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?

Let $x_1$ and $x_2$ be positive integers. On a straight line, $y_1$ white segments and $y_2$ black segments are given, with $y_1 \ge x_1$ and $y_2 \ge x_2$. Suppose that no two segments of the same colour intersect (and do not have common ends). Moreover, suppose that for any choice of $x_1$ white segments and $x_2$ black segments, some pair of selected segments will intersect. Prove that $(y_1-x_1)(y_2-x_2)<x_1x_2$. [i]Proposed by G. Chelnokov[/i]

Let $x_1\leq x_2\leq x_3\leq x_4$ be real numbers. Prove that there exist polynomials of degree two $P(x)$ and $Q(x)$ with real coefficients such that $x_1$, $x_2$, $x_3$ and $x_4$ are the roots of $P(Q(x))$ if and only if $x_1+x_4=x_2+x_3$.

Consider a triangle and 2 lines that each go through a corner and intersects the opposing segment, such that the areas are as on the attachment. Find the "?"

Pentagon $ABCDE$ has circle $S$ inscribed into it. Side $BC$ is tangent to $S$ at point $K$. If $AB=BC=CD$, prove that angle $EKB$ is a right angle.

If $n$ is a positive integer and $a, b$ are relatively prime positive integers, calculate $(a + b,a^n + b^n)$.

The isosceles right triangle $ABC$ has right angle at $C$ and area $12.5$. The rays trisecting $\angle{ACB}$ intersect $AB$ at $D$ and $E$. What is the area of $\triangle{CDE}$? $\textbf{(A) }\frac{5\sqrt{2}}{3}\qquad\textbf{(B) }\frac{50\sqrt{3}-75}{4}\qquad\textbf{(C) }\frac{15\sqrt{3}}{8}\qquad\textbf{(D) }\frac{50-25\sqrt{3}}{2}\qquad\textbf{(E) }\frac{25}{6}$

Let $XYZ$ be a right triangle of area $1$ m$^2$ . Consider the triangle $X'Y'Z'$ such that $X'$ is the symmetric of X wrt side $YZ$, $Y'$ is the symmetric of $Y$ wrt side $XZ$ and $Z' $ is the symmetric of $Z$ wrt side $XY$. Calculate the area of the triangle $X'Y'Z'$.

What’s the smallest integer $n>1$ such that $p \mid \left(n^{p-1}-1\right)$ for all integers $2 \leq p \leq 10?$ [i]Individual #6[/i]

Let $ABCD$ be a square and let the points $M$ on $BC$, $N$ on $CD$, $P$ on $DA$, be such that $\angle (AB,AM)=x,\angle (BC,MN)=2x,\angle (CD,NP)=3x$. 1) Show that for any $0\le x\le 22.5$, such a configuration uniquely exists, and that $P$ ranges over the whole segment $DA$; 2) Determine the number of angles $0\le x\le 22.5$ for which$\angle (DA,PB)=4x$. (Dan Schwarz)

Let $a_1,a_2,\cdots ,a_{10}$ be pairwise distinct natural numbers with their sum equal to 1995. Find the minimal value of $a_1a_2+a_2a_3+\cdots +a_9a_{10}+a_{10}a_1$.

Find all functions $f:\mathbb{R} \to \mathbb{R}$ satisfying \[xf(f(x)+y)=f(xy)+x^2\] for all $x,y \in \mathbb{R}$.

Professor Piraldo takes part in soccer matches with a lot of goals and judges a match in his own peculiar way. A match with score of $m$ goals to $n$ goals, $m\geq n$, is [i]tough[/i] when $m\leq f(n)$, where $f(n)$ is defined by $f(0) = 0$ and, for $n \geq 1$, $f(n) = 2n-f(r)+r$, where $r$ is the largest integer such that $r < n$ and $f(r) \leq n$. Let $\phi ={1+\sqrt 5\over 2}$. Prove that a match with score of $m$ goals to $n$, $m\geq n$, is tough if $m\leq \phi n$ and is not tough if $m \geq \phi n+1$.