Navvye/Olympiad-Problems-CSV huggingface dataset:
2014 All-Russian Olympiad
The sphere $ \omega $ passes through the vertex $S$ of the pyramid $SABC$ and intersects with the edges $SA,SB,SC$ at $A_1,B_1,C_1$  other than $S$. The sphere $ \Omega $ is the circumsphere of the pyramid $SABC$ and intersects with $ \omega $ circumferential,  lies on a plane which parallel to the plane $(ABC)$.

Points $A_2,B_2,C_2$ are symmetry points of the points $A_1,B_1,C_1$ respect to midpoints of the edges $SA,SB,SC$ respectively. Prove that the points $A$, $B$, $C$, $A_2$, $B_2$, and  $C_2$ lie on a sphere.
2014 All-Russian Olympiad
If the polynomials $f(x)$ and $g(x)$ are written on a blackboard then we can also write down the polynomials $f(x)\pm g(x)$, $f(x)g(x)$, $f(g(x))$ and $cf(x)$, where $c$ is an arbitrary real constant. The polynomials $x^3-3x^2+5$ and $x^2-4x$ are written on the blackboard. Can we write a nonzero polynomial of form $x^n-1$ after a finite number of steps?
2014 All-Russian Olympiad
Two players play a card game. They have a deck of $n$ distinct cards. About any two cards from the deck know which of them has a different (in this case, if $A$ beats $B$, and $B$ beats $C$, then it may be that $C$ beats $A$). The deck is split between players in an arbitrary manner.  In each turn the players over the top card from his deck and one whose card has a card from another player takes both cards and puts them to the bottom of your deck in any order of their discretion.  Prove that for any initial distribution of cards, the players can with knowing the location agree and act so that one of the players left without a card.



E. Lakshtanov
2013 All-Russian Olympiad
Given three distinct real numbers $a$, $b$, and $c$, show that at least two of the three following equations

$$(x-a)(x-b)=x-c$$

$$(x-c)(x-b)=x-a$$

$$(x-c)(x-a)=x-b$$

have real solutions.
2013 All-Russian Olympiad
Acute-angled triangle $ABC$ is inscribed into circle $\Omega$. Lines tangent to $\Omega$ at $B$ and $C$ intersect at $P$. Points $D$ and $E$ are on $AB$ and $AC$ such that $PD$ and $PE$ are perpendicular to $AB$ and $AC$ respectively. Prove that the orthocentre of triangle $ADE$ is the midpoint of $BC$.
2013 All-Russian Olympiad
$100$ distinct natural numbers $a_1, a_2, a_3, \ldots, a_{100}$ are written on the board. Then, under each number $a_i$, someone wrote a number $b_i$, such that $b_i$ is the sum of $a_i$ and the greatest common factor of the other $99$ numbers. What is the least possible number of distinct natural numbers that can be among $b_1, b_2, b_3, \ldots, b_{100}$?
2013 All-Russian Olympiad
$N$ lines lie on a plane, no two of which are parallel and no three of which are concurrent. Prove that there exists a non-self-intersecting broken line $A_0A_1A_2A_3...A_N$ with $N$ parts, such that on each of the $N$ lines lies exactly one of the $N$ segments of the line.
2013 All-Russian Olympiad
$2n$ real numbers with a positive sum are aligned in a circle. For each of the numbers, we can see there are two sets of $n$ numbers such that this number is on the end. Prove that at least one of the numbers has a positive sum for both of these two sets.
2013 All-Russian Olympiad
Peter and Basil together thought of ten quadratic trinomials. Then, Basil began calling consecutive natural numbers starting with some natural number. After each called number, Peter chose one of the ten polynomials at random and plugged in the called number. The results were recorded on the board. They eventually form a sequence. After they finished, their sequence was arithmetic. What is the greatest number of numbers that Basil could have called out?
2013 All-Russian Olympiad
Squares $CAKL$ and $CBMN$ are constructed on the sides of acute-angled triangle $ABC$, outside of the triangle. Line $CN$ intersects line segment $AK$ at $X$, while line $CL$ intersects line segment $BM$ at $Y$. Point $P$, lying inside triangle $ABC$, is an intersection of the circumcircles of triangles $KXN$ and $LYM$. Point $S$ is the midpoint of $AB$. Prove that angle $\angle ACS=\angle BCP$.
2013 All-Russian Olympiad
On a $55\times 55$ square grid, $500$ unit squares were cut out as well as $400$ L-shaped pieces consisting of 3 unit squares (each piece can be oriented in any way) [refer to the figure]. Prove that at least two of the cut out pieces bordered each other before they were cut out.

[asy]size(2.013cm);

draw ((0,0)--(0,1));

draw ((0,0)--(1,0));

draw ((0,1)--(.5,1));

draw ((.5,1)--(.5,0));

draw ((0,.5)--(1,.5));

draw ((1,.5)--(1,0));

draw ((1,.5)--(1,0));

[/asy]
2013 All-Russian Olympiad
Circle is divided into $n$ arcs by $n$ marked points on the circle. After that circle rotate an angle $ 2\pi k/n $ (for some positive integer $ k $), marked points moved to $n$  new points , dividing the circle into $ n $  new arcs. Prove that there is a new arc that lies entirely in the one of the old arсs.

(It is believed that the endpoints of arcs belong to it.)



I. Mitrophanov
2013 All-Russian Olympiad
Find all positive integers  $k$ such that for the first $k$ prime numbers $2, 3, \ldots, p_k$ there exist positive integers $a$ and $n>1$, such that $2\cdot 3\cdot\ldots\cdot p_k - 1=a^n$.



V. Senderov
2013 All-Russian Olympiad
Inside the inscribed quadrilateral $ABCD$ are marked points $P$ and $Q$, such that $\angle PDC + \angle PCB,$ $\angle PAB + \angle PBC,$ $\angle QCD + \angle QDA$ and $\angle QBA + \angle QAD$ are all equal to $90^\circ$. Prove that the line $PQ$ has equal angles with lines $AD$ and $BC$.



A. Pastor
2013 All-Russian Olympiad
Does exist natural $n$, such that for any non-zero digits $a$ and $b$ $$\overline {ab}\ |\ \overline {anb}\ ?$$

(Here by $ \overline {x \ldots y} $ denotes the number obtained by concatenation decimal digits $x$, $\dots$, $y$.)



V. Senderov
2013 All-Russian Olympiad
Peter and Vasil together thought of ten 5-degree polynomials. Then, Vasil began calling consecutive natural numbers starting with some natural number. After each called number, Peter chose one of the ten polynomials at random and plugged in the called number. The results were recorded on the board. They eventually form a sequence. After they finished, their sequence was arithmetic. What is the greatest number of numbers that Vasil could have called out?



A. Golovanov
2013 All-Russian Olympiad
The incircle of triangle $ ABC $ has centre $I$ and touches the sides $ BC $, $ CA $, $ AB $ at points $ A_1 $, $ B_1 $, $ C_1 $, respectively. Let $ I_a $, $ I_b $, $ I_c $ be excentres of triangle $ ABC $, touching the sides $ BC $, $ CA $, $ AB $ respectively.  The segments $ I_aB_1 $ and $ I_bA_1 $ intersect at $ C_2 $. Similarly, segments $ I_bC_1 $ and $ I_cB_1 $ intersect at $ A_2 $, and the segments $ I_cA_1 $ and  $ I_aC_1 $ at $ B_2 $.  Prove that $ I $ is the center of the circumcircle of the triangle $ A_2B_2C_2 $.



L. Emelyanov, A. Polyansky
2013 All-Russian Olympiad
A square with horizontal and vertical sides is drawn on the plane. It held several segments parallel to the sides, and there are no two segments which lie on one line or intersect at an interior point for both segments. It turned out that the segments cuts square into rectangles, and any vertical line intersecting the square and not containing segments of the partition intersects exactly $ k $ rectangles of the partition, and any horizontal line intersecting the square and not containing segments of the partition intersects exactly $\ell$ rectangles. How much the number of rectangles can be?



I. Bogdanov, D. Fon-Der-Flaass
2013 All-Russian Olympiad
Let  $P(x)$ and $Q(x)$ be (monic) polynomials with real coefficients (the first coefficient being equal to $1$), and   $\deg P(x)=\deg Q(x)=10$. Prove that  if  the equation $P(x)=Q(x)$ has no real solutions,  then  $ P(x+1)=Q(x-1) $ has a real solution.
2013 All-Russian Olympiad
The inscribed and exscribed sphere of a triangular pyramid $ABCD$ touch her face $BCD$ at different points $X$ and $Y$. Prove that the triangle $AXY$ is obtuse triangle.
2013 All-Russian Olympiad
Find all positive $k$ such that product of the first $k$ odd prime numbers, reduced by 1 is exactly degree of natural number (which more than one).
2013 All-Russian Olympiad
On each of the cards written in $2013$ by number, all of these $2013$ numbers are different. The cards are turned down by numbers. In a single move is allowed to point out the ten cards and in return will report one of the numbers written on them (do not know what). For what most $w$ guaranteed to be able to find $w$ cards for which we know what numbers are written on each of them?
2013 All-Russian Olympiad
$101$ distinct numbers are chosen among the integers between $0$ and $1000$. Prove that, among the absolute values ​​of their pairwise differences, there are ten different numbers not exceeding $100$.
2013 All-Russian Olympiad
Let $a,b,c,d$ be positive real numbers such that $ 2(a+b+c+d)\ge abcd $. Prove that $$ a^2+b^2+c^2+d^2 \ge abcd .$$
2013 All-Russian Olympiad
The head of the Mint wants to release 12 coins denominations (each - a natural number rubles) so that any amount from 1 to 6543 rubles could be paid without having to pass, using no more than 8 coins. Can he do it? (If the payment amount you can use a few coins of the same denomination.)
2013 All-Russian Olympiad
Let  $ \omega $ be the incircle of  the  triangle $ABC$ and with centre $I$.  Let $\Gamma $ be the circumcircle of the triangle $AIB$. Circles $ \omega $ and $ \Gamma $ intersect at the point $X$ and $Y$. Let $Z$ be the intersection of the common tangents of the circles $\omega$ and $\Gamma$. Show that the   circumcircle of the triangle $XYZ$ is tangent to the circumcircle of the triangle $ABC$.
2012 All-Russian Olympiad
Let $a_1,\ldots a_{11}$ be distinct positive integers, all at least $2$ and with sum $407$. Does there exist an integer $n$ such that the sum of the $22$ remainders after the division of $n$ by $a_1,a_2,\ldots ,a_{11},4a_1,4a_2,\ldots ,4a_{11}$ is $2012$?
2012 All-Russian Olympiad
A regular $2012$-gon is inscribed in a circle. Find the maximal $k$ such that we can choose $k$ vertices from given $2012$ and construct a convex $k$-gon without parallel sides.
2012 All-Russian Olympiad
Consider the parallelogram $ABCD$ with obtuse angle $A$. Let $H$ be the feet of perpendicular from $A$ to the side $BC$. The median from $C$ in triangle $ABC$ meets the circumcircle of triangle $ABC$ at the point $K$. Prove that points $K,H,C,D$ lie on the same circle.
2012 All-Russian Olympiad
The positive real numbers $a_1,\ldots ,a_n$ and $k$ are such that $a_1+\cdots +a_n=3k$, $a_1^2+\cdots +a_n^2=3k^2$ and $a_1^3+\cdots +a_n^3>3k^3+k$. Prove that the difference between some two of $a_1,\ldots,a_n$ is greater than $1$.
2012 All-Russian Olympiad
$101$ wise men stand in a circle. Each of them either thinks that the Earth orbits Jupiter or that Jupiter orbits the Earth. Once a minute, all the wise men express their opinion at the same time. Right after that, every wise man who stands between two people with a different opinion from him changes his opinion himself. The rest do not change. Prove that at one point they will all stop changing opinions.
2012 All-Russian Olympiad
The points $A_1,B_1,C_1$ lie on the sides sides $BC,AC$ and $AB$ of the triangle $ABC$ respectively. Suppose that $AB_1-AC_1=CA_1-CB_1=BC_1-BA_1$. Let $I_A, I_B, I_C$ be the incentres of triangles $AB_1C_1,A_1BC_1$ and $A_1B_1C$ respectively. Prove that the circumcentre of triangle $I_AI_BI_C$ is the incentre of triangle $ABC$.
2012 All-Russian Olympiad
Initially, ten consecutive natural numbers are written on the board. In one turn, you may pick any two numbers from the board (call them $a$ and $b$) and replace them with the numbers $a^2-2011b^2$ and $ab$. After several turns, there were no initial numbers left on the board. Could there, at this point, be again, ten consecutive natural numbers?
2012 All-Russian Olympiad
In a city's bus route system, any two routes share exactly one stop, and every route includes at least four stops. Prove that the stops can be classified into two groups such that each route includes stops from each group.
2012 All-Russian Olympiad
Let $a_1,\ldots ,a_{10}$ be distinct positive integers, all at least $3$ and with sum $678$. Does there exist a positive integer $n$ such that the sum of the $20$ remainders of $n$ after division by $a_1,a_2,\ldots ,a_{10},2a_1,2a_2,\ldots ,2a_{10}$ is $2012$?
2012 All-Russian Olympiad
The inscribed circle $\omega$ of the non-isosceles acute-angled triangle $ABC$ touches the side $BC$ at the point $D$. Suppose that $I$ and $O$ are the centres of inscribed circle and circumcircle of triangle $ABC$ respectively. The circumcircle of triangle $ADI$ intersects $AO$ at the points $A$ and $E$. Prove that $AE$ is equal to the radius $r$ of $\omega$.
2012 All-Russian Olympiad
Any two of the real numbers $a_1,a_2,a_3,a_4,a_5$ differ by no less than $1$. There exists some real number $k$ satisfying

$$a_1+a_2+a_3+a_4+a_5=2k$$$$a_1^2+a_2^2+a_3^2+a_4^2+a_5^2=2k^2$$

Prove that $k^2\ge 25/3$.
2012 All-Russian Olympiad
Initially there are $n+1$ monomials on the blackboard: $1,x,x^2, \ldots, x^n $. Every minute each of $k$ boys simultaneously write on the blackboard the sum of some two polynomials that were written before. After $m$ minutes among others there are the polynomials $S_1=1+x,S_2=1+x+x^2,S_3=1+x+x^2+x^3,\ldots ,S_n=1+x+x^2+ \ldots +x^n$ on the blackboard. Prove that $ m\geq \frac{2n}{k+1} $.
2012 All-Russian Olympiad
Does there exist natural numbers $a,b,c$ all greater than $10^{10}$ such that their product is divisible by each of these numbers increased by $2012$?
2012 All-Russian Olympiad
On a Cartesian plane, $n$ parabolas are drawn, all of which are graphs of quadratic trinomials. No two of them are tangent. They divide the plane into many areas, one of which is above all the parabolas. Prove that the border of this area has no more than $2(n-1)$ corners (i.e. the intersections of a pair of parabolas).
2012 All-Russian Olympiad
The point $E$ is the midpoint of the segment connecting the orthocentre of the scalene triangle $ABC$ and the point $A$. The incircle of triangle $ABC$ incircle is tangent to $AB$ and $AC$ at points $C'$ and $B'$ respectively. Prove that point $F$, the point symmetric to point $E$ with respect to line $B'C'$, lies on the line that passes through both the circumcentre and the incentre of triangle $ABC$.
2012 All-Russian Olympiad
Initially, there are $111$ pieces of clay on the table of equal mass. In one turn, you can choose several groups of an equal number of pieces and push the pieces into one big piece for each group. What is the least number of turns after which you can end up with $11$ pieces no two of which have the same mass?
2012 All-Russian Olympiad
A plane is coloured into black and white squares in a chessboard pattern. Then, all the white squares are coloured red and blue such that any two initially white squares that share a corner are different colours. (One is red and the other is blue.) Let $\ell$ be a line not parallel to the sides of any squares. For every line segment $I$ that is parallel to $\ell$, we can count the difference between the length of its red and its blue areas. Prove that for every such line $\ell$ there exists a number $C$ that exceeds all those differences that we can calculate.
2012 All-Russian Olympiad
Given is a pyramid $SA_1A_2A_3\ldots A_n$ whose base is convex polygon $A_1A_2A_3\ldots A_n$. For every $i=1,2,3,\ldots ,n$ there is a triangle $X_iA_iA_{i+1} $ congruent to triangle $SA_iA_{i+1}$ that lies on the same side from $A_iA_{i+1}$ as the base of that pyramid. (You can assume $a_1$ is the same as $a_{n+1}$.) Prove that these triangles together cover the entire base.
2012 All-Russian Olympiad
Given is the polynomial $P(x)$ and the numbers $a_1,a_2,a_3,b_1,b_2,b_3$ such that $a_1a_2a_3\not=0$. Suppose that for every $x$, we have

$$P(a_1x+b_1)+P(a_2x+b_2)=P(a_3x+b_3)$$

Prove that the polynomial $P(x)$ has at least one real root.
2012 All-Russian Olympiad
The points $A_1,B_1,C_1$ lie on the sides $BC,CA$ and $AB$ of the triangle $ABC$ respectively. Suppose that $AB_1-AC_1=CA_1-CB_1=BC_1-BA_1$. Let $O_A,O_B$ and $O_C$ be the circumcentres of triangles $AB_1C_1,A_1BC_1$ and $A_1B_1C$ respectively. Prove that the incentre of triangle $O_AO_BO_C$ is the incentre of triangle $ABC$ too.
2012 All-Russian Olympiad
On a circle there are $2n+1$ points, dividing it into equal arcs ($n\ge 2$). Two players take turns to erase one point. If after one player's turn, it turned out that all the triangles formed by the remaining points on the circle were obtuse, then the player wins and the game ends.

Who has a winning strategy: the starting player or his opponent?
2012 All-Russian Olympiad
For a positive integer $n$ define $S_n=1!+2!+\ldots +n!$. Prove that there exists an integer $n$ such that $S_n$ has a prime divisor greater than $10^{2012}$.
2011 All-Russian Olympiad
A quadratic trinomial $P(x)$ with the $x^2$ coefficient of one is such, that $P(x)$ and $P(P(P(x)))$ share a root. Prove that $P(0)*P(1)=0$.
2011 All-Russian Olympiad
Given is an acute angled triangle $ABC$. A circle going through $B$ and the triangle's circumcenter, $O$, intersects $BC$ and $BA$ at points $P$ and $Q$ respectively. Prove that the intersection of the heights of the triangle $POQ$ lies on line $AC$.
2011 All-Russian Olympiad
A convex 2011-gon is drawn on the board. Peter keeps drawing its diagonals in such a way, that each newly drawn diagonal intersected no more than one of the already drawn diagonals. What is the greatest number of diagonals that Peter can draw?
2011 All-Russian Olympiad
Do there exist any three relatively prime natural numbers so that the square of each of them is divisible by the sum of the two remaining numbers?
2011 All-Russian Olympiad
For some 2011 natural numbers, all the $\frac{2010\cdot 2011}{2}$ possible sums were written out on a board. Could it have happened that exactly one third of the written numbers were divisible by three and also exactly one third of them give a remainder of one when divided by three?
2011 All-Russian Olympiad
In the notebooks of Peter and Nick, two numbers are written. Initially, these two numbers are 1 and 2 for Peter and 3 and 4 for Nick. Once a minute, Peter writes a quadratic trinomial $f(x)$, the roots of which are the two numbers in his notebook, while Nick writes a quadratic trinomial $g(x)$ the roots of which are the numbers in his notebook. If the equation $f(x)=g(x)$ has two distinct roots, one of the two boys replaces the numbers in his notebook by those two roots. Otherwise, nothing happens. If Peter once made one of his numbers 5, what did the other one of his numbers become?
2011 All-Russian Olympiad
Let $ABC$ be an equilateral triangle. A point $T$ is chosen on $AC$ and on arcs $AB$ and $BC$ of the circumcircle of $ABC$, $M$ and $N$ are chosen respectively, so that $MT$ is parallel to $BC$ and $NT$ is parallel to $AB$. Segments $AN$ and $MT$ intersect at point $X$, while $CM$ and $NT$ intersect in point $Y$. Prove that the perimeters of the polygons $AXYC$ and $XMBNY$ are the same.
2011 All-Russian Olympiad
There are some counters in some cells of $100\times 100$ board. Call a cell nice if there are an even number of counters in adjacent cells. Can exactly one cell be nice?



K. Knop
2011 All-Russian Olympiad
In every cell of a table with $n$ rows and ten columns, a digit is written. It is known that for every row $A$ and any two columns, you can always find a row that has different digits from $A$ only when it intersects with two columns. Prove that $n\geq512$.
2011 All-Russian Olympiad
Nine quadratics, $x^2+a_1x+b_1, x^2+a_2x+b_2,...,x^2+a_9x+b_9$ are written on the board. The sequences $a_1, a_2,...,a_9$ and $b_1, b_2,...,b_9$ are arithmetic. The sum of all nine quadratics has at least one real root. What is the the greatest possible number of original quadratics that can have no real roots?
2011 All-Russian Olympiad
The graph $G$ is not $3$-coloured. Prove that $G$ can be divided into two graphs $M$ and $N$ such that $M$ is not $2$-coloured and $N$ is not $1$-coloured.



V. Dolnikov
2011 All-Russian Olympiad
Perimeter of triangle ABC is 4. Point X is marked at ray AB and point Y is marked at ray AC such that AX=AY=1. BC intersects XY at point M. Prove that perimeter of one of triangles ABM or ACM is 2.

(V. Shmarov).
2011 All-Russian Olympiad
Given are $10$ distinct real numbers. Kyle wrote down the square of the difference for each pair of those numbers in his notebook, while Peter wrote in his notebook the absolute value of the differences of the squares of these numbers. Is it possible for the two boys to have the same set of $45$ numbers in their notebooks?
2011 All-Russian Olympiad
Given is an acute triangle $ABC$. Its heights $BB_1$ and $CC_1$ are extended past points $B_1$ and $C_1$. On these extensions, points $P$ and $Q$ are chosen, such that angle $PAQ$ is right. Let $AF$ be a height of triangle $APQ$. Prove that angle $BFC$ is a right angle.
2011 All-Russian Olympiad
For positive integers $a>b>1$, define

$$x_n = \frac {a^n-1}{b^n-1}$$

Find the least $d$ such that for any $a,b$, the sequence $x_n$ does not contain $d$ consecutive prime numbers.



V. Senderov
2011 All-Russian Olympiad
A $2010\times 2010$ board is divided into corner-shaped figures of three cells. Prove that it is possible to mark one cell in each figure such that each row and each column will have the same number of marked cells.



I. Bogdanov & O. Podlipsky
2011 All-Russian Olympiad
Two natural numbers $d$ and $d'$, where $d'>d$, are both divisors of $n$. Prove that $d'>d+\frac{d^2}{n}$.
2011 All-Russian Olympiad
On side $BC$ of parallelogram $ABCD$ ($A$ is acute) lies point $T$ so that triangle $ATD$ is an acute triangle. Let $O_1$, $O_2$, and $O_3$ be the circumcenters of triangles $ABT$, $DAT$, and $CDT$ respectively. Prove that the orthocenter of triangle $O_1O_2O_3$ lies on line $AD$.
2011 All-Russian Olympiad
There are 999 scientists. Every 2 scientists are both interested in exactly 1 topic and for each topic there are exactly 3 scientists that are interested in that topic. Prove that it is possible to choose 250 topics such that every scientist is interested in at most 1 theme.



A. Magazinov
2011 All-Russian Olympiad
Ten cars are moving at the road. There are some cities at the road. Each car is moving with some constant speed through cities and with some different constant speed outside the cities (different cars may move with different speed). There are 2011 points at the road. Cars don't overtake at the points. Prove that there are 2 points such that cars pass through these points in the same order.



S. Berlov
2011 All-Russian Olympiad
Given are two distinct monic cubics $F(x)$ and $G(x)$. All roots of the equations $F(x)=0$, $G(x)=0$ and $F(x)=G(x)$ are written down. There are eight numbers written. Prove that the greatest of them and the least of them cannot be both roots of the polynomial $F(x)$.
2011 All-Russian Olympiad
There are more than $n^2$ stones on the table. Peter and Vasya play a game, Peter starts. Each turn, a player can take any prime number less than $n$ stones, or any multiple of $n$ stones, or $1$ stone. Prove that Peter always can take the last stone (regardless of Vasya's strategy).



S Berlov
2011 All-Russian Olympiad
Let $P(a)$ be the largest prime positive divisor of $a^2 + 1$. Prove that exist infinitely many positive integers $a, b, c$ such that $P(a)=P(b)=P(c)$.



A. Golovanov
2011 All-Russian Olympiad
Let $N$ be the midpoint of arc $ABC$ of the circumcircle of triangle $ABC$, let $M$ be the midpoint of $AC$ and let $I_1, I_2$ be the incentres of triangles $ABM$ and $CBM$. Prove that points $I_1, I_2, B, N$ lie on a circle.



M. Kungojin
2010 All-Russian Olympiad
There are $24$ different pencils, $4$ different colors, and $6$ pencils of each color. They were given to $6$ children in such a way that each got $4$ pencils. What is the least number of children that you can randomly choose so that you can guarantee that you have pencils of all colors.



P.S. for 10 grade gives same problem with $40$ pencils, $10$ of each color and $10$ children.