Navvye/Olympiad-Problems-CSV huggingface dataset:
2019 Gulf Math Olympiad
Let $ABCD$ be a trapezium (trapezoid) with $AD$ parallel to $BC$  and $J$ be the intersection of the diagonals $AC$ and $BD$. Point $P$ a chosen on the side $BC$ such that the distance from $C$ to the line $AP$ is equal to the distance from $B$ to the line $DP$.



The following three questions 1,  2 and 3 are independent, so that a condition in one question does not apply in another question.



1.Suppose that  $Area( \vartriangle AJB) =6$ and that $Area(\vartriangle BJC) = 9$. Determine $Area(\vartriangle APD)$.

2. Find all points $Q$ on the plane of the trapezium such that  $Area(\vartriangle AQB) = Area(\vartriangle DQC)$.

3.	Prove that $PJ$ is the angle bisector of $\angle APD$.
2019 Gulf Math Olympiad
1. Find $N$, the smallest positive  multiple of $45$ such that all of its digits are either $7$ or $0$.

2. Find $M$, the smallest positive multiple of $32$ such that all of its digits are either $6$ or $1$.

3. How many elements of the set $\{1,2,3,...,1441\}$ have a positive multiple such that all of its digits are either $5$ or $2$?
2019 Gulf Math Olympiad
Consider the set $S = \{1,2,3, ...,1441\}$.

1.	Nora counts thoses subsets of $S$ having exactly two elements, tbe sum of which is even. Rania counts those subsets of $S$ having exactly two elements, the sum of which is odd. Determine the numbers counted by Nora and Rania.

2.	Let $t$ be the number of subsets of $S$ which have at least two elements and the product of the elements is even. Determine the greatest power of $2$ which divides $t$.

3.	Ahmad counts the subsets of $S$ having $77$ elements such that in each subset the sum of the elements is even. Bushra counts the subsets of $S$ having $77$ elements such that in each subset the sum of the elements is odd. Whose number is bigger? Determine the difference between the numbers found by Ahmad and Bushra.
2019 Gulf Math Olympiad
Consider the sequence $(a_n)_{n\ge 1}$ defined by $a_n=n$ for $n\in \{1,2,3.4,5,6\}$, and for $n \ge 7$: $$a_n={\lfloor}\frac{a_1+a_2+...+a_{n-1}}{2}{\rfloor}$$where ${\lfloor}x{\rfloor}$ is the greatest integer less than or equal to $x$. For example : ${\lfloor}2.4{\rfloor} = 2,  {\lfloor}3{\rfloor} = 3$ and ${\lfloor}\pi {\rfloor}= 3$.



For all integers $n \ge 2$, let $S_n = \{a_1,a_1,...,a_n\}- \{r_n\}$ where $r_n$ is the remainder when $a_1 + a_2 + ... + a_n$ is divided by $3$. The minus $-$  denotes the ''remove it if it is there'' notation. For example : $S_4 = {2,3,4}$ because $r_4= 1$ so $1$ is removed from $\{1,2,3,4\}$. However $S_5=  \{1,2,3,4,5\}$ betawe $r_5 = 0$ and $0$ is not in the set $\{1,2,3,4,5\}$.

1.	Determine $S_7,S_8,S_9$ and $S_{10}$.

2.	We say that a set $S_n$ for $n\ge 6$   is well-balanced if it can be partitioned into three pairwise disjoint subsets with equal sum. For example : $S_6 = \{1,2,3,4,5,6\} =\{1,6\}\cup \{2,5\}\cup \{3,4\}$ and $1 +6 = 2 + 5 = 3 + 4$. Prove that $S_7,S_8,S_9$ and $S_{10}$  are well-balanced .

3.	Is the set $S_{2019}$ well-balanced? Justify your answer.
2017 Gulf Math Olympiad
1- Find a pair $(m,n)$ of positive integers such that $K = |2^m-3^n|$ in all of this cases :



$a) K=5$

$b) K=11$

$c) K=19$



2-Is there a pair $(m,n)$ of positive integers such that : $$|2^m-3^n| = 2017$$3-Every prime number less than $41$ can be represented in the form $|2^m-3^n|$ by taking an Appropriate pair $(m,n)$

of positive integers. Prove that the number $41$ cannot be represented in the form $|2^m-3^n|$ where $m$ and $n$ are positive integers



4-Note that $2^5+3^2=41$ . The number $53$ is the least prime number that cannot be represented as a sum or an difference of a power of $2$ and a power of $3$ . Prove that the number $53$ cannot be represented in any of the forms $2^m-3^n$ , $3^n-2^m$ , $2^m-3^n$ where $m$ and $n$ are positive integers
2017 Gulf Math Olympiad
One country consists of islands $A_1,A_2,\cdots,A_N$,The ministry of transport decided to build some bridges such that anyone will can travel by car from any of the islands $A_1,A_2,\cdots,A_N$ to any another island by one or more of these bridges. For technical reasons the only bridges that can be built is between $A_i$ and $A_{i+1}$ where $i = 1,2,\cdots,N-1$ , and between $A_i$ and $A_N$ where $i<N$.



We say that a plan to build some bridges is good if it is satisfies the above conditions , but when we remove any bridge it will not satisfy this conditions. We assume that there is $a_N$ of good plans. Observe that $a_1 = 1$ (The only good plan is to not build any bridge) , and $a_2 = 1$ (We build one bridge).



1-Prove that $a_3 = 3$

2-Draw at least $5$ different good plans in the case that $N=4$ and the islands are the vertices of a square

3-Compute $a_4$

4-Compute $a_6$

5-Prove that there is a positive integer $i$ such that $1438$ divides $a_i$
2017 Gulf Math Olympiad
Let $C_1$ and $C_2$ be two different circles , and let their radii be $r_1$ and $r_2$ , the two circles are passing through the two points $A$ and $B$

(i)Let $P_1$ on $C_1$ and $P_2$ on $C_2$ such that the line $P_1P_2$ passes through $A$. Prove that $P_1B \cdot  r_2 = P_2B \cdot r_1$



(ii)Let $DEF$ be a triangle that it's inscribed in $C_1$ , and let $D'E'F'$ be a triangle that is inscribed in $C_2$ . The lines $EE'$,$DD'$ and $FF'$ all pass through $A$ . Prove that the triangles $DEF$ and $D'E'F'$ are similar



(iii)The circle $C_3$ also passes through $A$ and $B$ . Let $l$ be a line that passes through $A$ and cuts circles $C_i$ in $M_i$ with $i = 1,2,3$ . Prove that the value of$$\frac{M_1M_2}{M_1M_3}$$is constant regardless of the position of $l$ Provided that $l$ is different from $AB$
2017 Gulf Math Olympiad
1 - Prove that  $55 < (1+\sqrt{3})^4 < 56$ .



2 - Find the largest power of $2$ that divides $\lceil(1+\sqrt{3})^{2n}\rceil$ for the positive integer $n$
2016 Gulf Math Olympiad
Consider sequences $a_0$,$a_1$,$a_2$,$\cdots$ of non-negative integers defined by selecting any $a_0$,$a_1$,$a_2$ (not all 0) and for each $n$ $\geq$ 3 letting

$a_n$ = |$a_n-1$ - $a_n-3$|



1-In the particular case that $a_0$ = 1,$a_1$ = 3 and $a_2$ = 2, calculate the beginning of the sequence, listing

$a_0$,$a_1$,$\cdots$,$a_{19}$,$a_{20}$.



2-Prove that for each sequence, there is a constant $c$ such that $a_i$ $\leq$ $c$ for all $i$ $\geq$ 0. Note that the constant $c$ my depend on the numbers $a_0$,$a_1$ and $a_2$



3-Prove that, for each choice of $a_0$,$a_1$ and $a_2$, the resulting sequence is eventually periodic.



4-Prove that, the minimum length p of the period described in (3) is the same for all permitted starting values

$a_0$,$a_1$,$a_2$ of the sequence
2016 Gulf Math Olympiad
Let $x$ be a real number that satisfies $x^1 + x^{-1} = 3$

Prove that $x^n + x^{-n}$ is an positive integer , then prove that the positive integer $x^{3^{1437}}+x^{3^{-1437}}$ is divisible by at least $1439 \times 2^{1437}$ positive integers
2016 Gulf Math Olympiad
Consider the acute-angled triangle $ABC$. Let $X$ be a point on the side

$BC$, and $Y$ be a point on the side $CA$. The circle $k_1$ with diameter $AX$ cuts

$AC$ again at $E'$ .The circle $k_2$ with diameter $BY$ cuts $BC$ again at $B'$.

(i) Let $M$ be the midpoint of $XY$ . Prove that $A'M = B'M$.

(ii) Suppose that $k_1$ and $k_2$ meet at $P$ and $Q$. Prove that the orthocentre

of $ABC$ lies on the line $PQ$.
2016 Gulf Math Olympiad
4. Suppose that four people A, B, C and D decide to play games of tennis

doubles. They might first play the team A and B against the team C

and D. Next A and C might play B and D. Finally A and D might play

B and C. The advantage of this arrangement is that two conditions are

satisfied.

(a) Each player is on the same team as each other player exactly once.

(b) Each player is on the opposing team to each other player exactly

twice.

Is it possible to arrange a collection of tennis matches satisfying both

condition (a) and condition (b) in the following circumstances?

(i) There are five players.

(ii) There are seven players.

(iii) There are nine players.
2015 Gulf Math Olympiad
a) Suppose that $n$ is an odd integer. Prove that $k(n-k)$ is divisible by $2$ for all positive integers $k$.



b) Find an integer $k$ such that $k(100-k)$ is not divisible by $11$.



c) Suppose that $p$ is an odd prime, and $n$ is an integer.

Prove that there is an integer $k$ such that $k(n-k)$ is not divisible by $p$.



d) Suppose that $p,q$ are two different odd primes, and $n$ is an integer.

Prove that there is an integer $k$ such that $k(n-k)$ is not divisible by any of $p,q$.
2015 Gulf Math Olympiad
a) Let $UVW$ , $U'V'W'$  be two triangles such that $ VW = V'W' , UV = U'V' , \angle WUV = \angle W'U'V'.$

Prove that the angles $\angle VWU , \angle V'W'U'$ are equal or supplementary.



b) $ABC$ is a triangle where $\angle A$ is obtuse. take a point $P$ inside the triangle , and extend $AP,BP,CP$ to meet  the sides $BC,CA,AB$ in $K,L,M$ respectively. Suppose that $PL = PM .$

1)  If $AP$ bisects $\angle A$ , then prove that $AB = AC$ .

2) Find the angles of the triangle $ABC$ if you know that $AK,BL,CM$ are angle bisectors of the triangle $ABC$ and that $2AK = BL$.
2015 Gulf Math Olympiad
We have a large supply of black, white, red and green hats.

And we want to give $8$ of these hats to $8$ students that are sitting around a round table.

Find the number of ways of doing that in each of these cases (assuming for the purposes of this problem that students will notchange their places, and that hats of the same color are identical)

a) Each hat to be used must be either red or green.

b) Exactly two hats of each color are to be used

c) Exactly two hats of each color are to be used, and every two hats of the same color are to be given to two adjacent students.

d) Exactly two hats of each color are to be used, and no two hats of the same color are to be given to two adjacent students.

e) There are no restrictions on the number of hats of each color that are to be used, but no two hats of the same color are to be given to two adjacent students.
2015 Gulf Math Olympiad
a) We have a geometric sequence of $3$ terms. If the sum of these terms is $26$ , and their sum of squares is $364$ , find the terms of the sequence.



b) Suppose that $a,b,c,u,v,w$ are positive real numbers , and each of $a,b,c$ and $u,v,w$ are geometric sequences. Suppose also that $a+u,b+v,c+w$ are an arithmetic sequence. Prove that $a=b=c$ and $u=v=w$



c) Let $a,b,c,d$ be real numbers (not all zero), and let $f(x,y,z)$ be the polynomial in three variables defined by$$f(x,y,z) = axyz + b(xy + yz + zx) + c(x+y+z) + d$$.Prove that $f(x,y,z)$ is reducible if and only if $a,b,c,d$ is a geometric sequence.
2014 Gulf Math Olympiad
A sequence $a_0,a_1,a_2,\cdots$ satisfies the conditions $a_0 = 0$ , $a_{n-1}^2 - a_{n-1} = a_n^2 + a_n$

1) determine the two possible values of $a_1$ . then determine all possible values of $a_2$ .

2)for each $n$, prove that $a_{n+1}=a_n+1$ or $a_{n+1} = -a_n$

3)Describe the possible values of $a_{1435}$

4)Prove that the values that you got in (3) are correct
2014 Gulf Math Olympiad
Ahmad and Salem play the following game. Ahmad writes two integers (not necessarily different) on a board. Salem writes their sum and product. Ahmad does the same thing: he writes the sum and product of the two numbers which Salem has just written.

They continue in this manner, not stopping unless the two players write the same two numbers one after the other (for then they are stuck!). The order of the two numbers which each player writes is not important.

Thus if Ahmad starts by writing $3$ and $-2$, the first five moves (or steps) are as shown:

(a)	Step 1 (Ahmad) $3$ and $-2$

(b)	Step 2 (Salem) $1$ and $-6$

(c)	Step 3 (Ahmad) $-5$ and $-6$

(d)	Step 4 (Salem) $-11$ and $30$

(e)	Step 5 (Ahmad) $19$ and $-330$



(i)	Describe all pairs of numbers that Ahmad could write, and ensure that Salem must write the same numbers, and so the game stops at step 2.

(ii)	What pair of integers should Ahmad write so that the game finishes at step 4?

(iii)	Describe all pairs of integers which Ahmad could write at step 1, so that the game will finish after finitely many steps.

(iv)	Ahmad and Salem decide to change the game. The first player writes three numbers on the board, $u, v$ and $w$. The second player then writes the three numbers $u + v + w,uv + vw + wu$ and $uvw$, and they proceed as before, taking turns, and using this new rule describing how to work out the next three numbers. If Ahmad goes first, determine all collections of three numbers which he can write down, ensuring that Salem has to write the same three numbers at the next step.
2014 Gulf Math Olympiad
(i) $ABC$ is a triangle with a right angle at $A$, and $P$ is a point on the hypotenuse $BC$.

The line $AP$ produced beyond $P$ meets the line through $B$ which is perpendicular to $BC$ at $U$.

Prove that $BU = BA$ if, and only if, $CP = CA$.

(ii) $A$ is a point on the semicircle $CB$, and points $X$ and $Y$ are on the line segment $BC$.

The line $AX$, produced beyond $X$, meets the line through $B$ which is perpendicular to $BC$ at $U$.

Also the line $AY$, produced beyond $Y$, meets the line through $C$ which is perpendicular to $BC$ at $V$.

Given that $BY = BA$ and $CX = CA$, determine the angle $\angle VAU$.
2014 Gulf Math Olympiad
The numbers from $1$ to $64$ must be written on the small squares of a chessboard, with a different number in each small square. Consider the $112$ numbers you can make by adding the numbers in two small squares which have a common edge. Is it possible to write the numbers in the squares so that these $112$ sums are all different?
2013 Gulf Math Olympiad
Let $a_1,a_2,\ldots,a_{2n}$ be positive real numbers such that $a_ja_{n+j}=1$ for the values $j=1,2,\ldots,n$.



a. Prove that either the average of the numbers

$a_1,a_2,\ldots,a_n$ is at least 1 or the average of

the numbers $a_{n+1},a_{n+2},\ldots,a_{2n}$ is at least 1.



b. Assuming that $n\ge2$, prove that there exist two distinct numbers $j,k$ in the set $\{1,2,\ldots,2n\}$ such that

$$|a_j-a_k|<\frac{1}{n-1}.$$
2013 Gulf Math Olympiad
In triangle $ABC$, the bisector of angle $B$ meets the opposite side $AC$ at $B'$. Similarly, the bisector

of angle $C$ meets the opposite side $AB$ at $C'$ . Prove that $A=60^{\circ}$ if, and only if, $BC'+CB'=BC$.
2013 Gulf Math Olympiad
There are $n$ people standing on a circular track. We want to perform a number of moves so that we end up with a situation where the distance between every two neighbours is the same. The move that is allowed consists in selecting two people and asking one of them to walk a distance $d$ on the circular track clockwise, and asking the other to walk the same distance on the track anticlockwise. The two people selected and the quantity $d$ can vary from move to move.



Prove that it is possible to reach the desired situation (where the distance between every two neighbours is the same) after at most $n-1$ moves.
2013 Gulf Math Olympiad
Let $m,n$ be integers. It is known that there are integers $a,b$ such that $am+bn=1$ if, and only if, the greatest common divisor of $m,n$ is 1. You are not required to prove this.



Now suppose that $p,q$ are different odd primes. In each case determine if there are integers $a,b$ such that $ap+bq=1$ so that the given condition is satisfied:



a.

$p$ divides $b$ and $q$ divides $a$;

b. $p$ divides $a$ and $q$ divides $b$;

c. $p$ does not divide $a$ and $q$ does not divide $b$.
2012 Gulf Math Olympiad
Let $X,\ Y$ and $Z$ be the midpoints of sides $BC,\ CA$, and $AB$ of the triangle $ABC$, respectively. Let $P$ be a point inside the triangle. Prove that the quadrilaterals $AZPY,\ BXPZ$, and $CYPX$ have equal areas if, and only if, $P$ is the centroid of $ABC$.
2012 Gulf Math Olympiad
Prove that if $a, b, c$ are positive real numbers, then the least possible value of $$6a^3 + 9b^3 + 32c^3 + \frac{1}{4abc}$$

is $6$. For which values of $a, b$ and $c$ is equality attained?
2012 Gulf Math Olympiad
Consider a $3\times7$ grid of squares. Each square may be coloured green or white.



(a) Is it possible to find a colouring so that no subrectangle has all four corner squares of the same colour?

(b) Is it possible for a

$4\times 6$ grid?





Subrectangles must have their corners at grid-points of the original diagram. The corner squares of a subrectangle must be different. The original diagram is a subrectangle of itself.
2012 Gulf Math Olympiad
Fawzi cuts a spherical cheese completely into (at least three) slices of equal thickness. He starts at one end, making successive parallel cuts, working through the cheese until the slicing is complete. The discs exposed by the first two cuts have integral areas.



(i) Prove that all the discs that he cuts have integral areas.

(ii) Prove that the original sphere had integral surface area if, and only if, the area of the second disc that he exposes is even.
2009 Hungary-Israel Binational
For a given prime $ p > 2$ and positive integer $ k$ let $$ S_k = 1^k + 2^k + \ldots + (p - 1)^k$$ Find those values of $ k$ for which $ p \, |\, S_k$.
2009 Hungary-Israel Binational
Denote the three real roots of the cubic $ x^3 - 3x - 1 = 0$ by $ x_1$, $ x_2$, $ x_3$ in order of increasing magnitude. (You may assume that the equation in fact has three distinct real roots.) Prove that $ x_3^2 - x_2^2 = x_3 - x_1$.
2009 Hungary-Israel Binational
Does there exist a pair $ (f; g)$ of strictly monotonic functions, both from $ \mathbb{N}$ to $ \mathbb{N}$, such that $$ f(g(g(n))) < g(f(n))$$ for every $ n \in\mathbb{N}$?
2009 Hungary-Israel Binational
Given is the convex quadrilateral $ ABCD$. Assume that there exists a point $ P$ inside the quadrilateral for which the triangles $ ABP$ and $ CDP$ are both isosceles right triangles with the right angle at the common vertex $ P$. Prove that there exists a point $ Q$ for which the triangles $ BCQ$ and $ ADQ$ are also isosceles right triangles with the right angle at the common vertex $ Q$.
2009 Hungary-Israel Binational
Let $ x$, $ y$ and $ z$ be non negative numbers. Prove that $$ \frac{x^2+y^2+z^2+xy+yz+zx}{6}\le \frac{x+y+z}{3}\cdot\sqrt{\frac{x^2+y^2+z^2}{3}}$$
2009 Hungary-Israel Binational
(a) Do there exist 2009 distinct positive integers such that their sum is divisible by each of the given numbers?



(b) Do there exist 2009 distinct positive integers such that their sum is divisible by the sum of any two of the given numbers?
2008 Hungary-Israel Binational
Find the largest value of n, such that there exists a polygon with n sides, 2 adjacent sides of length 1, and all his diagonals have an integer length.
2008 Hungary-Israel Binational
For every natural number $ t$, $ f(t)$ is the probability that if a fair coin is tossed $ t$ times, the number of times we get heads is 2008 more than the number of tails. What is the value of $ t$ for which $ f(t)$ attains its maximum? (if there is more than one, describe all of them)
2008 Hungary-Israel Binational
Prove that: $ \sum_{i=1}^{n^2} \lfloor \frac{i}{3} \rfloor= \frac{n^2(n^2-1)}{6}$

For all $ n \in N$.
2008 Hungary-Israel Binational
The sequence $ a_n$ is defined as follows: $ a_0=1, a_1=1, a_{n+1}=\frac{1+a_{n}^2}{a_{n-1}}$.

Prove that all the terms of the sequence are integers.
2008 Hungary-Israel Binational
P and Q are 2 points in the area bounded by 2 rays, e and f, coming out from a point O. Describe how to construct, with a ruler and a compass only, an isosceles triangle ABC, such that his base AB is on the ray e, the point C is on the ray f, P is on AC, and Q on BC.
2007 Hungary-Israel Binational
You have to organize a fair procedure to randomly select someone  from $ n$ people  so that every one of them would be chosen with the probability $ \frac{1}{n}$. You are allowed to choose two real numbers $ 0<p_1<1$ and $ 0<p_2<1$ and order two coins which satisfy the following requirement: the probability of tossing "heads" on the first coin $ p_1$ and the probability of tossing "heads" on the second coin is $ p_2$. Before starting the procedure, you are supposed to announce an upper bound on the total number of times that the two coins are going to be flipped altogether. Describe a procedure that achieves this goal under the given conditions.
2007 Hungary-Israel Binational
Let $ a,b,c,d$ be real numbers, such that $ a^2\le 1, a^2 + b^2\le 5, a^2 + b^2 + c^2\le 14, a^2 + b^2 + c^2 + d^2\le 30$. Prove that $ a + b + c + d\le 10$.
2007 Hungary-Israel Binational
Let $ AB$ be the diameter of a given circle with radius $ 1$ unit, and let $ P$ be a given point on $ AB$. A line through $ P$ meets the circle at points $ C$ and $ D$, so a convex quadrilateral $ ABCD$ is formed. Find the maximum possible area of the quadrilateral.
2007 Hungary-Israel Binational
A given rectangle $ R$ is divided into $mn$ small rectangles by straight lines parallel to its sides. (The distances between the parallel lines may not be equal.) What is the minimum number of appropriately selected rectangles’ areas that should be known in order to determine the area of $ R$?
2007 Hungary-Israel Binational
Given is an ellipse $ e$ in the plane. Find the locus of all points $ P$ in space such that the cone of apex $ P$ and directrix $ e$ is a right circular cone.
2007 Hungary-Israel Binational
Let $ t \ge 3$  be a given real number and assume that the polynomial $ f(x)$ satisfies $|f(k)-t^k|<1$, for $ k=0,1,2,\ldots ,n$. Prove that the degree of $f(x)$ is at least $n$.
2006 Hungary-Israel Binational
If natural numbers $ x$, $ y$, $ p$, $ n$, $ k$ with $ n > 1$ odd and $ p$ an odd prime satisfy $ x^n + y^n = p^k$, prove that $ n$ is a power of $ p$.
2006 Hungary-Israel Binational
A block of size $ a\times b\times c$ is composed of $ 1\times 1\times 2$ domino blocks. Assuming that each of the three possible directions of domino blocks occurs equally many times, what are the possible values of $ a$, $ b$, $ c$?
2006 Hungary-Israel Binational
Let $ \mathcal{H} = A_1A_2\ldots A_n$ be a convex $ n$-gon. For $ i = 1, 2, \ldots, n$, let $ A'_{i}$ be the point symmetric to $ A_i$ with respect to the midpoint of $ A_{i - 1}A_{i + 1}$ (where $ A_{n + 1} = A_1$). We say that the vertex $ A_i$ is good if $ A'_{i}$ lies inside $ \mathcal{H}$. Show that at least $ n - 3$ vertices of $ \mathcal{H}$ are good.
2006 Hungary-Israel Binational
A point $ P$ inside a circle is such that there are three chords of the same length passing through $ P$. Prove that $ P$ is the center of the circle.
2006 Hungary-Israel Binational
If $ x$, $ y$, $ z$ are nonnegative real numbers with the sum $ 1$, find the maximum value of $ S = x^2(y + z) + y^2(z + x) + z^2(x + y)$ and $ C = x^2y + y^2z + z^2x$.
2006 Hungary-Israel Binational
A group of $ 100$ students numbered $ 1$ through $ 100$ are playing the following game. The judge writes the numbers $ 1$, $ 2$, $ \ldots$, $ 100$ on $ 100$ cards, places them on the table in an arbitrary order and turns them over. The students $ 1$ to $ 100$ enter the room one by one, and each of them flips $ 50$ of the cards. If among the cards flipped by student $ j$ there is card $ j$, he gains one point. The flipped cards are then turned over again. The students cannot communicate during the game nor can they see the cards flipped by other students. The group wins the game if each student gains a point. Is there a strategy giving the group more than $ 1$ percent of chance to win?
2005 Hungary-Israel Binational
Squares $ABB_{1}A_{2}$ and $BCC_{1}B_{2}$ are externally drawn on the hypotenuse $AB$ and on the leg $BC$ of a right triangle $ABC$ . Show that the lines $CA_{2}$ and $AB_{2}$ meet on the perimeter of a square with the vertices on the perimeter of triangle $ABC .$
2005 Hungary-Israel Binational
Let $f$ be an increasing mapping from the family of subsets of a given ﬁnite set $H$ into itself, i.e. such that for every $X \subseteq Y\subseteq H$  we have $f (X )\subseteq f (Y )\subseteq H .$  Prove that there exists a subset $H_{0}$ of $H$ such that $f (H_{0}) = H_{0}.$
2005 Hungary-Israel Binational
Find all sequences $x_{1},x_{2},...,x_{n}$ of distinct positive integers such that

$\frac{1}{2}=\sum_{i=1}^{n}\frac{1}{x_{i}^{2}}$.
2005 Hungary-Israel Binational
Does there exist a sequence of $2005$ consecutive positive integers that

contains exactly $25$ prime numbers?