The bisectors $AA_1$ and $CC_1$ of the right triangle $ABC$ ($\angle B = 90^o$) intersect at point $I$. The line passing through the point $C_1$ and perpendicular on the line $AA_1$ intersects the line that passes through $A_1$ and is perpendicular on $CC_1$, at the point $K$. Prove that the midpoint of the segment $KI$ lies on segment $AC$.

Suppose that two circles $\alpha, \beta$ with centers $P,Q$, respectively , intersect orthogonally at $A$,$B$. Let $CD$ be a diameter of $\beta$ that is exterior to $\alpha$. Let $E,F$ be points on $\alpha$ such that $CE,DF$ are tangent to $\alpha$ , with $C,E$ on one side of $PQ$ and $D,F$ on the other side of $PQ$. Let $S$ be the intersection of $CF,AQ$ and $T$ be the intersection of $DE,QB$. Prove that $ST$ is parallel to $CD$ and is tangent to $\alpha$

Let $AD$ be the common chord of two equal-sized circles $O_1$ and $O_2$. Let $B$ and $C$ be points on $O_1$ and $O_2$, respectively, so that $D$ lies on the segment $BC$. Assume that $AB = 15, AD = 13$ and $BC = 18$, what is the ratio between the inradii of $\vartriangle ABD$ and $\vartriangle ACD$?

Find the last five digits of $1^{100}+2^{100}+3^{100}+...+999999^{100}$

Solve the system of equations $\begin{cases} x^3+y = x^2+1\\ 2y^3+z=2y^2+1 \\ 3z^3+x=3z^2+1 \end{cases}$

The tetrahedron $ABCD$ has its vertices on the fixed sphere $S$. Prove that $AB^{2}+AC^{2}+AD^{2}-BC^{2}-BD^{2}-CD^{2}$ is minimum iff $AB\perp AC,AC\perp AD,AD\perp AB$.

An integer $N$ is the product of two consecutive integers. (a) Prove that we can add two digits to the right of this number and obtain a perfect square. (b) Prove that this can be done in only one way if $N > 12$

Find the Betti numbers of the surface $x_1^2+\cdots+x_k^2-y_1^2-\cdots-y_l^2=1$ and the set $x_1^2+\cdots+x_k^2\le1+y_1^2+\cdots+y_l^2$ in a $(k+l)$-dimensional linear space.

In a group of $199$ persons, each person is a friend of exactly $100$ other persons in the group. All friendships are mutual, and we do not count a person as a friend of himself/herself. For which integers $k > 1$ is the existence of $k$ persons, all being friends of each other, guaranteed?

Let $\mathbb{S}$ is the set of prime numbers that less or equal to 26. Is there any $a_1, a_2, a_3, a_4, a_5, a_6 \in \mathbb{N}$ such that $$ gcd(a_i,a_j) \in \mathbb{S} \qquad \text {for } 1\leq i \ne j \leq 6$$ and for every element $p$ of $\mathbb{S}$ there exists a pair of $ 1\leq k \ne l \leq 6$ such that $$s=gcd(a_k,a_l)?$$

We are given $n$ intervals $[l_1,r_1],[l_2,r_2],[l_3,r_3],\dots, [l_n,r_n]$ in the number line. We can divide the intervals into two sets such that no two intervals in the same set have overlaps. Prove that there are at most $n-1$ pairs of overlapping intervals.

The keystone arch is an ancient architectural feature. It is composed of congruent isosceles trapezoids fitted together along the non-parallel sides, as shown. The bottom sides of the two end trapezoids are horizontal. In an arch made with $ 9$ trapezoids, let $ x$ be the angle measure in degrees of the larger interior angle of the trapezoid. What is $ x$? [asy]unitsize(4mm); defaultpen(linewidth(.8pt)); int i; real r=5, R=6; path t=r*dir(0)--r*dir(20)--R*dir(20)--R*dir(0); for(i=0; i<9; ++i) { draw(rotate(20*i)*t); } draw((-r,0)--(R+1,0)); draw((-R,0)--(-R-1,0));[/asy]$ \textbf{(A)}\ 100 \qquad \textbf{(B)}\ 102 \qquad \textbf{(C)}\ 104 \qquad \textbf{(D)}\ 106 \qquad \textbf{(E)}\ 108$

Compute the smallest nonnegative integer that can be written as the sum of 2020 distinct integers.

Let $I$ be the incenter of an acute riangle $ABC$. Let $C_A(A, AI)$ be the circle with center $A$ and radius $AI$. Circles $C_B(B, BI)$, $C_C(C, CI) $ are defined in an analogous way. Let $X, Y, Z$ be the intersection points of $C_B$ with $C_C$, $C_C$ with $C_A$, $C_A$ with $C_B$ respectively (different than $I$) . Show that if the radius of the circle that passes through the points $X, Y, Z$ is equal to the radius of the circle that passes through points $A$, $B$ and $C$ then triangle $ABC$ is equilateral.

Suppose a circle passes through the feet of the symmedians of a non-isosceles triangle $ABC$ , and is tangent to one of the sides. Show that $a^2 +b^2, b^2 + c^2 , c^2 + a^2$ are in geometric progression when taken in some order

Find the smallest positive integer solution to the equation $2^{2^k}\equiv k\pmod{29}$.

If $a,b,c,$ and $d$ are non-zero numbers such that $c$ and $d$ are the solutions of $x^2+ax+b=0$ and $a$ and $b$ are the solutions of $x^2+cx+d=0$, then $a+b+c+d$ equals $\textbf{(A) }0\qquad\textbf{(B) }-2\qquad\textbf{(C) }2\qquad\textbf{(D) }4\qquad \textbf{(E) }(-1+\sqrt{5})/2$

In a triangle $ABC$, let $D$, $E$ and $F$ be the feet of the altitudes from $A$, $B$ and $C$ respectively. Let $D'$, $E'$ and $F'$ be the second intersection of lines $AD$, $BE$ and $CF$ with the circumcircle of $ABC$. Show that the triangles $DEF$ and $D'E'F'$ are similar.

Let $a_1, a_2, \dots$ be an arithmetic sequence and $b_1, b_2, \dots$ be a geometric sequence. Suppose that $a_1 b_1 = 20$, $a_2 b_2 = 19$, and $a_3 b_3 = 14$. Find the greatest possible value of $a_4 b_4$.

Let $S = \mathbb Z \times \mathbb Z$. A subset $P$ of $S$ is called [i]nice[/i] if [list] [*] $(a, b) \in P \implies (b, a) \in P$ [*] $(a, b)$, $(c, d)\in P \implies (a + c, b - d) \in P$ [/list] Find all $(p, q) \in S$ so that if $(p, q) \in P$ for some [i]nice[/i] set $P$ then $P = S$.

Find the minimum value attained by $\sum_{m=1}^{100} \gcd(M - m, 400)$ for $M$ an integer in the range $[1746, 2017]$.

$(USS 5)$ Given $5$ points in the plane, no three of which are collinear, prove that we can choose $4$ points among them that form a convex quadrilateral.

We have a polyhedron such that an ant can walk from one vertex to another, traveling only along edges, and traversing every edge exactly once. What is the smallest possible total number of vertices, edges, and faces of this polyhedron?

Let $G$ be a complete directed graph with $2024$ vertices and let $k \leq 10^5$ be a positive integer. Angel and Boris play the following game: Angel colors $k$ of the edges in red and puts a pawn in one of the vertices. After that in each move, first Angel moves the pawn to a neighboring vertex and then Boris has to flip one of the non-colored edges. Boris wins if at some point Angel can't make a move. Find, depending on $G$ and $k$, whether or not Boris has a winning strategy.

Principal Skinner is thinking of two integers $m$ and $n$ and bets Superintendent Chalmers that he will not be able to determine these integers with a single piece of information. Chalmers asks Skinner the numerical value of $mn+13m+13n-m^2-n^2$. From the value of this expression alone, he miraculously determines both $m$ and $n$. What is the value of the above expression.