Let $O$ denote the origin and let $\gamma$ be the circle with center $(1,0)$ and radius $1$ in the Cartesian system of coordinates. Let $\lambda$ be a real number from the interval $(0,2)$, and let the line $x=\lambda$ intersect the circle $\gamma$ at points $P$ and $Q$. The lines $OP$ and $OQ$ intersect the line $x=2-\lambda$ at the points $P'$ and $Q'$, respectively. Let $\mathcal G$ denote the locus of such points $P'$ and $Q'$ as $\lambda$ varies over the interval $(0,2)$. Prove that there exist points $R$ and $S$ different from the origin in the plane such that for every $A\in \mathcal G$ there exists a point $A'$ on line $OA$ satisfying \[ A'R^2=(A'S-OS)^2=A'A\cdot A'O.\] [i]Proposed by: Áron Bán-Szabó, Budapest[/i]

Inside triangle \(ABC\), point \(P\) is marked. Point \(Q\) is on segment \(AB\), and point \(R\) is on segment \(AC\) such that the circumcircles of triangles \(BPQ\) and \(CPR\) are tangent to line \(AP\). Lines are drawn through points \(B\) and \(C\) passing through the center of the circumcircle of triangle \(BPC\), and through points \(Q\) and \(R\) passing through the center of the circumcircle of triangle \(PQR\). Prove that there exists a circle tangent to all four drawn lines.

Define a sequence $<x_n>$ by $x_0 = 0$ and $$\large x_n = \left\{ \begin{array}{ll} x_{n-1} + \frac{3^r-1}{2} & if \,\,n = 3^{r-1}(3k + 1)\\ & \\ x_{n-1} - \frac{3^r+1}{2} & if \,\, n = 3^{r-1}(3k + 2)\\ \end{array} \right. $$ where $k, r$ are integers. Prove that every integer occurs exactly once in the sequence.

Find all polynomials $ p\in\mathbb Z[x]$ such that $ (m,n)\equal{}1\Rightarrow (p(m),p(n))\equal{}1$

Players Aino and Eino take turns choosing numbers from the set $\{0,..., n\}$ with $n\in \Bbb{N}$ being fixed in advance. The game ends when the numbers picked by one of the players include an arithmetic progression of length $4.$ The one who obtains the progression wins. Prove that for some $n,$ the starter of the game wins. Find the smallest such $n.$

Let $p,q$ be positive integers. For any $a,b\in\mathbb{R}$ define the sets $$P(a)=\bigg\{a_n=a \ + \ n \ \cdot \ \frac{1}{p} : n\in\mathbb{N}\bigg\}\text{ and }Q(b)=\bigg\{b_n=b \ + \ n \ \cdot \ \frac{1}{q} : n\in\mathbb{N}\bigg\}.$$ The [i]distance[/i] between $P(a)$ and $Q(b)$ is the minimum value of $|x-y|$ as $x\in P(a), y\in Q(b)$. Find the maximum value of the distance between $P(a)$ and $Q(b)$ as $a,b\in\mathbb{R}$.

A class of $10$ students took a math test. Each problem was solved by exactly $7$ of the students. If the first nine students each solved $4$ problems, how many problems did the tenth student solve?

Does there exist a sequence $ F(1), F(2), F(3), \ldots$ of non-negative integers that simultaneously satisfies the following three conditions? [b](a)[/b] Each of the integers $ 0, 1, 2, \ldots$ occurs in the sequence. [b](b)[/b] Each positive integer occurs in the sequence infinitely often. [b](c)[/b] For any $ n \geq 2,$ \[ F(F(n^{163})) \equal{} F(F(n)) \plus{} F(F(361)). \]

Several positive integers $a_0 , a_1 , a_2 , ... , a_n$ are written on a board. On a second board, we write the amount $b_0$ of numbers written on the first board, the amount $b_1$ of numbers on the first board exceeding $1$, the amount $b_2$ of numbers greater than $2$, and so on as long as the $b$s are still positive. Then we stop, so that we do not write any zeros. On a third board we write the numbers $c_0 , c_1 , c_2 , ...$. using the same rules as before, but applied to the numbers $b_0 , b_1 , b_2 , ...$ of the second board. Prove that the same numbers are written on the first and the third boards. (H. Lebesgue - A Kanel)

Prove that there is no inclined plane such that any tetrahedron placed arbitrarily with a certain face on the plane will not fall over. It means the following: Given a plane $ \pi $ and a line $ l $ not perpendicular to it. Prove that there is a tetrahedron $ T $ such that for each of its faces $ S $ there is in the plane $ \pi $ a triangle $ ABC $ congruent to $ S $ and there is a point $ D $ such that the tetrahedron $ ABCD $ congruent to $ T $ and the line parallel to $ l $ passing through the center of gravity of the tetrahedron $ ABCD $ does not intersect the triangle $ ABC $. Note. The center of gravity of a tetrahedron is the intersection point of the segments connecting the centers of gravity of the faces of this tetrahedron with the opposite vertices (it is known that such a point always exists).

Pauline Bunyan can shovel snow at the rate of $20$ cubic yards for the first hour, $19$ cubic yards for the second, $18$ for the third, etc., always shoveling one cubic yard less per hour than the previous hour. If her driveway is $4$ yards wide, $10$ yards long, and covered with snow $3$ yards deep, then the number of hours it will take her to shovel it clean is closest to $\text{(A)}\ 4 \qquad \text{(B)}\ 5 \qquad \text{(C)}\ 6 \qquad \text{(D)}\ 7 \qquad \text{(E)}\ 12$

Let $a$, $b$, $c$ be the sides of an acute triangle $\triangle ABC$ , then for any $x, y, z \geq 0$, such that $xy+yz+zx=1$ holds inequality:$$a^2x + b^2y + c^2z \geq 4F$$ where $F$ is the area of the triangle $\triangle ABC$

Let $f_1(x) = \frac{2}{3}-\frac{3}{3x+1}$, and for $n \ge 2$, define $f_n(x) = f_1(f_{n-1} (x))$. The value of x that satisfies $f_{1001}(x) = x - 3$ can be expressed in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

The quadrilateral $ABCD$ is a square of sidelength $1$, and the points $E, F, G, H$ are the midpoints of the sides. Determine the area of quadrilateral $PQRS$. [img]https://1.bp.blogspot.com/--fMGH2lX6Go/XzcDqhgGKfI/AAAAAAAAMXo/x4NATcMDJ2MeUe-O0xBGKZ_B4l_QzROjACLcBGAsYHQ/s0/2000%2BMohr%2Bp1.png[/img]

Let $ABCD$ be a cyclic quadrilateral and let the intersection point of lines $AB$ and $CD$ be $E$. Let the points $K$ and $L$ be arbitrary points on sides $CD$ and $AB$ respectively, which satisfy the conditions $$\angle KAD = \angle KBC \quad \text{and} \quad \angle LDA = \angle LCB.$$ Prove that $EK = EL$.

Ben flips a coin 10 times and then records the absolute difference between the total number of heads and tails he's flipped. He then flips the coin one more time and records the absolute difference between the total number of heads and tails he's flipped. If the probability that the second number he records is greater than the first can be expressed as $\frac{a}{b}$ for positive integers $a,b$ with $\gcd (a,b) = 1$, then find $a + b$. [i]Proposed by Vismay Sharan[/i]

Determine the number of permutations $a_1, a_2, \dots, a_n$ of $1, 2, \dots, n$ such that for every positive integer $k$ with $1 \le k \le n$, there exists an integer $r$ with $0 \le r \le n - k$ which satisfies \[ 1 + 2 + \dots + k = a_{r+1} + a_{r+2} + \dots + a_{r+k}. \]

Two pirates divide the loot, consisting of two bags of coins and a diamond, according to the following rules. First the first pirate takes take a few coins from any bag and transfer them from this bag in the other the same number of coins. Then the second pirate does the same (choosing the bag from which he takes the coins at his discretion) and etc. until you can take coins according to these rules. The pirate who takes the coins last gets the diamond. Who will get the diamond if is each of the pirates trying to get it? Give your answer depending on the initial number of coins in the bags.

How many permutations $a_1, a_2, \ldots, a_n$ of $\{1, 2, . . ., n \}$ are sorted into increasing order by at most three repetitions of the following operation: Move from left to right and interchange $a_i$ and $a_{i+1}$ whenever $a_i > a_{i+1}$ for $i$ running from $1$ up to $n - 1 \ ?$

Let $ABCD$ be a convex cyclic quadilateral. Suppose $P$ is a point in the plane of the quadilateral such that the sum of its distances from the vertices of $ABCD$ is the least. If $$\{PC, PB, PC, PD\} = \{3, 4, 6, 8\}$$, what is the maxumum possible area of $ABCD$?

Let $ r$, $ s$, and $ t$ be the three roots of the equation \[ 8x^3\plus{}1001x\plus{}2008\equal{}0.\]Find $ (r\plus{}s)^3\plus{}(s\plus{}t)^3\plus{}(t\plus{}r)^3$.

Find the number of rearrangements of the nine letters $\text{AAABBBCCC}$ where no three consecutive letters are the same. For example, count $\text{AABBCCABC}$ and $\text{ACABBCCAB}$ but not $\text{ABABCCCBA}.$

Let $x_1, x_2 , x_3, .... x_n$ be $n$ real numbers such that $0 < x_j < \frac{1}{2}$. Prove that \[ \frac{ \prod\limits_{j=1}^{n} x_j } { \left( \sum\limits_{j=1}^{n} x_j \right)^n} \leq \frac{ \prod\limits_{j=1}^{n} (1-x_j) } { \left( \sum\limits_{j=1}^{n} (1 - x_j) \right)^n} \]

Given that the $32$-digit integer $$64 \ 312 \ 311 \ 692 \ 944 \ 269 \ 609 \ 355 \ 712 \ 372 \ 657$$ is the product of $6$ consecutive primes, compute the sum of these $6$ primes.

For each natural number $k$ find the least number $n$ such that in every tournament with $n$ vertices, there exists a vertex with in-degree and out-degree at least $k$. (Tournament is directed complete graph.)