Given a line $\ell$ and a ray $p$ on a plane with its origin on this line. Two fixed circles (not necessarily equal) are constructed, inscribed in the two formed angles. On ray $p$, point $A$ is taken so that the tangents from $A$ to the given circles, different from $p$, intersect line $\ell$ at points $B$ and $C$, and at the same time triangle $ABC$ contains the given circles. Find the locus of the centers of the circles inscribed in triangle $ABC$ (as $A$ moves).

Given real numbers $x,y,z,t\in (0,\pi /2]$ such that $$\cos^2 (x)+\cos^2 (y) +\cos^2 (z) +\cos^2 (t)=1.$$ What is the minimum possible value of $$\cot (x) +\cot (y) +\cot (z) +\cot (t)?$$

In the isosceles triangle $ABC$, $ (AB = BC)$ the bisector $AD$ was drawn, and in the triangle $ABD$ the bisector $DE$ was drawn. Find the values of the angles of the triangle $ABC$, if it is known that the bisectors of the angles $ABD$ and $AED$ intersect on the line $AD$. (Fedak Ivan)

Determine all pairs $(p, q)$ of primes for which $p^{q+1}+q^{p+1}$ is a perfect square.

Suppose that the function $ g : (0,1) \rightarrow \mathbb{R}$ can be uniformly approximated by polynomials with nonnegative coefficients. Prove that $ g$ must be analytic. Is the statement also true for the interval $ (\minus{}1,0)$ instead of $ (0,1)$? [i]J. Kalina, L. Lempert[/i]

A square matrix is sum-magic if the sum of all elements in each row, column and major diagonal is constant. Similarly, a square matrix is product-magic if the product of all elements in each row, column and major diagonal is constant. Determine if there exist $3\times 3$ matrices of real numbers which are both sum-magic and product-magic.

Let $k$ and $n$ be integers with $1\leq k<n$. Alice and Bob play a game with $k$ pegs in a line of $n$ holes. At the beginning of the game, the pegs occupy the $k$ leftmost holes. A legal move consists of moving a single peg to any vacant hole that is further to the right. The players alternate moves, with Alice playing first. The game ends when the pegs are in the $k$ rightmost holes, so whoever is next to play cannot move and therefore loses. For what values of $n$ and $k$ does Alice have a winning strategy?

An equilateral triangle is inscribed in the ellipse whose equation is $x^2+4y^2=4.$ One vertex of the triangle is $(0,1),$ one altitude is contained in the $y$-axis, and the length of each side is $\sqrt{\frac mn},$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$

For all positive integers $n$ and $k$, define $F(n,k) = \sum_{r = 1}^n r^{2k - 1}$. Prove that $F(n,1)$ divides $F(n,k)$.

[b]3.[/b] Let $n$ be a positive integer having at least one prime factor with expoente $\geq 2$. Show that $n$ has as many factorizations into an odd number of factors as into an even number of factors. (Factorizations into the same factors arranged in different order are considered different.)[b](N. 10)[/b]

Find the least positive integer $k$ such that when $\frac{k}{2023}$ is written in simplest form, the sum of the numerator and denominator is divisible by $7$. [i]Proposed byMuztaba Syed[/i]

A rectangular floor that is $10$ feet wide and $17$ feet long is tiled with $170$ one-foot square tiles. A bug walks from one corner to the opposite corner in a straight line. Including the first and the last tile, how many tiles does the bug visit? $\textbf{(A) } 17 \qquad\textbf{(B) } 25 \qquad\textbf{(C) } 26 \qquad\textbf{(D) } 27 \qquad\textbf{(E) } 28$

A circle with center $O$ is drawn in the first quadrant of the 2D Cartesian plane (the quadrant with both positive $x$ and $y$ values) such that it lies tangent to the $x$ and $y$-axes. A line is drawn with slope $m>1$ and passing through the origin; the line intersects the circle at two points $A$ and $B$, with $A$ closer to the origin than $B$. Suppose that $ABO$ is an equilateral triangle. Compute $m$.

Wanda the Worm likes to eat Pascal's triangle. One day, she starts at the top of the triangle and eats $\textstyle\binom{0}{0}=1$. Each move, she travels to an adjacent positive integer and eats it, but she can never return to a spot that she has previously eaten. If Wanda can never eat numbers $a,b,c$ such that $a+b=c$, prove that it is possible for her to eat 100,000 numbers in the first 2011 rows given that she is not restricted to traveling only in the first 2011 rows. (Here, the $n+1$st row of Pascal's triangle consists of entries of the form $\textstyle\binom{n}{k}$ for integers $0\le k\le n$. Thus, the entry $\textstyle\binom{n}{k}$ is considered adjacent to the entries $\textstyle\binom{n-1}{k-1}$, $\textstyle\binom{n-1}{k}$, $\textstyle\binom{n}{k-1}$, $\textstyle\binom{n}{k+1}$, $\textstyle\binom{n+1}{k}$, $\textstyle\binom{n+1}{k+1}$.) [i]Linus Hamilton.[/i]

Let $f: Z^+ \to Z^+ \cup \{0\}$ a function that meets the following conditions: a) $f (a b) = f (a) + f (b)$, b) $f (a) = 0$ provided that the digits of the unit of $a$ are $7$, c) $f (10) = 0$. Find $f (2016).$

Given $s,t$ are non-zero integers, $(x,y) $ is an integer pair , A transformation is to change pair $(x,y)$ into pair $(x+t,y-s)$ . If the two integers in a certain pair becoems relatively prime after several tranfomations , then we call the original integer pair "a good pair" . (1) Is $(s,t)$ a good pair ? (2) Prove :for any $s$ and $t$ , there exists pair $(x,y)$ which is " a good pair".

$(SWE 6)$ Prove that there are infinitely many positive integers that cannot be expressed as the sum of squares of three positive integers.

Juan and Mary play a two-person game in which the winner gains 2 points and the loser loses 1 point. If Juan won exactly 3 games and Mary had a final score of 5 points, how many games did they play? $\textbf{(A)}\ 7 \qquad \textbf{(B)}\ 8 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 5 \qquad \textbf{(E)}\ 11$

Consider a triangle $ABC$ and two congruent triangles $A_1B_1C_1$ and $A_2B_2C_2$ which are respectively similar to $ABC$ and inscribed in it: $A_i,B_i,C_i$ are located on the sides of $ABC$ in such a way that the points $A_i$ are on the side opposite to $A$, the points $B_i$ are on the side opposite to $B$, and the points $C_i$ are on the side opposite to $C$ (and the angle at A are equal to angles at $A_i$ etc.). The circumcircles of $A_1B_1C_1$ and $A_2B_2C_2$ intersect at points $P$ and $Q$. Prove that the line $PQ$ passes through the orthocenter of $ABC$.

Given a rectangle $ABCD$. Points $E$ and $F$ lie on sides $BC$ and $CD$ respectively so that the area of triangles $ABE$, $ECF$, $FDA$ is equal to $1$. Determine the area of triangle $AEF$.

The centers $O_1$; $O_2$; $O_3$ of three nonintersecting circles of equal radius are positioned at the vertices of a triangle. From each of the points $O_1$; $O_2$; $O_3$ one draws tangents to the other two given circles. It is known that the intersection of these tangents form a convex hexagon. The sides of the hexagon are alternately colored red and blue. Prove that the sum of the lengths of the red sides equals the sum of the lengths of the blue sides. [i]D. Tereshin[/i]

Let $P$ be a non-constant polynomial with integer coefficients such that if $n$ is a perfect power, so is $P(n)$. Prove that $P(x) = x$ or $P$ is a perfect power of a polynomial with integer coefficients. A perfect power is an integer $n^k$, where $n \in \mathbb Z$ and $k \ge 2$. A perfect power of a polynomial is a polynomial $P(x)^k$, where $P$ has integer coefficients and $k \ge 2$.

Let $ABC$ be a scalene triangle with incenter $I$ and incircle $\omega$. Let the tangency points of $\omega$ to $BC,AC\text{ and } AB$ be $D,E,F$ respectively. Let the line $EF$ intersect the circumcircle of $ABC$ at the points $G, H$. Assume that $E$ lies between the points $F$ and $G$. Let $\Gamma$ be a circle that passes through $G$ and $H$ and that is tangent to $\omega$ at the point $M$ which lies on different semi-planes with $D$ with respect to the line $EF$. Let $\Gamma$ intersect $BC$ at points $K$ and $L$ and let the second intersection point of the circumcircle of $ABC$ and the circumcircle of $AKL$ be $N$. Prove that the intersection point of $NM$ and $AI$ lies on the circumcircle of $ABC$ if and only if the intersection point of $HB$ and $GC$ lies on $\Gamma$.

Find the number of permutations $(a_1, a_2, . . . , a_{2013})$ of $(1, 2, \dots , 2013)$ such that there are exactly two indices $i \in \{1, 2, \dots , 2012\}$ where $a_i < a_{i+1}$.

We have a number of towns, with bus routes between some of them (each bus route goes from a town to another town without any stops). It is known that you can get from any town to any other by bus (possibly changing buses several times). Mr. Ivanov bought one ticket for each of the bus routes (a ticket allows single travel in either direction, but not returning on the same route). Mr. Petrov bought n tickets for each of the bus routes. Both Ivanov and Petrov started at town A. Ivanov used up all his tickets without buying any new ones and finished his travel at town B. Petrov, after using some of his tickets, got stuck at town X: he can not leave it without buying a new ticket. Prove that X is either A or B.