Quadrilateral $ABCD$ inscribed in circle with center $O$. Let lines $AD$ and $BC$ intersects at $M$, lines $AB$ and $CD$- at $N$, lines $AC$ and $BD$ -at $P$, lines $OP$ and $MN$ at $K$. Proved that $ \angle AKP = \angle PKC$. As I know, this problem was very short solution by polars, but in olympiad for this solution gives maximum 4 balls (in marking schemes written, that needs to prove all theorems about properties of polars)

In a convex pentagon $ ABCDE$ side $ AB$ is perpendicular to $ CD$ and side $ BC$ is perpendicular to $ DE$. Prove that if $ AB \equal{} AE \equal{} ED \equal{} 1$, then $ BC \plus{} CD < 1$.

You are given a convex hexagon with parallel opposite sides. For each pair of opposite sides, a line is drawn parallel to these sides and equidistant from them. Prove that the three lines thus obtained intersect at one point if and only if the lengths of the opposite sides are equal. [i]Proposed by Nazar Serdyuk[/i]

Let $n$ be a positive integer, and define $f(n)=1!+2!+\ldots+n!$. Find polynomials $P$ and $Q$ such that $$f(n+2)=P(n)f(n+1)+Q(n)f(n)$$for all $n\ge1$.

Given points $A_1 (x_1, y_1, z_1), A_2 (x_2, y_2, z_2), .., A_n (x_n, y_n, z_n)$ let $P (x, y, z)$ be the point which minimizes $\Sigma ( |x - x_i| + |y -y_i| + |z -z_i| )$. Give an example (for each $n > 4$) of points $A_i $ for which the point $P$ lies outside the convex hull of the points $A_i$.

The figure below shows a square and four equilateral triangles, with each triangle having a side lying on a side of the square, such that each triangle has side length 2 and the third vertices of the triangles meet at the center of the square. The region inside the square but outside the triangles is shaded. What is the area of the shaded region? [asy] pen white = gray(1); pen gray = gray(0.5); draw((0,0)--(2sqrt(3),0)--(2sqrt(3),2sqrt(3))--(0,2sqrt(3))--cycle); fill((0,0)--(2sqrt(3),0)--(2sqrt(3),2sqrt(3))--(0,2sqrt(3))--cycle, gray); draw((sqrt(3)-1,0)--(sqrt(3),sqrt(3))--(sqrt(3)+1,0)--cycle); fill((sqrt(3)-1,0)--(sqrt(3),sqrt(3))--(sqrt(3)+1,0)--cycle, white); draw((sqrt(3)-1,2sqrt(3))--(sqrt(3),sqrt(3))--(sqrt(3)+1,2sqrt(3))--cycle); fill((sqrt(3)-1,2sqrt(3))--(sqrt(3),sqrt(3))--(sqrt(3)+1,2sqrt(3))--cycle, white); draw((0,sqrt(3)-1)--(sqrt(3),sqrt(3))--(0,sqrt(3)+1)--cycle); fill((0,sqrt(3)-1)--(sqrt(3),sqrt(3))--(0,sqrt(3)+1)--cycle, white); draw((2sqrt(3),sqrt(3)-1)--(sqrt(3),sqrt(3))--(2sqrt(3),sqrt(3)+1)--cycle); fill((2sqrt(3),sqrt(3)-1)--(sqrt(3),sqrt(3))--(2sqrt(3),sqrt(3)+1)--cycle, white); [/asy] $\textbf{(A) } 4\qquad\textbf{(B) }12 - 4\sqrt{3} \qquad\textbf{(C) } 3\sqrt{3} \qquad \textbf{(D) }4\sqrt{3}\qquad \textbf{(E) }16 - \sqrt{3}$

Determine the locus of points $M$ in the plane of a given rhombus $ABCD$ such that $MA\cdot MC+MB\cdot MD=AB^2$.

Let $ ABCD $ be a rhombus with center $ O. $ $ P $ is a point lying on the side $ AB. $ Let $ I, $ $ J, $ and $ L $ be the incenters of triangles $ PCD, $ $ PAD, $ and $PBC, $ respectively. Let $ H $ and $ K $ be orthocenters of triangles $ PLB $ and $ PJA, $ respectively. Prove that $ OI \perp HK. $ [i]Proposed by buratinogigle[/i]

Evaluate $ \int_e^{e^2} \frac {(\ln x)^7\minus{}7!}{(\ln x)^8}\ dx.$ Sorry, I have deleted my first post because that was wrong. kunny

Arnold has plates weighing $5$, $15$, $25$, $35$, or $45$ pounds. He lifts a barbell, which consists of a $45$-pound bar and any number of plates that he has. Vlad looks at Arnold's bar and is impressed to see him bench-press $600$ pounds. Unfortunately, Vlad mistook each plate on Arnold's bar for the plate one size heavier, and Arnold was actually lifting $470$ pounds. How many plates did Arnold have on the bar? [i]2018 CCA Math Bonanza Team Round #2[/i]

For any positive integer $n$ the sum $\displaystyle 1+\frac 12+ \cdots + \frac 1n$ is written in the form $\displaystyle \frac{P(n)}{Q(n)}$, where $P(n)$ and $Q(n)$ are relatively prime. a) Prove that $P(67)$ is not divisible by 3; b) Find all possible $n$, for which $P(n)$ is divisible by 3.

Let $\alpha_1,\ldots,\alpha_d,\beta_1,\ldots,\beta_e\in\mathbb{C}$ be such that the polynomials $f_1(x) =\prod_{i=1}^d(x-\alpha_i)$ and $f_2(x) =\prod_{i=1}^e(x-\beta_i)$ have integer coefficients. Suppose that there exist polynomials $g_1, g_2 \in\mathbb{Z}[x]$ such that $f_1g_1 +f_2g_2 = 1$. Prove that $\left|\prod_{i=1}^d \prod_{j=1}^e (\alpha_i - \beta_j)\right|=1$

Find all positive integers $n$ such that the unit segments of an $n \times n$ grid of unit squares can be partitioned into groups of three such that the segments of each group share a common vertex.

Let f be a function from the set Z of all integers into itself, that satisfies the condition for all $m \in Z$, $$f(f(m)) =-m. \ \ (1)$$ Then: (a) $f$ is a mutually unique mapping, i.e. a simple mapping of the set $Z$ onto the set $Z$ , (b) for all $m \in Z$ holds that $f(-m) = -f(m)$ , (c) $f(m) = 0$ if and only if $m = 0$ . Prove these statements and construct an example of a mapping f that satisfies condition (1).

Find all positive integers $(m, n)$ such that $3 \cdot 2^n + 1 = m^2$.

Written on a blackboard are $n$ nonnegative integers whose greatest common divisor is $1$. A [i]move[/i] consists of erasing two numbers $x$ and $y$, where $x\ge y$, on the blackboard and replacing them with the numbers $x-y$ and $2y$. Determine for which original $n$-tuples of numbers on the blackboard is it possible to reach a point, after some number of moves, where $n-1$ of the numbers of the blackboard are zeroes.

Let $S$ denote the set of subsets of $\{1,2,\ldots,2017\}$. For two sets $A$ and $B$ of integers, define $A\circ B$ as the [i]symmetric difference[/i] of $A$ and $B$. (In other words, $A\circ B$ is the set of integers that are an element of exactly one of $A$ and $B$.) Let $N$ be the number of functions $f:S\rightarrow S$ such that $f(A\circ B)=f(A)\circ f(B)$ for all $A,B\in S$. Find the remainder when $N$ is divided by $1000$. [i]Proposed by Michael Ren[/i]

Consider an isosceles triangle $ABC$ with $AB=AC$, and a circle $\omega$ which is tangent to the sides $AB$ and $AC$ of this triangle and intersects the side $BC$ at the points $K$ and $L$. The segment $AK$ intersects the circle $\omega$ at a point $M$ (apart from $K$). Let $P$ and $Q$ be the reflections of the point $K$ in the points $B$ and $C$, respectively. Show that the circumcircle of triangle $PMQ$ is tangent to the circle $\omega$.

Three friends are trying to meet for lunch at a cafe. Each friend will arrive independently at random between $1\!:\!00$ pm and $2\!:\!00$ pm. Each friend will only wait for $5$ minutes by themselves before leaving. However, if another friend arrives within those $5$ minutes, the pair will wait $15$ minutes from the time the second friend arrives. If the probability that the three friends meet for lunch can be expressed in simplest form as $\tfrac{m}{n}$, what is $m+n$?

Let $ABC$ be an isosceles triangle with $AC = BC$ and $\angle ACB < 60^\circ$. We denote the incenter and circumcenter by $I$ and $O$, respectively. The circumcircle of triangle $BIO$ intersects the leg $BC$ also at point $D \ne B$. (a) Prove that the lines $AC$ and $DI$ are parallel. (b) Prove that the lines $OD$ and $IB$ are mutually perpendicular. (Walther Janous)

Let $ABC$ be a triangle, $AD$ be the altitude from $A$ on to $BC$. Draw perpendiculars $DD_1$ and $DD_2$ from $D$ on to $AB$ and $AC$ respectively and let $p(A)$ be the length of the segment $D_1D_2$. Similarly define $p(B)$ and $p(C)$. Prove that $\frac{p(A)p(B)p(C)}{s^3}\le \frac18$ , where s is the semi-perimeter of the triangle $ABC$.

Yang has a standard $6$-sided die, a standard $8$-sided die, and a standard $10$-sided die. He tosses these three dice simultaneously. The probability that the three numbers that show up form the side lengths of a right triangle can be expressed as $\frac{m}{n}$, for relatively prime positive integers $m$ and $n$. Find $100m+n$. [i]Proposed by Yannick Yao[/i]

Let $\omega$ be the circumcircle of isosceles triangle $ABC$ ($AB=AC$). Points $P$ and $Q$ lie on $\omega$ and $BC$ respectively such that $AP=AQ$ .$AP$ and $BC$ intersect at $R$. Prove that the tangents from $B$ and $C$ to the incircle of $\triangle AQR$ (different from $BC$) are concurrent on $\omega$. [i]Proposed by Ali Zamani, Hooman Fattahi[/i]

In a group of 2017 persons, any pair of persons has exactly one common friend (other than the pair of persons). Determine the smallest possible value of the difference between the numbers of friends of the person with the most friends and the person with the least friends in such a group.

Define $f(x)=2x+3$ and suppose that $g(x+2)=f(f(x-1)\cdot f(x+1)+f(x))$. Find $g(6)$.