In convex pentagon $ABCDE$ points $A_1$, $B_1$, $C_1$, $D_1$, $E_1$ are intersections of pairs of diagonals $(BD, CE)$, $(CE, DA)$, $(DA, EB)$, $(EB, AC)$ and $(AC, BD)$ respectively. Prove that if four of quadrilaterals $AB_{1}A_{1}B$, $BC_{1}B_{1}C$, $CD_{1}C_{1}D$, $DE_{1}D_{1}E$ and $EA_{1}E_{1}A$ are cyclic then the fifth one is also cyclic.

Toner Drum and Celery Hilton are both running for president. A total of $129$ million people cast their vote in a random order, with exactly $63$ million and $66$ million voting for Toner Drum and Celery Hilton, respectively. The Combinatorial News Network displays the face of the leading candidate on the front page of their website. If the two candidates are tied, both faces are displayed. What is the probability that Toner Drum's face is never displayed on the front page? [i]2017 CCA Math Bonanza Individual Round #13[/i]

A chessboard with some unit squares marked is called a $\textit{good board}$ if for any pair of rows \((s, t)\), a rook placed on a marked square in row \(s\) can reach a marked square in row \(t\) in several moves by only moving to marked squares above, below, or to the right of its current position. Consider a chessboard with 220 rows and 12 columns, where exactly 9 unit squares in each row are marked. Regardless of how the marked squares are chosen, if it is possible to delete \(k\) columns and rearrange the remaining columns to form a $\textit{good board}$ determine the maximum possible value of \(k\).

Positive integers $a$, $b$, and $c$ satisfy $a^2 +b^2 = c^3 -1$ where $c \le 40$. Find the sum of all distinct possible values of $c$.

Prove that the number of incidences of $n$ distinct points on $n$ distinct lines in plane is $\mathcal O (n^{\frac{4}{3}})$. Find a configuration for which $\Omega (n^{\frac{4}{3}})$ incidences happens.

Let $ABC$ be a triangle inscribed in circle $C(O,R)$. Let $M$ ba apoint on the arc $BC$ . Let $D,E,Z$ be the feet of the perpendiculars drawn from $M$ on lines $AB,AC,BC$ respectively. Prove that $\frac{(BC)^2}{(MZ)^2} \ge 8\frac{R U_a}{(MD)\cdot(ME)}$ where $U_a$ is the altitude drawn on $BC$.

Given an integer $a_1$($a_1 \neq -1$), find a real number sequence $\{ a_n \}$($a_i \neq 0, i=1,2,\cdots,5$) such that $x_1,x_2,\cdots,x_5$ and $y_1,y_2,\cdots,y_5$ satisfy $b_{i1}x_1+b_{i2}x_2+\cdots +b_{i5}x_5=2y_i$, $i=1,2,3,4,5$, then $x_1y_1+x_2y_2+\cdots+x_5y_5=0$, where $b_{ij}=\prod_{1 \leq k \leq i} (1+ja_k)$.

A circle centred at $I$ is tangent to the sides $BC, CA$, and $AB$ of an acute-angled triangle $ABC$ at $A_1, B_1$, and $C_1$, respectively. Let $K$ and $L$ be the incenters of the quadrilaterals $AB_1IC_1$ and $BA_1IC_1$, respectively. Let $CH$ be an altitude of triangle $ABC$. Let the internal angle bisectors of angles $AHC$ and $BHC$ meet the lines $A_1C_1$ and $B_1C_1$ at $P$ and $Q$, respectively. Prove that $Q$ is the orthocenter of the triangle $KLP$. Kolmogorov Cup 2018, Major League, Day 3, Problem 1; A. Zaslavsky

Let $ABC$ be a triangle with $AB=7$, $BC=9$, and $CA=4$. Let $D$ be the point such that $AB\parallel CD$ and $CA\parallel BD$. Let $R$ be a point within triangle $BCD$. Lines $\ell$ and $m$ going through $R$ are parallel to $CA$ and $AB$ respectively. Line $\ell$ meets $AB$ and $BC$ at $P$ and $P^\prime$ respectively, and $m$ meets $CA$ and $BC$ at $Q$ and $Q^\prime$ respectively. If $S$ denotes the largest possible sum of the areas of triangle $BPP^\prime$, $RP^\prime Q^\prime$, and $CQQ^\prime$, determine the value of $S^2$.

Simplify $\sin (x-y) \cos y + \cos (x-y) \sin y$. $ \textbf{(A)}\ 1\qquad\textbf{(B)}\ \sin x\qquad\textbf{(C)}\ \cos x\qquad\textbf{(D)}\ \sin x \cos 2y\qquad\textbf{(E)}\ \cos x \cos 2y $

Let $S = \{1,2,3,\dots,2014\}$. What is the largest subset of $S$ that contains no two elements with a difference of $4$ or $7$?

Find all polynomials \(P(x)\) with real coefficients such that for all nonzero real numbers \(x\), \[P(x)+P\left(\frac1x\right) =\frac{P\left(x+\frac1x\right) +P\left(x-\frac1x\right)}2.\] [i]Proposed by Holden Mui[/i]

Estimate the value of the $2020$th prime number $p$ such that $p + 2$ is also prime. If $E > 0$ is your estimate and $A$ is the correct answer, you will receive $25 \min \left( \frac{E}{A}, \frac{A}{E}\right)^2$ points, rounded to the nearest integer. (An estimate less than or equal to $0$ will receive $0$ points.

Consider the following graphs of position [i]vs.[/i] time. [asy] size(500); picture pic; // Rectangle draw(pic,(0,0)--(20,0)--(20,15)--(0,15)--cycle); label(pic,"0",(0,0),S); label(pic,"2",(4,0),S); label(pic,"4",(8,0),S); label(pic,"6",(12,0),S); label(pic,"8",(16,0),S); label(pic,"10",(20,0),S); label(pic,"-15",(0,2),W); label(pic,"-10",(0,4),W); label(pic,"-5",(0,6),W); label(pic,"0",(0,8),W); label(pic,"5",(0,10),W); label(pic,"10",(0,12),W); label(pic,"15",(0,14),W); label(pic,rotate(90)*"x (m)",(-2,7),W); label(pic,"t (s)",(11,-2),S); // Tick Marks draw(pic,(4,0)--(4,0.3)); draw(pic,(8,0)--(8,0.3)); draw(pic,(12,0)--(12,0.3)); draw(pic,(16,0)--(16,0.3)); draw(pic,(20,0)--(20,0.3)); draw(pic,(4,15)--(4,14.7)); draw(pic,(8,15)--(8,14.7)); draw(pic,(12,15)--(12,14.7)); draw(pic,(16,15)--(16,14.7)); draw(pic,(20,15)--(20,14.7)); draw(pic,(0,2)--(0.3,2)); draw(pic,(0,4)--(0.3,4)); draw(pic,(0,6)--(0.3,6)); draw(pic,(0,8)--(0.3,8)); draw(pic,(0,10)--(0.3,10)); draw(pic,(0,12)--(0.3,12)); draw(pic,(0,14)--(0.3,14)); draw(pic,(20,2)--(19.7,2)); draw(pic,(20,4)--(19.7,4)); draw(pic,(20,6)--(19.7,6)); draw(pic,(20,8)--(19.7,8)); draw(pic,(20,10)--(19.7,10)); draw(pic,(20,12)--(19.7,12)); draw(pic,(20,14)--(19.7,14)); // Path add(pic); path A=(0,14)--(20,14); draw(A); label("I.",(8,-4),3*S); path B=(0,6)--(20,6); picture pic2=shift(30*right)*pic; draw(shift(30*right)*B); label("II.",(38,-4),3*S); add(pic2); path C=(0,12)--(20,14); picture pic3=shift(60*right)*pic; draw(shift(60*right)*C); label("III.",(68,-4),3*S); add(pic3); [/asy] Which of the graphs could be the motion of a particle in the given potential? (A) $\text{I}$ (B) $\text{III}$ (C) $\text{I and II}$ (D) $\text{I and III}$ (E) $\text{I, II, and III}$

The natural numbers $X$ and $Y$ are obtained from each other by permuting the digits. Prove that the sums of the digits of the numbers $5X$ and $5Y$ coincide.

A value of $ x$ satisfying the equation $ x^2 \plus{} b^2 \equal{} (a \minus{} x)^2$ is: $ \textbf{(A)}\ \frac{b^2 \plus{} a^2}{2a}\qquad \textbf{(B)}\ \frac{b^2 \minus{} a^2}{2a}\qquad \textbf{(C)}\ \frac{a^2 \minus{} b^2}{2a}\qquad \textbf{(D)}\ \frac{a \minus{} b}{2}\qquad \textbf{(E)}\ \frac{a^2 \minus{} b^2}{2}$

There are $110$ guinea pigs for each of the $110$ species, arranging as a $110\times 110$ array. Find the maximum integer $n$ such that, no matter how the guinea pigs align, we can always find a column or a row of $110$ guinea pigs containing at least $n$ different species.

Find all functions $f :\mathbb{N} \rightarrow \mathbb{N}$ such that $f(m + f(n)f(m)) = nf(m) + m$ holds for all $m,n \in \mathbb{N}$.

Let $I$ be the center of the circle of the inscribed triangle $ABC$ and $M$ be the center of its side $BC$. If $|AI| = |MI|$, prove that there are two of the sides of triangle $ABC$, of which one is twice of the other.

Three lines were drawn through the point $X$ in space. These lines crossed some sphere at six points. It turned out that the distances from point $X$ to some five of them are equal to $2$ cm, $3$ cm, $4$ cm, $5$ cm, $6$ cm. What can be the distance from point $X$ to the sixth point? (Alexey Panasenko)

Let $d$ be a positive integer. The seqeunce $a_1, a_2, a_3,...$ of positive integers is defined by $a_1 = 1$ and $a_{n + 1} = n\left \lfloor \frac{a_n}{n} \right \rfloor+ d$ for $n = 1,2,3, ...$ . Prove that there exists a positive integer $N$ so that the terms $a_N,a_{N + 1}, a_{N + 2},...$ form an arithmetic progression. Note: If $x$ is a real number, $\left \lfloor x \right \rfloor $ denotes the largest integer that is less than or equal to $x$.

The kindergarten group is standing in the column of pairs. The number of boys equals the number of girls in each of the two columns. The number of mixed (boy and girl) pairs equals to the number of the rest pairs. Prove that the total number of children in the group is divisible by eight.

There are $24$ different pencils, $4$ different colors, and $6$ pencils of each color. They were given to $6$ children in such a way that each got $4$ pencils. What is the least number of children that you can randomly choose so that you can guarantee that you have pencils of all colors. P.S. for 10 grade gives same problem with $40$ pencils, $10$ of each color and $10$ children.

An acute scalene triangle $ABC$ with circumcircle $\Omega$ is given. The altitude from $B$ intersects side $AC$ at $B_1$ and circle $\Omega$ at $B_2$. The circle with diameter $B_1B_2$ intersects circle $\Omega$ again at $B_3$. Similarly, the altitude from $C$ intersects side $AB$ at $C_1$ and circle $\Omega$ at $C_2$. The circle with diameter $C_1C_2$ intersects circle $\Omega$ again at $C_3$. Let $X$ be the intersection of lines $B_1B_3$ and $C_1C_3$, and let $Y$ be the intersection of lines $B_3C$ and $C_3B$. Prove that line $XY$ bisects side $BC$.

Let $ABCD$ be a unit square, and let there be two unit circles centered at $C$ and $D$. Let $P$ be the point of intersection of the two circles inside the square. Compute $\angle APB$ in degrees.