM$, $CLPK$ has an inscribed circle.
2019 Caucasus Mathematical Olympiad
Determine if there exist pairwise distinct positive integers $a_1,a_2,\ldots,a_{101}$, $b_1$, $b_2$, \ldots, $b_{101}$ satisfying the following property: for each non-empty subset $S$ of $\{1,2,\ldots,101\}$  the sum  $\sum\limits_{i\in S}a_i$  divides  $\left( 100!+\sum\limits_{i\in S}b_i \right)$.
2018 Caucasus Mathematical Olympiad
Let $a$, $b$, $c$ be real numbers, not all of them are equal. Prove that $a+b+c=0$ if and only if $a^2+ab+b^2=b^2+bc+c^2=c^2+ca+a^2$.
2018 Caucasus Mathematical Olympiad
On a chessboard $8\times 8$, $n>6$ Knights are placed so that for any 6 Knights there are two Knights that attack each other. Find the greatest possible value of $n$.
2018 Caucasus Mathematical Olympiad
Suppose that $a,b,c$ are positive integers such that $a^b$ divides $b^c$, and $a^c$ divides $c^b$. Prove that $a^2$ divides $bc$.
2018 Caucasus Mathematical Olympiad
By centroid of a quadrilateral $PQRS$ we call a common point of two lines through the midpoints of its opposite sides. Suppose that $ABCDEF$ is a hexagon inscribed into the circle $\Omega$ centered at $O$. Let  $AB=DE$, and  $BC=EF$. Let $X$, $Y$, and $Z$ be centroids of $ABDE$, $BCEF$; and $CDFA$, respectively. Prove that $O$ is the orthocenter of triangle $XYZ$.
2018 Caucasus Mathematical Olympiad
Baron Munсhausen discovered the following theorem: "For any positive integers $a$ and  $b$ there exists a positive integer $n$ such that $an$ is a perfect square, while $bn$ is a perfect cube". Determine if the statement of Baron’s theorem is correct.
2018 Caucasus Mathematical Olympiad
Given a convex quadrilateral $ABCD$ with $\angle BCD=90^\circ$. Let $E$ be the midpoint of $AB$. Prove that $2EC \leqslant AD+BD$.
2018 Caucasus Mathematical Olympiad
Given a positive integer $n>1$. In the cells of an $n\times n$ board, marbles are placed one by one. Initially there are no marbles on the board. A marble could be placed in a free cell neighboring (by side) with at least two cells which are still free. Find the greatest possible number of marbles that could be placed on the board according to these rules.
2018 Caucasus Mathematical Olympiad
Let $a, b, c$ be the lengths of sides of a triangle. Prove the inequality

$$(a+b)\sqrt{ab}+(a+c)\sqrt{ac}+(b+c)\sqrt{bc} \geq (a+b+c)^2/2.$$
2018 Caucasus Mathematical Olympiad
A tetrahedron is given. Determine whether it is possible to put some 10 consecutive positive integers at 4 vertices and at 6 midpoints of the edges so that the number at the midpoint of each edge is equal to the arithmetic mean of two numbers at the endpoints of this edge.
2018 Caucasus Mathematical Olympiad
Let $I$ be the incenter of an acute-angled triangle $ABC$. Let $P$, $Q$, $R$ be points on sides $AB$, $BC$, $CA$ respectively, such that $AP=AR$, $BP=BQ$ and $\angle PIQ = \angle BAC$. Prove that $QR \perp AC$.
2018 Caucasus Mathematical Olympiad
For $2n$ positive integers a matching (i.e. dividing them into $n$ pairs) is called {\it non-square} if the product of two numbers in each pair is not a perfect square. Prove that if there is a non-square matching, then there are at least $n!$ non-square matchings.

(By $n!$ denote the product $1\cdot 2\cdot 3\cdot \ldots \cdot n$.)
2018 Caucasus Mathematical Olympiad
Morteza places a function $[0,1]\to [0,1]$ (that is a function with domain [0,1] and values from [0,1]) in each cell of an $n \times n$ board. Pavel wants to place a function $[0,1]\to [0,1]$ to the left of each row and below each column (i.e. to place $2n$ functions in total) so that the following condition holds for any cell in this board:

If $h$ is the function in this cell, $f$ is the function below its column, and $g$ is the function to the left of its row, then $h(x) = f(g(x))$ for all $x \in [0, 1]$.

Prove that Pavel can always fulfil his plan.
2018 Caucasus Mathematical Olympiad
Baron Munсhausen discovered the following theorem: "For any positive integers $a$ and  $b$ there exists a positive integer $n$ such that $an$ is a perfect cube, while $bn$ is a perfect fifth power". Determine if the statement of Baron’s theorem is correct.
2018 Caucasus Mathematical Olympiad
Two graphs $G_1$ and $G_2$ of quadratic polynomials intersect at points $A$ and $B$. Let $O$ be the vertex of $G_1$. Lines $OA$ and $OB$ intersect $G_2$ again at points $C$ and $D$. Prove that $CD$ is parallel to the $x$-axis.
2018 Caucasus Mathematical Olympiad
In an acute-angled triangle $ABC$, the altitudes from $A,B,C$ meet the sides of $ABC$ at $A_1$, $B_1$, $C_1$, and meet the circumcircle of $ABC$ at $A_2$, $B_2$, $C_2$, respectively. Line  $A_1 C_1$ intersects the circumcircles of triangles $AC_1 C_2$ and $CA_1 A_2$ at points $P$ and $Q$ ($Q\neq A_1$, $P\neq C_1$). Prove that the circle $PQB_1$ touches the line $AC$.
2018 Caucasus Mathematical Olympiad
In the cells of an $8\times 8$ board, marbles are placed one by one. Initially there are no marbles on the board. A marble could be placed in a free cell neighboring (by side) with at least three cells which are still free. Find the greatest possible number of marbles that could be placed on the board according to these rules.
2017 Caucasus Mathematical Olympiad
Basil needs to solve an exercise on summing two fractions $\dfrac{a}{b}$ and $\dfrac{c}{d}$, where  $a$, $b$, $c$, $d$ are some non-zero real numbers. But instead of summing he performed multiplication  (correctly). It appears that Basil's answer coincides with the correct answer to given exercise. Find the value of  $\dfrac{b}{a} + \dfrac{d}{c}$.
2017 Caucasus Mathematical Olympiad
On Mars a basketball team consists of 6 players. The coach of the team Mars can select  any line-up of 6 players among 100 candidates. The coach considers some line-ups as appropriate while the other line-ups are not (there exists at least one appropriate line-up). A set of 5 candidates is called perspective if one more candidate could be added to it to obtain an appropriate line-up. A candidate is called universal if he completes each perspective set of 5 candidates (not containing him) upto an appropriate line-up. The coach has selected a line-up of 6 universal candidates. Determine if it follows that this line-up is appropriate.
2017 Caucasus Mathematical Olympiad
Find the least positive integer  $n$ satisfying the following statement: for eash pair of positive integers $a$ and $b$ such that $36$ divides  $a+b$ and $n$ divides $ab$ it follows that $36$ divides both  $a$ and $b$.
2017 Caucasus Mathematical Olympiad
In an acute traingle $ABC$ with  $AB< BC$ let $BH_b$ be its altitude, and let $O$ be the circumcenter. A line through $H_b$ parallel to $CO$ meets $BO$ at $X$. Prove that $X$ and the midpoints of $AB$ and $AC$ are collinear.
2017 Caucasus Mathematical Olympiad
In a football tournament $20$ teams participated, each pair of teams played exactly one game. For the win the team is awarded $3$ points, for the draw -- $1$ point, for the lose no points awarded. The total number of points of all teams in the tournament is $554$. Prove that there exist $7$ teams each having at least one draw.
2017 Caucasus Mathematical Olympiad
A triangle is cut by $3$ cevians from its $3$ vertices into $7$ pieces: $4$ triangles and $3$ quadrilaterals. Determine if it is possible that all $3$ quadrilaterals are inscribed.
2017 Caucasus Mathematical Olympiad
$10$ distinct numbers are given. Professor Odd had calculated all possible products of $1$, $3$, $5$, $7$, $9$ numbers among given numbers, and wrote down the sum of all these products. Similarly, Professor Even had calculated all possible products of $2$, $4$, $6$, $8$, $10$ numbers among given numbers, and wrote down the sum of all these products. It appears that Odd's sum is greater than Even's sum by $1$. Prove that one of $10$ given numbers is equal to $1$.
2017 Caucasus Mathematical Olympiad
$100$ points are marked in the plane so that  no three of marked points are collinear. One of marked points is red, and the others are blue. A triangle with vertices at blue points is called good if the red point lies inside it. Determine if it is possible that the number of good triangles is not less than the half of the total number of traingles with vertices at blue points.
2017 Caucasus Mathematical Olympiad
Two points $A$ and $B$ lie on two branches of hyperbola given by equation $y=\frac1x$. Let $A_x$ and $A_y$ be projections of $A$ onto coordinate axis, similarly, let $B_x$ and $B_y$ be projections of $B$ onto coordinate axis. Prove that triangles $AB_xB_y$ and $BA_xA_y$ have equal areas.
2017 Caucasus Mathematical Olympiad
On Mars a basketball team consists of 6 players. The coach of the team Mars can select any line-up of 6 players among 100 candidates. The coach considers some line-ups as appropriate while the other line-ups are not (there exists at least one appropriate line-up). A set of 5 candidates is called perspective if one more candidate could be added to it to obtain an appropriate line-up. A candidate is called universal if he completes each perspective set of 5 candidates (not containing him) upto an appropriate line-up. The coach has selected a line-up of 6 universal candidates. Determine if it follows that this line-up is appropriate.
2017 Caucasus Mathematical Olympiad
Determine if there exist $101$ positive integers (not necessarily distinct) such that their product is equal to the sum of all their pairwise LCM.
2017 Caucasus Mathematical Olympiad
Given real numbers $a$, $b$, $c$ satisfy inequality $\left| \frac{a^2+b^2-c^2}{ab} \right|<2$. Prove that they also satisfy equalities $\left| \frac{b^2+c^2-a^2}{bc} \right|<2$ and $\left|  \frac{c^2+a^2-b^2}{ca} \right| <2$.
2017 Caucasus Mathematical Olympiad
$8$ ants are placed on the edges of the unit cube. Prove that there exists a pair of ants at a distance not exceeding $1$.
2017 Caucasus Mathematical Olympiad
Given a table in a form of the regular $1000$-gon with sidelength $1$. A Beetle initially is in one of its vertices. All $1000$ vertices are numbered in some order by numbers $1$, $2$, $\ldots$, $1000$ so that initially the Beetle is in the vertex $1$.  The  Beetle can move only along the edges of $1000$-gon and only clockwise. He starts to move from vertex $1$ and he is moving without stops until he reaches vertex $2$ where he has a stop. Then he continues his journey clockwise from vertex $2$ until he reaches the vertex $3$ where he has a stop, and so on. The Beetle finishes his journey at vertex $1000$. Find the number of ways to enumerate all vertices so that the total length of the Beetle's journey is equal to $2017$.
2015 Caucasus Mathematical Olympiad
Does there exist a four-digit positive integer with different non-zero digits, which has the following property: if we add the same number written in the reverse order, then we get a number divisible by $101$?
2015 Caucasus Mathematical Olympiad
There are $9$ cards with the numbers $1, 2, 3, 4, 5, 6, 7, 8$ and $9$. What is the largest number of these cards can be decomposed in a certain order in a row, so that in any two adjacent cards, one of the numbers is divided by the other?
2015 Caucasus Mathematical Olympiad
Petya bought one cake, two cupcakes and three bagels, Apya bought three cakes and a bagel, and Kolya bought six cupcakes. They all paid the same amount of money for purchases. Lena bought two cakes and two bagels. And how many cupcakes could be bought for the same amount spent to her?
2015 Caucasus Mathematical Olympiad
There are $26$ students in the class.

They agreed that each of them would either be a liar (liars always lie) or a knight (knights always tell the truth).

When they came to the class and sat down for desks, each of them said: “I am sitting next to a liar.”

Then some students moved for other desks. After that, everyone says: “ I am sitting next to a knight .”

Is this possible?

Every time exactly two students sat at any desk.
2015 Caucasus Mathematical Olympiad
What is the smallest number of $3$-cell corners needed to be painted in a $6\times 6$ square so that it was impossible to paint more than one corner of it?

(The painted corners should not overlap.)
2015 Caucasus Mathematical Olympiad
Find some four different natural numbers with the following property: if you add to the product of any two of them the product of the two remaining numbers. you get a prime number.
2015 Caucasus Mathematical Olympiad
In the convex quadrilateral $ABCD$, point $K$ is the midpoint of $AB$, point $L$ is the midpoint of $BC$, point $M$ is the midpoint of CD, and point $N$ is the midpoint of $DA$. Let $S$ be a point lying inside the quadrilateral $ABCD$ such that $KS = LS$ and  $NS = MS$ .Prove that $\angle KSN =  \angle MSL$.
2015 Caucasus Mathematical Olympiad
The workers laid a floor of size $n \times n$ with tiles of two types: $2 \times  2$ and $3 \times  1$.

It turned out that they were able to completely lay the floor in such a way that the same number of tiles of each type was used. Under what conditions could this happen?

(You can’t cut tiles and also put them on top of each other.)
2015 Caucasus Mathematical Olympiad
The sum of the numbers $a,b$ and $c$ is zero, and their product is negative.

Prove that the number  $\frac{a^2+b^2}{c}+\frac{b^2+c^2}{a}+\frac{c^2+a^2}{b}$  is positive.
2015 Caucasus Mathematical Olympiad
On the table are $300$ coins. Petya, Vasya and Tolya play the next game. They go in turn in the following order: Petya, Vasya, Tolya, Petya. Vasya, Tolya, etc. In one move, Petya can take $1, 2, 3$, or $4$ coins from the table, Vasya, $1$ or $2$ coins, and Tolya, too, $1$ or $2$ coins. Can Vasya and Tolya agree so that, as if Petya were playing, one of them two will take the last coin off the table?
2015 Caucasus Mathematical Olympiad
At the round table, $10$ people are sitting, some of them are knights, and the rest are liars (knights always say pride, and liars always lie) .  It is clear thath I have at least one knight and at least one liar.

What is the largest number of those sitting at the table can say: ''Both of my neighbors are knights '' ?

(A statement that is at least partially false is considered false.)
2015 Caucasus Mathematical Olympiad
Let $a$ and $b$ be arbitrary distinct numbers.

Prove that the equation $(x +a) (x+b)=2x+a+b$ has two different roots.
2015 Caucasus Mathematical Olympiad
Let $AL$ be the angle bisector of the acute-angled triangle $ABC$. and $\omega$ be the circle circumscribed about it. Denote by $P$ the intersection point of the extension of the altitude $BH$ of the triangle $ABC$ with the circle $\omega$ . Prove that if $\angle BLA= \angle BAC$, then $BP = CP$.
2015 Caucasus Mathematical Olympiad
Is there a nine-digit number without zero digits, the remainder of dividing which on each of its digits is different?