A circle inscribed in triangle $ABC$ has center $O$ and touches side $AC$ at point $K$. A second circle also has center $O$, intersects all sides of triangle $ABC$. Let $E$ and $F$ be the corresponding points of intersection with sides $AB$ and $BC$, closest to vertex $B$; $B_1$ and $B_2$ are the points of its intersection with side $AC$, and $B_1$ is closer to $A$. Prove that points $B$, $K$ and point $P$, the intersections of the segments $B_2E$ and $B_1F$ lie on the same straight line.

In convex hexagon $AXBYCZ$, sides $AX$, $BY$ and $CZ$ are parallel to diagonals $BC$, $XC$ and $XY$, respectively. Prove that $\triangle ABC$ and $\triangle XYZ$ have the same area. [i]Proposed by Evan Chen[/i]

A geometric sequence consists of $11$ terms. The arithmetic mean of the first $6$ terms is $63$, and the arithmetic mean of the last $6$ terms is $2016$. Find the $7$th term in the sequence. [i]Proposed by Powell Zhang[/i]

In a scalene triangle $ABC$, the points $E$ and $F$ are the foot of altitudes drawn from $B$ and $C$, respectively. The points $X$ and $Y$ are the reflections of the vertices $B$ and $C$ to the line $EF$, respectively. Let the circumcircles of the $\triangle ABC$ and $\triangle AEF$ intersect at $T$ for the second time. Show that the four points $A, X, Y, T$ lie on a single circle.

The width of a long winding river is not greater than $1$ km. This means by definition that from any point of each bank of the river one can reach the other bank swimming $1$ km or less. Is it true that a boat can move along the river so that its distances from both banks are never greater than (a) $0.7$ km? (b) $0.8$ km? (Grigory Kondakov, Moscow) You may assume that the banks consist of segments and arcs of circles.

$A$ is a point lying outside a circle $(O)$. The tangents from $A$ drawn to $(O)$ meet the circle at $B,C.$ Let $P,Q$ be points on the rays $AB, AC$ respectively such that $PQ$ is tangent to $(O).$ The parallel lines drawn through $P,Q$ parallel to $CA, BA,$ respectively meet $BC$ at $E,F,$ respectively. $(a)$ Show that the straight lines $EQ$ always pass through a fixed point $M,$ and $FP$ always pass through a fixed point $N.$ $(b)$ Show that $PM\cdot QN$ is constant.

Let $ABC$ be an acute-angled triangle with $\angle BAC = 60^{\circ}$ and with incenter $I$ and circumcenter $O$. Let $H$ be the point diametrically opposite(antipode) to $O$ in the circumcircle of $\triangle BOC$. Prove that $IH=BI+IC$.

What is the largest amount of complex $ z $ solutions a system can have? $ | z-1 || z + 1 | = 1 $ $ Im (z) = b? $ (where $ b $ is a real constant)

Let be a point $ P $ inside a triangle $ ABC. $ Prove that the following relations are equivalent: $ \text{(i)} $ Any collinear triple of points $ (E,P,F) $ with $ E,F $ on $ AB,AC, $ respectively, verifies the equality $$ \frac{1}{AE} +\frac{1}{AF} =\frac{AB+BC+CA}{AB\cdot AC} $$ $ \text{(ii)} P $ is the incircle of $ ABC $

Victor has four types of coins: gold, silver, bronze, and copper. All coins of the same type have the same weight, which is an integer number of grams. Victor performs two weighings: - He takes 6 gold coins, 13 silver coins, 3 bronze coins, and 7 copper coins, and the total weight on the scale is 162 grams. - He takes 15 gold coins, 5 silver coins, and 11 bronze coins, and the total weight on the scale is 110 grams. Determine the weight of each type of coin.

Find the value of $ a$ for which the area of the figure surrounded by $ y \equal{} e^{ \minus{} x}$ and $ y \equal{} ax \plus{} 3\ (a < 0)$ is minimized.

Is there a function $f$ from the positive integers to themselves such that $$f(a)f(b) \geq f(ab)f(1)$$ with equality [b]if and only if[/b] $(a-1)(b-1)(a-b)=0$?

Compute the sum of the squares of the first $100$ terms of an arithmetic progression, given that their sum is $-1$ and that the sum of those among them having an even index is $1$.

The number of real solutions to the equation \[ \frac{x}{100} = \sin x \] is $\text{(A)} \ 61 \qquad \text{(B)} \ 62 \qquad \text{(C)} \ 63 \qquad \text{(D)} \ 64 \qquad \text{(E)} \ 65$

Find the smallest natural number $n$ that the following statement holds : Let $A$ be a finite subset of $\mathbb R^{2}$. For each $n$ points in $A$ there are two lines including these $n$ points. All of the points lie on two lines.

Let $n \geq 2$ be a positive integer. A total of $2n$ balls are coloured with $n$ colours so that there are two balls of each colour. These balls are put inside $n$ cylindrical boxes with two balls in each box, one on top of the other. Phoe Wa Lone has an empty cylindrical box and his goal is to sort the balls so that balls of the same colour are grouped together in each box. In a [i]move[/i], Phoe Wa Lone can do one of the following: [list] [*]Select a box containing exactly two balls and reverse the order of the top and the bottom balls. [*]Take a ball $b$ at the top of a non-empty box and either put it in an empty box, or put it in the box only containing the ball of the same colour as $b$. [/list] Find the smallest positive integer $N$ such that for any initial placement of the balls, Phoe Wa Lone can always achieve his goal using at most $N$ moves in total.

Does there exists a base in which the numbers of the form: \[ 10101, 101010101, 1010101010101,\cdots \] are all prime numbers?

Let points $A, B$ and $C$ lie on the parabola $\Delta$ such that the point $H$, orthocenter of triangle $ABC$, coincides with the focus of parabola $\Delta$. Prove that by changing the position of points $A, B$ and $C$ on $\Delta$ so that the orthocenter remain at $H$, inradius of triangle $ABC$ remains unchanged. [i]Proposed by Mahdi Etesamifard[/i]

The trapezium ${ABCD}$, where ${AB}$ and ${CD}$ are parallel and ${AD < CD}$, is inscribed in the circle ${c}$. Let ${DP}$ be a chord of the circle, parallel to ${AC}$. Assume that the tangent to ${c}$ at ${D}$ meets the line ${AB}$ at ${E}$ and that ${PB}$ and ${DC}$ meet at ${Q}$. Show that ${EQ = AC}$.

Suppose $ \cos x \equal{} 0$ and $ \cos (x \plus{} z) \equal{} 1/2$. What is the smallest possible positive value of $ z$? $ \textbf{(A) } \frac {\pi}{6}\qquad \textbf{(B) } \frac {\pi}{3}\qquad \textbf{(C) } \frac {\pi}{2}\qquad \textbf{(D) } \frac {5\pi}{6}\qquad \textbf{(E) } \frac {7\pi}{6}$

Centuries ago, the pirate Captain Blackboard buried a vast amount of treasure in a single cell of a $2 \times 4$ grid-structured island. Treasure was buried in a single cell of an $M\times N$ ($2\le M$, $N$) grid. You and your crew have reached the island and have brought special treasure detectors to find the cell with the treasure For each detector, you can set it up to scan a specific subgrid $[a,b]\times[c,d]$ with $1\le a\le b\le 2$ and $1\le c\le d\le 4$. Running the detector will tell you whether the treasure is in the region or not, though it cannot say where in the region the treasure was detected. You plan on setting up $Q$ detectors, which may only be run simultaneously after all $Q$ detectors are ready. What is the minimum $Q$ required to gaurantee to determine the location of the Blackboard’s legendary treasure?

The set $N$ has been partitioned into two sets A and $B$. Show that for every $n \in N$ there exist distinct integers $a, b > n$ such that $a, b, a + b$ either all belong to $A$ or all belong to $B$.

Assume that $C$ is a convex subset of $\mathbb R^{d}$. Suppose that $C_{1},C_{2},\dots,C_{n}$ are translations of $C$ that $C_{i}\cap C\neq\emptyset$ but $C_{i}\cap C_{j}=\emptyset$. Prove that \[n\leq 3^{d}-1\] Prove that $3^{d}-1$ is the best bound. P.S. In the exam problem was given for $n=3$.

[b]p1.[/b] A function $f$ defined on the set of positive numbers satisfies the equality $$f(xy) = f(x) + f(y), x, y > 0.$$ Find $f(2007)$ if $f\left( \frac{1}{2007} \right) = 1$. [b]p2.[/b] The plane is painted in two colors. Show that there is an isosceles right triangle with all vertices of the same color. [b]p3.[/b] Show that the number of ways to cut a $2n \times 2n$ square into $1\times 2$ dominoes is divisible by $2$. [b]p4.[/b] Two mirrors form an angle. A beam of light falls on one mirror. Prove that the beam is reflected only finitely many times (even if the angle between mirrors is very small). [b]p5.[/b] A sequence is given by the recurrence relation $a_{n+1} = (s(a_n))^2 +1$, where $s(x)$ is the sum of the digits of the positive integer $x$. Prove that starting from some moment the sequence is periodic. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

In the triangle $ABC , \angle A = \alpha, BC = a$. The inscribed circle touches the lines $AB$ and $AC$ at points $M$ and $P$. Find the length of the chord cut by the line $MP$ in a circle with diameter $BC$.