Two circles $\gamma_1$ and $\gamma_2$ intersect at $A$ and $B$. A line $r$ through $B$ meets $\gamma_1$ at $C$ and $\gamma_2$ at $D$ so that $B$ is between $C$ and $D$. Let $s$ be the line parallel to $AD$ which is tangent to $\gamma_1$ at $E$, at the smaller distance from $AD$. Line $EA$ meets $\gamma_2$ in $F$. Let $t$ be the tangent to $\gamma_2$ at $F$. $(a)$ Prove that $t$ is parallel to $AC$. $(b)$ Prove that the lines $r,s,t$ are concurrent.

Let $s(n)$ denote the number of positive divisors of positive integer $n$. What is the largest prime divisor of the sum of numbers $(s(k))^3$ for all positive divisors $k$ of $2014^{2014}$? $ \textbf{(A)}\ 5 \qquad\textbf{(B)}\ 7 \qquad\textbf{(C)}\ 11 \qquad\textbf{(D)}\ 13 \qquad\textbf{(E)}\ \text{None of the preceding} $

Three squares $BCDE,CAFG$ and $ABHI$ are constructed outside the triangle $ABC$. Let $GCDQ$ and $EBHP$ be parallelograms. Prove that $APQ$ is an isosceles right triangle.

Is there a heptagon and a point $P$ inside it such that any vertex of the heptagon has its distance from $P$ equal to the length of the side opposite the vertex? [i]A side and a vertex are said to be opposite if the side is the fourth from the vertex page (in any direction).[/i]

Let $a$ and $b$ be the roots of the polynomial $x^2+2020x+c$. Given that $\frac{a}{b}+\frac{b}{a}=98$, compute $\sqrt c$.

Let $n$ be a positive integer. Determine the number of sequences $a_0, a_1, \ldots, a_n$ with terms in the set $\{0,1,2,3\}$ such that $$n=a_0+2a_1+2^2a_2+\ldots+2^na_n.$$

Let $n$ be a positive square free integer, $S$ is a subset of $[n]:=\{1,2,\ldots ,n\}$ such that $|S|\ge n/2.$ Prove that there exists three elements $a,b,c\in S$ (can be same), satisfy $ab\equiv c\pmod n.$ [i]Created by Zhenhua Qu[/i]

Prove that for any triangle each exradius is less than four times the circumradius.

$200$ natural numbers are written in a row. For any two adjacent numbers of the row, the right one is either $9$ times greater than the left one, $2$ times smaller than the left one. Can the sum of all these 200 numbers be equal to $24^{2022}$?

Let $O$ be the circumcenter of the triangle $ABC$. Points $D,E$ are on sides $AC,AB$ and points $P,Q,R,S$ are given in plane such that $P,C$ and $R,C$ are on different sides of $AB$ and pints $Q,B$ and $S,B$ are on different sides of $AC$ such that $R,S$ lie on circumcircle of $DAP,EAQ$ and $\triangle BCE \sim \triangle ADQ , \triangle CBD \sim \triangle AEP$(In that order), $\angle ARE=\angle ASD=\angle BAC$, If $RS\| PQ$ prove that $RE ,DS$ are concurrent on $AO$. [i]Proposed by Alireza Dadgarnia[/i]

If $ a$, $ b$, $ c$, $ d$, and $ e$ are constants such that every $ x > 0$ satisfies \[ \frac{5x^4 \minus{} 8x^3 \plus{} 2x^2 \plus{} 4x \plus{} 7}{(x \plus{} 2)^4} \equal{} a \plus{} \frac{b}{x \plus{} 2} \plus{} \frac{c}{(x \plus{} 2)^2} \plus{} \frac{d}{(x \plus{} 2)^3} \plus{} \frac{e}{(x \plus{} 2)^4} \, ,\] then what is the value of $ a \plus{} b \plus{} c \plus{} d \plus{} e$?

Square $ABCD$ has side length $1$. Point $P$, $Q$, $R$, and $S$ each lie on a side of $ABCD$ such that $APQCRS$ is an equilateral convex hexagon with side length $s$. What is $s$? $\textbf{(A) } \frac{\sqrt{2}}{3} \qquad \textbf{(B) } \frac{1}{2} \qquad \textbf{(C) } 2-\sqrt{2} \qquad \textbf{(D) } 1-\frac{\sqrt{2}}{4} \qquad \textbf{(E) } \frac{2}{3}$

a) A castle is equilateral triangle with the side of $100$ metres. It is divided onto $100$ triangle rooms. Each wall between the rooms is $10$ metres long and contain one door. You are inside and are allowed to pass through every door not more than once. Prove that you can visit not more than $91$ room (not exiting the castle). b) Every side of the triangle is divided onto $k$ parts by the lines parallel to the sides. And the triangle is divided onto $k^2$ small triangles. Let us call the "chain" such a sequence of triangles, that every triangle in it is included only once, and the consecutive triangles have the common side. What is the greatest possible number of the triangles in the chain?

Find integer solutions of the equation $x^3 - 2y^3 - 4z^3 = 0$.

Let $$ A = \log (1) + \log 2 + \log(3) + \cdots + \log(2023) $$ and $$ B = \log(1/1) + \log(1/2) + \log(1/3) + \cdots + \log(1/2023). $$ What is the value of $A + B\ ?$ $($logs are logs base $10)$ $$ \mathrm a. ~ 0\qquad \mathrm b.~1\qquad \mathrm c. ~{-\log(2023!)} \qquad \mathrm d. ~\log(2023!) \qquad \mathrm e. ~{-2023} $$

The bisector of angle $BAD$ in the parallelogram $ABCD$ intersects the lines $BC$ and $CD$ at the points $K$ and $L$ respectively. It is known that $ABCD$ is not a rhombus. Prove that the centre of the circle passing through the points $C, K$ and $L$ lies on the circle passing through the points $B, C$ and $D$.

A clock has an hour, minute, and second hand, all of length $1$. Let $T$ be the triangle formed by the ends of these hands. A time of day is chosen uniformly at random. What is the expected value of the area of $T$? [i]Proposed by Dylan Toh[/i]

For which values of $n$ does there exist a convex polyhedron with $n$ edges?

From the symmetry center of two congruent intersecting circles, two rays are drawn that intersect the circles at four non-collinear points. Prove that these points lie on one circle.

Let $I$ be the incenter of a triangle $ABC$ with the property $\angle ABC - \angle BAC=30^{\circ}$. Line $CI$ intersects the circumcircle of $ABC$ at $C_1$. It turned out that $C_1$ lies on a common tangent line of circumcircles of triangles $ABC$ and $BCI$. Find the angles of triangle $ABC$.

Prove that among all possible triangles whose vertices are $3,5$ and $7$ apart from a given point $P$, the ones with the largest perimeter have $P$ as incentre.

We will assume that the sides of a square are reflective and we will designate them with the names of the four cardinal points. Marking a point on the side $N$ , determine in which direction a ray of light should exit (into the interior of the square) so that it returns to it after having undergone $n$ reflections on the side $E$ , another $n$ on the side $W$ , $m$ on the $S$ and $m - 1$ on the $N$, where $n$ and $m$ are known natural numbers. What happens if m and $n$ are not prime to each other? Calculate the length of the light ray considered as a function of $m$ and $n$, and of the length of the side of the square.

An infinite set $S$ of positive numbers is called thick, if in every interval of the form $\left [1/(n+1),1/n\right]$ (where $n$ is an arbitrary positive integer) there is a number which is the difference of two elements from $S$. Does there exist a thick set such that the sum of its elements is finite? Proposed by [i]Gábor Szűcs[/i], Szikszó

The six faces of a hexahedron are quadrilaterals. Prove that if seven its vertices lie on a sphere, then the eighth vertex also lies on the sphere.

Find [the decimal form of] the largest prime divisor of $100111011_6$.