On a circumference, points $A$ and $B$ are on opposite arcs of diameter $CD$. Line segments $CE$ and $DF$ are perpendicular to $AB$ such that $A-E-F-B$ (i.e., $A$, $E$, $F$ and $B$ are collinear on this order). Knowing $AE=1$, find the length of $BF$.

Let $E(n)$ denote the sum of the even digits of $n$. For example, $E(5681) = 6+8 = 14$. Find $E(1) + E(2) + E(3) + \cdots + E(100)$. $\text{(A)}\ 200 \qquad \text{(B)}\ 360 \qquad \text{(C)}\ 400 \qquad \text{(D)}\ 900 \qquad \text{(E)}\ 2250$

Find the number of two-digit positive integers whose digits total $7$. $\textbf{(A)}\ 6\qquad \textbf{(B)}\ 7\qquad \textbf{(C)}\ 8\qquad \textbf{(D)}\ 9\qquad \textbf{(E)}\ 10$

The convex polygon $A_1A_2...A_n$ is given in the plane. Denote by $T_k$ $(k \le n)$ the convex $k$-gon of the largest area, with vertices at the points $A_1, A_2, ..., A_n$ and by $T_k(A+1)$ the convex k-gon of the largest area with vertices at the points $A_1, A_2, ..., A_n$ in which one of the vertices is in $A_1$. Set the relationship between the order of arrangement in the sequence $A_1, A_2, ..., A_n$ vertices: 1) $T_3$ and $T_3 (A_2)$ 2) $T_k$ and $T_k (A_1) $ 3) $T_k$ and $T_{k+1}$

Consider 2009 cards which are lying in sequence on a table. Initially, all cards have their top face white and bottom face black. The cards are enumerated from 1 to 2009. Two players, Amir and Ercole, make alternating moves, with Amir starting. Each move consists of a player choosing a card with the number $k$ such that $k < 1969$ whose top face is white, and then this player turns all cards at positions $k,k+1,\ldots,k+40.$ The last player who can make a legal move wins. (a) Does the game necessarily end? (b) Does there exist a winning strategy for the starting player? [i]Also compare shortlist 2009, combinatorics problem C1.[/i]

250 numbers are chosen among positive integers not exceeding 501. Prove that for every integer $ t$ there are four chosen numbers $ a_1$, $ a_2$, $ a_3$, $ a_4$, such that $ a_1 \plus{} a_2 \plus{} a_3 \plus{} a_4 \minus{} t$ is divisible by 23. [i]Author: K. Kokhas[/i]

$n$ is a natural number, $f_n(x)=\frac{x^{n+1}-x^{-n-1}}{x-x^{-1}}(x\neq0,\pm1)$, let $y=x+\frac{1}{x}$. [b](a)[/b] Prove that $f_{n+1}(x)=yf_n(x)-f_{n-1}(x)$ [b](b)[/b] Prove with mathematical induction: $f_n(x)=\begin{cases} y^n-\text{C}_{n-1}^{1}y^{n-2}+\cdots+(-1)^i\text{C}_{n-i}^{i}y^{n-2i}+\cdots+(-1)^{\frac{n}{2}}(i=1,2,\cdots,\frac{n}{2},n\text{ is even})\\ y^n-\text{C}_{n-1}^{1}y^{n-2}+\cdots+(-1)^i\text{C}_{n-i}^{i}y^{n-2i}+\cdots+(-1)^{\frac{n-1}{2}}\text{C}_{\frac{n+1}{2}}^{\frac{n-1}{2}}y(i=1,2,\cdots,\frac{n-1}{2},n\text{ is odd}) \end{cases}$.

$\text{ }$ [asy] unitsize(11); for(int i=0; i<6; ++i) { if(i<5) draw( (i, 0)--(i,5) ); else draw( (i, 0)--(i,2) ); if(i < 3) draw((0,i)--(5,i)); else draw((0,i)--(4,i)); } [/asy] We are dividing the above figure into parts with shapes: [asy] unitsize(11); draw((0,0)--(0,2)); draw((1,0)--(1,2)); draw((2,1)--(2,2)); draw((0,0)--(1,0)); draw((0,1)--(2,1)); draw((0,2)--(2,2)); [/asy][asy] unitsize(11); draw((0,0)--(0,2)); draw((1,0)--(1,2)); draw((2,1)--(2,2)); draw((3,1)--(3,2)); draw((0,0)--(1,0)); draw((0,1)--(3,1)); draw((0,2)--(3,2)); [/asy] After that division, find the number of [asy] unitsize(11); draw((0,0)--(0,2)); draw((1,0)--(1,2)); draw((2,1)--(2,2)); draw((0,0)--(1,0)); draw((0,1)--(2,1)); draw((0,2)--(2,2)); [/asy] shaped parts.

Given a convex quadrilateral $ABCD$. Let $R_a, R_b, R_c$ and $R_d$ be the circumradii of triangles $DAB, ABC, BCD, CDA$. Prove that inequality $R_a < R_b < R_c < R_d$ is equivalent to $180^o - \angle CDB < \angle CAB < \angle CDB$ . (O.Musin)

Let $I$ be the center of the inscribed circle of $ABC$ and let $I_A$ be the center of the exscribed circle touching the side $BC$. Let $M$ be the midpoint of the side $BC$, and $N$ be the midpoint of the arc $BAC$ of the circumscribed circle of $ABC$ . The point $T$ is symmetric to the point $N$ wrt point $A$. Prove that the points $I_A,M,I,T$ lie on the same circle. (Danilo Hilko)

Consider the following operation. Given a positive integer $n$, if $n$ is a multiple of $3$, then you replace $n$ by $\dfrac{n}3$. If $n$ is not a multiple of $3$, then you replace $n$ by $n + 10$. Then continue this process. For example, beginning with $n = 4$, this procedure gives $4 \to 14 \to 24 \to 8 \to 18 \to 6 \to 2 \to 12 \to \cdots$. Suppose you start with $n = 100$. What value results if you perform this operation exactly $100$ times? $\textbf{(A) }10\qquad\textbf{(B) }20\qquad\textbf{(C) }30\qquad\textbf{(D) }40\qquad\textbf{(E) }50$

For a certain triangle all of its altitudes are integers whose sum is less than 20. If its Inradius is also an integer Find all possible values of area of the triangle.

positive reals $a_1, a_2, . . . $ satisfying (i) $a_{n+1}=a_1^2\cdot a_2^2 \cdot . . . \cdot a_n^2-3$(all positive integers $n$) (ii) $\frac{1}{2}(a_1+\sqrt{a_2-1})$ is positive integer. prove that $\frac{1}{2}(a_1 \cdot a_2 \cdot . . . \cdot a_n + \sqrt{a_{n+1}-1})$ is positive integer

On a rectangular board with $m \times n$ squares ($m, n \ge 3$) there are dominoes ($2 \times 1$ or $1\times 2$ tiles), which do not overlap and do not extend beyond the board. Every domino covers exactly two squares of the board. Assume that the dominos cover the has the property that no more dominos can be added to the board and that the four corner spaces of the board are not all empty. Prove that at least $2/3$ of the squares of the board are covered with dominos.

Find the maximal number of edges a connected graph $G$ with $n$ vertices may have, so that after deleting an arbitrary cycle, $G$ is not connected anymore.

Given a triangle $ABC$ with $AD$ is the $A-$symmedian $(D$ is on the side $BC).$ Prove that: $\dfrac{DB}{DC}=\dfrac{AB^2}{AC^2}.$

Find the maximum possible value of the inradius of a triangle whose vertices lie in the interior, or on the boundary, of a unit square.

Source: 2017 Canadian Open Math Challenge, Problem B2 ----- There are twenty people in a room, with $a$ men and $b$ women. Each pair of men shakes hands, and each pair of women shakes hands, but there are no handshakes between a man and a woman. The total number of handshakes is $106$. Determine the value of $a \cdot b$.

Given a triangle $ABC$. On its sides $BC$ , $CA$ and $AB$ , the points $A_1$ , $B_1$ and $C_1$ are taken, respectively , such that $2 \angle B_1 A_1 C_1 + \angle BAC = 180^o$ , $2 \angle A_1 C_1 B_1 + \angle ACB = 180^o$ , $2 \angle C_1 B_1 A_1 + \angle CBA = 180^o$ . Find the locus of the centers of the circles circumscribed about the triangles $A_1 B_1 C_1$ (all possible such triangles are considered).

Given a triangle $ABC$, the perpendicular bisector of the side $AC$ intersects the angle bisector of the triangle $AK$ at the point $P$, $M$ - such a point that $\angle MAC = \angle PCB$, $\angle MPA = \angle CPK$, and points $M$ and $K$ lie on opposite sides of the line $AC$. Prove that the line $AK$ bisects the segment $BM$. (Anton Trygub)

$3n$ ($n \ge 2, n \in N$) people attend a gathering, in which any two acquaintances have exactly $n$ common acquaintances, and any two unknown people have exactly $2n$ common acquaintances. If three people know each other, it is called a [i]Taoyuan Group[/i]. (1) Find the number of all Taoyuan groups; (2) Prove that these $3n$ people can be divided into three groups, with $n$ people in each group, and the three people obtained by randomly selecting one person from each group constitute a Taoyuan group. Note: Acquaintance means that two people know each other, otherwise they are not acquaintances. Two people who know each other are called acquaintances.

If $\sin x+\cos x=1/5$ and $0\le x<\pi$, then $\tan x$ is $\textbf{(A) }-\frac{4}{3}\qquad\textbf{(B) }-\frac{3}{4}\qquad\textbf{(C) }\frac{3}{4}\qquad\textbf{(D) }\frac{4}{3}\qquad$ $\textbf{(E) }\text{not completely determined by the given information}$

Let $ABC$ be a non-isosceles triangle with incenter $I$. Let $D$ be a point on the segment $BC$ such that the circumcircle of $BID$ intersects the segment $AB$ at $E\neq B$, and the circumcircle of $CID$ intersects the segment $AC$ at $F\neq C$. The circumcircle of $DEF$ intersects $AB$ and $AC$ at the second points $M$ and $N$ respectively. Let $P$ be the point of intersection of $IB$ and $DE$, and let $Q$ be the point of intersection of $IC$ and $DF$. Prove that the three lines $EN, FM$ and $PQ$ are parallel. [i]Proposed by Saudi Arabia[/i]

Let $ABCD$ be a circumscribed quadrilateral with incircle $\gamma$. Let $AB\cap CD=E, AD\cap BC=F, AC\cap EF=K, BD\cap EF=L$. Let a circle with diameter $KL$ intersect $\gamma$ at one of the points $X$. Prove that $(EXF)$ is tangent to $\gamma$.

Let $ABC$ be an isosceles triangle with $AB = AC$ and let $n$ be a natural number, $n>1$. On the side $AB$ we consider the point $M$ such that $n \cdot AM = AB$. On the side $BC$ we consider the points $P_1, P_2, ....., P_ {n-1}$ such that $BP_1 = P_1P_2 = .... = P_ {n-1} C = \frac{1}{n} BC$. Show that: $\angle {MP_1A} + \angle {MP_2A} + .... + \angle {MP_ {n-1} A} = \frac{1} {2} \angle {BAC}$.