Let $ A_1A_2...A_n$ be a convex polygon. Show that there exists an index $ j$ such that the circum-circle of the triangle $ A_j A_{j \plus{} 1} A_{j \plus{} 2}$ covers the polygon (here indices are read modulo n).

A point $K$ is taken inside parallelogram $ABCD$ so that the midpoint of $AD$ is equidistant from $K$ and $C$, and the midpoint of $CD$ is equidistant form $K$ and $A$. Let $N$ be the midpoint of $BK$. Prove that the angles $NAK$ and $NCK$ are equal.

Find all prime numbers $ a,b,c$ that fulfill the equality: $ (a\minus{}2)!\plus{}2b!\equal{}22c\minus{}1$

Let $f,g,h : \mathbb R \to \mathbb R$ be continuous functions and \(X\) be a random variable such that $E(g(X)h(X))=0$ and $E(g(X)^2) \neq 0 \neq E(h(X)^2)$. Prove that $$E(f(X)^2) \geq \frac{E(f(X)g(X))^2}{E(g(X)^2)} + \frac{E(f(X)h(X))^2}{E(h(X)^2)}.$$ You may assume that all expected values exist. [i]Proposed by Cristi Calin[/i]

Prove that the product of the 99 numbers $ \frac{k^3\minus{}1}{k^3\plus{}1},k\equal{}2,3,\ldots,100$ is greater than $ 2/3$.

A donkey suffers an attack of hiccups and the first hiccup happens at $\text{4:00}$ one afternoon. Suppose that the donkey hiccups regularly every $5$ seconds. At what time does the donkey’s $\text{700th}$ hiccup occur? $\textbf{(A) }$ $15$ seconds after $\text{4:58}$ $\textbf{(B) }$ $20$ seconds after $\text{4:58}$ $\textbf{(C)}$ $25$ seconds after $\text{4:58}$ $\textbf{(D) }$ $30$ seconds after $\text{4:58}$ $\textbf{(E) }$ $35$ seconds after $\text{4:58}$

Let $ABC$ be a triangle with $|AC| \ne |BC|$. Let $P$ and $Q$ be the intersection points of the line $AB$ with the internal and external angle bisectors at $C$, so that $P$ is between $A$ and $B$. Prove that if $M$ is any point on the circle with diameter $PQ$, then $\angle AMP = \angle BMP$.

A set of points has [i]point symmetry[/i] if a reflection in some point maps the set to itself. Let $\cal P$ be a solid convex polyhedron whose orthogonal projections onto any plane have point symmetry. Prove that $\cal P$ has point symmetry. [i]Proposed by Ethan Tan[/i]

A coin is biased in such a way that on each toss the probability of heads is $\frac{2}{3}$ and the probability of tails is $\frac{1}{3}$. The outcomes of the tosses are independent. A player has the choice of playing Game A or Game B. In Game A she tosses the coin three times and wins if all three outcomes are the same. In Game B she tosses the coin four times and wins if both the outcomes of the first and second tosses are the same and the outcomes of the third and fourth tosses are the same. How do the chances of winning Game A compare to the chances of winning Game B? $\textbf{(A)} \text{ The probability of winning Game A is }\frac{4}{81}\text{ less than the probability of winning Game B.} $ $\textbf{(B)} \text{ The probability of winning Game A is }\frac{2}{81}\text{ less than the probability of winning Game B.}$ $\textbf{(C)} \text{ The probabilities are the same.}$ $\textbf{(D)} \text{ The probability of winning Game A is }\frac{2}{81}\text{ greater than the probability of winning Game B.}$ $\textbf{(E)} \text{ The probability of winning Game A is }\frac{4}{81}\text{ greater than the probability of winning Game B.}$

Prove that there exists an integer $n \geq 1$, such that number of all pairs $(a, b)$ of positive integers, satisfying $$\frac{1}{a-b}-\frac{1}{a}+\frac{1}{b}=\frac{1}{n}$$ exceeds $2024.$

Assume that $ S\equal{}(a_1, a_2, \cdots, a_n)$ consists of $ 0$ and $ 1$ and is the longest sequence of number, which satisfies the following condition: Every two sections of successive $ 5$ terms in the sequence of numbers $ S$ are different, i.e., for arbitrary $ 1\le i<j\le n\minus{}4$, $ (a_i, a_{i\plus{}1}, a_{i\plus{}2}, a_{i\plus{}3}, a_{i\plus{}4})$ and $ (a_j, a_{j\plus{}1}, a_{j\plus{}2}, a_{j\plus{}3}, a_{j\plus{}4})$ are different. Prove that the first four terms and the last four terms in the sequence are the same.

Consider two concentric circles of radius $17$ and $19.$ The larger circle has a chord, half of which lies inside the smaller circle. What is the length of the chord in the larger circle? $\textbf{(A)}\ 12\sqrt{2} \qquad\textbf{(B)}\ 10\sqrt{3} \qquad\textbf{(C)}\ \sqrt{17 \cdot 19} \qquad\textbf{(D)}\ 18 \qquad\textbf{(E)}\ 8\sqrt{6}$

Arjun eats twice as many chocolates as Theo, and Wen eats twice as many chocolates as Arjun. If Arjun eats $6$ chocolates, compute the total number of chocolates that Arjun, Theo, and Wen eat.

Find the number of pairs of integer solutions $(x,y)$ that satisfy the equation \[(x-y+2)(x-y-2) = -(x-2)(y-2)\]

Let $AL$ be the inner bisector of triangle $ABC$. The circle centered at $B$ with radius $BL$ meets the ray $AL$ at points $L$ and $E$, and the circle centered at $C$ with radius $CL$ meets the ray $AL$ at points $L$ and $D$. Show that $AL^2 = AE\times AD$. [i](Proposed by Mykola Moroz)[/i]

Point $P$ is chosen in the plane of triangle $ABC$ such that $\angle{ABP}$ is congruent to $\angle{ACP}$ and $\angle{CBP}$ is congruent to $\angle{CAP}$. Show $P$ is the orthocentre.

We consider tilings of a rectangular $m \times n$-board with $1\times2$-tiles. The tiles can be placed either horizontally, or vertically, but they aren't allowed to overlap and to be placed partially outside of the board. All squares on theboard must be covered by a tile. (a) Prove that for every tiling of a $4 \times 2010$-board with $1\times2$-tiles there is a straight line cutting the board into two pieces such that every tile completely lies within one of the pieces. (b) Prove that there exists a tiling of a $5 \times  2010$-board with $1\times 2$-tiles such that there is no straight line cutting the board into two pieces such that every tile completely lies within one of the pieces.

Triangle $ABC$ is isosceles, and $\angle ABC=x^{\circ}$. If the sum of the possible measures of $\angle BAC$ is $240^{\circ}$, find $x$.

A ship tries to land in the fog. The crew does not know the direction to the land. They see a lighthouse on a little island, and they understand that the distance to the lighthouse does not exceed 10 km (the exact distance is not known). The distance from the lighthouse to the land equals 10 km. The lighthouse is surrounded by reefs, hence the ship cannot approach it. Can the ship land having sailed the distance not greater than 75 km? ([i]The waterside is a straight line, the trajectory has to be given before the beginning of the motion, after that the autopilot navigates the ship[/i].)

Let $c$ be the inscribed circle of the triangle $ABC$, $d$ a line tangent to $c$ which does not pass through the vertices of triangle $ABC$. Prove the existence of points $A_1,B_1, C_1$, respectively, on the lines $BC,CA,AB$ satisfying the following two properties: $(i)$ Lines $AA_1,BB_1$, and $CC_1$ are parallel. $(ii)$ Lines $AA_1,BB_1$, and $CC_1$ meet $d$ respectively at points $A' ,B'$, and $C'$ such that \[\frac{A'A_1}{A' A}=\frac{B'B_1}{B 'B}=\frac{C'C_1}{C'C}\]

If every $k-$element subset of $S=\{1,2,\dots , 32\}$ contains three different elements $a,b,c$ such that $a$ divides $b$, and $b$ divides $c$, $k$ must be at least ? $ \textbf{(A)}\ 17 \qquad\textbf{(B)}\ 24 \qquad\textbf{(C)}\ 25 \qquad\textbf{(D)}\ 29 \qquad\textbf{(E)}\ \text{None} $

Suppose there are $n$ points in a plane no three of which are collinear with the property that if we label these points as $A_1,A_2,\ldots,A_n$ in any way whatsoever, the broken line $A_1A_2\ldots A_n$ does not intersect itself. Find the maximum value of $n$. [i]Dinu Serbanescu, Romania[/i]

Let $x,y,z$ be non-negative real numbers such that $xyz=1$. Prove that $$(x^3+2y)(y^3+2z)(z^3+2x) \ge 27.$$

Is it possible that two cross-sections of a tetrahedron by two different cutting planes are two squares, one with a side of length no greater than $1$ and another with a side of length at least $100$? Mikhail Evdokimov

Let $BB_1$ and $CC_1$ be the altitudes of acute-angled triangle $ABC$, and $A_0$ is the midpoint of $BC$. Lines $A_0B_1$ and $A_0C_1$ meet the line passing through $A$ and parallel to $BC$ in points $P$ and $Q$. Prove that the incenter of triangle $PA_0Q$ lies on the altitude of triangle $ABC$.