Alison has an analog clock whose hands have the following lengths: $a$ inches (the hour hand), $b$ inches (the minute hand), and $c$ inches (the second hand), with $a < b < c$. The numbers $a$, $b$, and $c$ are consecutive terms of an arithmetic sequence. The tips of the hands travel the following distances during a day: $A$ inches (the hour hand), $B$ inches (the minute hand), and $C$ inches (the second hand). The numbers $A$, $B$, and $C$ (in this order) are consecutive terms of a geometric sequence. What is the value of $\frac{B}{A}$?

Define sequence $\{a_n\}$ as following: $a_0=0$, $a_1=1$, and $a_{i}=2a_{i-1}-a_{i-2}+2$ for all $i\geq 2$. Determine the value of $a_{1000}.$ [i] Proposed by Yannick Yao [/i]

Each of the numbers $1, 2, 3,... 25$ is arranged in a $5$ by $5$ table. In each row they appear in increasing order (left to right). Find the maximal and minimal possible sum of the numbers in the third column. (Folklore)

Triangle $ABC$ is assumed. The point $T$ is the second intersection of the symmedian of vertex $A$ with the circumcircle of the triangle $ABC$ and the point $D \neq A$ lies on the line $AC$ such that $BA=BD$. The line that at $D$ tangents to the circumcircle of the triangle $ADT$, intersects the circumcircle of the triangle $DCT$ for the second time at $K$. Prove that $\angle BKC = 90^{\circ}$(The symmedian of the vertex $A$, is the reflection of the median of the vertex $A$ through the angle bisector of this vertex).

Find all real numbers $x$ satisfying the equation \[\left\lfloor \frac{20}{x+18}\right\rfloor+\left\lfloor \frac{x+18}{20}\right\rfloor=1.\]

When a positive integer $x$ is divided by a positive integer $y$, the quotient is $u$ and the remainder is $v$, where $u$ and $v$ are integers. What is the remainder when $x+2uy$ is divided by $y$? ${{ \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 2u \qquad\textbf{(C)}\ 3u \qquad\textbf{(D)}\ v }\qquad\textbf{(E)}\ 2v } $

Find the geometric locus of the center of a rectangle whose vertices lie on the perimeter of a given triangle.

Three real numbers $x$, $y$, and $z$ are chosen independently and uniformly at random from the interval $[0,1]$. Compute the probability that $x$, $y$, and $z$ can be the side lengths of a triangle.

In a magic square $n \times n$ composed from the numbers $1,2,\cdots,n^2$, the centers of any two squares are joined by a vector going from the smaller number to the bigger one. Prove that the sum of all these vectors is zero. (A magic square is a square matrix such that the sums of entries in all its rows and columns are equal.)

The exact quantity of gas needed for a car to complete a single loop around a track is distributed among $n$ containers placed along the track. Prove that there exists a position starting at which the car, beginning with an empty tank of gas, can complete a loop around the track without running out of gas. The tank of gas is assumed to be large enough.

Which configurations of five (not necessarily distinct) points $p_1 ,\ldots, p_5$ on the circle $x^2 +y^2 =1$ maximize the sum of the ten distances $$\sum_{i<j} d(p_i, p_j)?$$

At a mathematical competition $n$ students work on $6$ problems each one with three possible answers. After the competition, the Jury found that for every two students the number of the problems, for which these students have the same answers, is $0$ or $2$. Find the maximum possible value of $n$.

A right-angled triangle has perimeter $60$ and the altitude of the hypotenuse has a length $12$. Determine the lengths of the sides.

Given a piece of graph paper with a letter assigned to each vertex of every square such that on every segment connecting two vertices that have the same letter and are on the same line of the mesh, there is at least one vertex with another letter. What is the least number of distinct letters needed to plot such a picture, along the sides of the cells?

In the Cartesian plane $XOY$, there is a rhombus $ABCD$ whose side lengths are all $4$ and $|OB|=|OD|=6$, where $O$ is the origin. [b](1)[/b] Prove that $|OA|\cdot |OB|$ is a constant. [b](2)[/b] Find the locus of $C$ if $A$ is a point on the semicircle \[(x-2)^2+y^2=4 \quad (2\le x\le 4).\]

Alice has four boxes, $327$ blue balls, and $2022$ red balls. The blue balls are labeled $1$ to $327$. Alice first puts each of the balls into a box, possibly leaving some boxes empty. Then, a random label between $1$ and $327$ (inclusive) is selected, Alice finds the box the ball with the label is in, and selects a random ball from that box. What is the maximum probability that she selects a red ball?

Given a function $g:[0,1] \to \mathbb{R}$ satisfying the property that for every non empty dissection of the trivial $[0,1]$ to subsets $A,B$ we have either $\exists x \in A; g(x) \in B$ or $\exists x \in B; g(x) \in A$ and we have furthermore $g(x)>x$ for $x \in [0,1]$. Prove that there exist infinite $x \in [0,1]$ with $g(x)=1$. [i]Proposed by Ali Zamani [/i]

Triangle $ABC$ has incenter $I$ and excircles $\Omega_A$, $\Omega_B$, and $\Omega_C$. Let $\ell_A$ be the line through the feet of the tangents from $I$ to $\Omega_A$, and define lines $\ell_B$ and $\ell_C$ similarly. Prove that the orthocenter of the triangle formed by lines $\ell_A$, $\ell_B$, and $\ell_C$ coincides with the Nagel point of triangle $ABC$. (The Nagel point of triangle $ABC$ is the intersection of segments $AT_A$, $BT_B$, and $CT_C$, where $T_A$ is the tangency point of $\Omega_A$ with side $BC$, and points $T_B$ and $T_C$ are defined similarly.) Proposed by [i]Nikolai Beluhov[/i], Bulgaria

Let $ABC$ be an acute-angled triangle with circumcircle $\omega$. A circle $\Gamma$ is internally tangent to $\omega$ at $A$ and also tangent to $BC$ at $D$. Let $AB$ and $AC$ intersect $\Gamma$ at $P$ and $Q$ respectively. Let $M$ and $N$ be points on line $BC$ such that $B$ is the midpoint of $DM$ and $C$ is the midpoint of $DN$. Lines $MP$ and $NQ$ meet at $K$ and intersect $\Gamma$ again at $I$ and $J$ respectively. The ray $KA$ meets the circumcircle of triangle $IJK$ again at $X\neq K$. Prove that $\angle BXP = \angle CXQ$. [i]Kian Moshiri, United Kingdom[/i]

In $\triangle{PQR}$, $PQ=4$, $PR=5$, and $QR=6$. Assume that an equilateral hexagon $ABCDEF$ is able to be drawn inside $\triangle{PQR}$ so that $\overline{AB}$ is parallel to $\overline{QR}$, $\overline{CD}$ is parallel to $\overline{PQ}$, $\overline{EF}$ is parallel to $\overline{RP}$, $\overline{BC}$ lies on $\overline{RP}$, $\overline{DE}$ lies on $\overline{QR}$, and $\overline{AF}$ lies on $\overline{PQ}$. Find the area of hexagon $ABCDEF$.

Brackets are to be inserted into the expression $10 \div 9 \div 8 \div 7 \div 6 \div 5 \div 4 \div 3 \div 2$ so that the resulting number is an integer. (a) Determine the maximum value of this integer. (b) Determine the minimum value of this integer.

The circumcenter of an acute-triangle $ABC$ with $|AB|<|BC|$ is $O$, $D$ and $E$ are midpoints of $|AB|$ and $|AC|$, respectively. $OE$ intersects $BC$ at $K$, the circumcircle of $OKB$ intersects $OD$ second time at $L$. $F$ is the foot of altitude from $A$ to line $KL$. Show that the point $F$ lies on the line $DE$

We say that a point inside a triangle is good if the lengths of the cevians passing through this point are inversely proportional to the respective side lengths. Find all the triangles for which the number of good points is maximal. (A.Zaslavsky, B.Frenkin)

Suppose $a$ is a real number such that $3a + 6$ is the greatest integer less than or equal to $a$ and $4a + 9$ is the least integer greater than or equal to $a$. Compute $a$.

Integers $n$ and $k$ are given, with $n\ge k\ge2$. You play the following game against an evil wizard. The wizard has $2n$ cards; for each $i=1,\ldots,n$, there are two cards labeled $i$. Initially, the wizard places all cards face down in a row, in unknown order. You may repeatedly make moves of the following form: you point to any $k$ of the cards. The wizard then turns those cards face up. If any two of the cards match, the game is over and you win. Otherwise, you must look away, while the wizard arbitrarily permutes the $k$ chosen cards and then turns them back face-down. Then, it is your turn again. We say this game is [i]winnable[/i] if there exist some positive integer $m$ and some strategy that is guaranteed to win in at most $m$ moves, no matter how the wizard responds. For which values of $n$ and $k$ is the game winnable?