An angle with vertex $A$ is given on the plane. Points $K$ and $P$ are selected on its sides so that $AK + AP = a$, where $a$ is a given segment. Let $M$ be a point on the plane such that the triangle $KPM$ is isosceles with the base $KP$ and the angle at the vertex $M$ equal to the given angle. Find the locus of points $M$ (as $K$ and $P$ move).

Find the number of positive divisors of $20^{22}$ that are perfect squares or perfect cubes.

Find all fourth degree polynomial $ p(x)$ such that the following four conditions are satisfied: (i) $ p(x)\equal{}p(\minus{}x)$ for all $ x$, (ii) $ p(x)\ge0$ for all $ x$, (iii) $ p(0)\equal{}1$ (iv) $ p(x)$ has exactly two local minimum points $ x_1$ and $ x_2$ such that $ |x_1\minus{}x_2|\equal{}2$.

Two eight-sided dice each have faces numbered $ 1$ through $ 8$. When the dice are rolled, each face has an equal probability of appearing on the top. What is the probability that the product of the two top numbers is greater than their sum? $ \textbf{(A)}\ \frac{1}{2}\qquad \textbf{(B)}\ \frac{47}{64}\qquad \textbf{(C)}\ \frac{3}{4}\qquad \textbf{(D)}\ \frac{55}{64}\qquad \textbf{(E)}\ \frac{7}{8}$

In triangle $ABC$, points $M$, $N$, and $P$ lie on sides $\overline{AC}$, $\overline{AB}$, and $\overline{BC}$, respectively. If $\angle{ABC}=42^\circ$, $\angle{MAN}=91^\circ$, and $\angle{NMA}=47^\circ$, compute $\frac{CB}{BP}$. [i]2015 CCA Math Bonanza Lightning Round #3.2[/i]

If $a_1, a_2,...., a_{n-1}, a_n$, are positive integers, prove that: $\frac{\prod_{i=1}^n(a_i^2+3a_i+1)}{a_1a_2....a_{n-1}a_n}\ge 2^{2n}$

Each citizen in a town knows at least $ 30$% of the remaining citizens. A citizen votes in elections if he/she knows at least one candidate. Prove that it is possible to schedule elections with two candidates for the mayor of the city so that at least $ 50$% of the citizen can vote.

Suppose $x,y,z\in \mathbb R^+$. Prove that \[\frac {x}{\sqrt{yz+4xy+4xz}}+\frac {y}{\sqrt{zx+4yz+4yx}}+\frac {z}{\sqrt{xy+4zx+4zy}}\geq 1\].

Point $D$ on side $BC$ of acute triangle ABC is such that $AB=AD$. The circumcircle of triangle $ABD$ intersects side $AC$ at points $A$ and $K$. Line $DK$ intersects the perpendicular drawn from $B$ on $AC$, at the point $L$. Prove that $CL= BC$

Let $ABC$ be a triangle and $AD,BE,CF$ be cevians concurrent at a point $P$. Suppose each of the quadrilaterals $PDCE,PEAF$ and $PFBD$ has both circumcircle and incircle. Prove that $ABC$ is equilateral and $P$ coincides with the center of the triangle.

The circumference of a circle is divided into eight arcs by a convex quadrilateral $ABCD$ with four arcs lying inside the quadrilateral and the remaining four lying outside it. The lengths of the arcs lying inside the quadrilateral are denoted by $p,q,r,s$ in counter-clockwise direction. Suppose $p+r = q+s$. Prove that $ABCD$ is cyclic.

In the hexagon $ABCDEF$, all angles are equally large. The side lengths satisfy $AB = CD = EF = 3$ and $BC = DE = F A = 2$. The diagonals $AD$ and $CF$ intersect each other in the point $G$. The point $H$ lies on the side $CD$ so that $DH = 1$. Prove that triangle $EGH$ is equilateral.

There are $5$ people at a party. For each pair of people, there is a $1/2$ chance they are friends, independent of all other pairs. Find the expected number of pairs of people who have a mutual friend, but are not friends themselves. [i]Proposed by Patrick Xue[/i]

Let $S$ be a set of $49$-digit numbers $n$, with the property that each of the digits $1, 2, 3, \dots, 7$ appears in the decimal expansion of $n$ seven times (and $8, 9$ and $0$ do not appear). Show that no two distinct elements of $S$ divide each other.

The sequence $1, 2, \dots, 2023, 2024$ is written on a whiteboard. Every second, Megavan chooses two integers $a$ and $b$, and four consecutive numbers on the whiteboard. Then counting from the left, he adds $a$ to the 1st and 3rd of those numbers, and adds $b$ to the 2nd and 4th of those numbers. Can he achieve the sequence $2024, 2023, \dots, 2, 1$ in a finite number of moves? [i](Proposed by Avan Lim Zenn Ee)[/i]

In the acute triangle $ABC$ on $AC$ point $P$ is chosen such that $2AP=BC$. Points $X$ and $Y$ are symmetric to $P$ wrt $A$ and $C$ respectively. It turned out that $BX=BY$. Find angle $C$.

The area of the largest square that can be inscribed in a regular hexagon with sidelength $1$ can be expressed as $a-b\sqrt{c}$ where $c$ is not divisible by the square of any prime. Find $a+b+c$.

Let $p$ be a prime, $A$ is an infinite set of integers. Prove that there is a subset $B$ of $A$ with $2p-2$ elements, such that the arithmetic mean of any pairwise distinct $p$ elements in $B$ does not belong to $A$.

Let $a,b $ be two relatively prime positive integers.Also let $m,n $ be positive integers with $n> m $.\\ Prove that\\ $lcm [am+b,a (m+1)+b,...,an+b]\ge (n+1)\cdot \binom {n}{m}$ [i]Proposed by Navid Safaei[/i]

Let $ABC$ be a right triangle with the hypotenus $BC.$ Let $BE$ be the bisector of the angle $\angle ABC.$ The circumcircle of the triangle $BCE$ cuts the segment $AB$ again at $F.$ Let $K$ be the projection of $A$ on $BC.$ The point $L$ lies on the segment $AB$ such that $BL=BK.$ Prove that $\frac{AL}{AF}=\sqrt{\frac{BK}{BC}}.$

In a quadrilateral $ABCD$ let $E$ be the intersection of the two diagonals, I the center of the parallelogram whose vertices are the midpoints of the four sides of the quadrilateral, and K the center of the parallelogram whose sides pass through the points. divide the four sides of the quadrilateral into three equal parts (see illustration ). [img]https://cdn.artofproblemsolving.com/attachments/1/c/8f2617103edd8361b8deebbee13c6180fa848b.png[/img] a) Prove that $\overrightarrow{EK} =\frac43 \overrightarrow{EI}$. b) Prove that $$\lambda_A \overrightarrow{KA} +\lambda_B \overrightarrow{KB} + \lambda_C \overrightarrow{KC} + \lambda_D \overrightarrow{KD} = \overrightarrow{0}$$ , where $$\lambda_A=1+\frac{S(ADB)}{S(ABCD)},\lambda_B=1+\frac{S(BCA)}{S(ABCD)},\lambda_C=1+\frac{S(CDB)}{S(ABCD)},\lambda_D=1+\frac{S(DAC)}{S(ABCD)}$$ , where $S$ is the area symbol.

On a $2023 \times 2023$ board, there are beetles on some of the cells, with at most one beetle per cell. After one minute, each beetle moves a cell to the right or to the left or to the top or to the bottom. After each further minute, the beetles continue to move to adjacent fields, but they always make a $90^\circ$ turn, i.e. when a beetle just moved to the right or to the left, it now moves to the top or to the bottom, and vice versa. What is the minimal number of beetles on the board such that no matter where they start and how they move (according to the rules), at some point two beetles will end up in the cell?

Points $D$, $E$, $F$ are taken respectively on sides $AB$, $BC$, and $CA$ of triangle $ABC$ so that $AD:DB=BE:CE=CF:FA=1:n$. The ratio of the area of triangle $DEF$ to that of triangle $ABC$ is: $\textbf{(A)}\ \frac{n^2-n+1}{(n+1)^2}\qquad \textbf{(B)}\ \frac{1}{(n+1)^2}\qquad \textbf{(C)}\ \frac{2n^2}{(n+1)^2}\qquad \textbf{(D)}\ \frac{n^2}{(n+1)^2}\qquad \textbf{(E)}\ \frac{n(n-1)}{n+1}$

Define a [i]domino[/i] to be an ordered pair of [i]distinct[/i] positive integers. A [i]proper sequence[/i] of dominoes is a list of distinct dominoes in which the first coordinate of each pair after the first equals the second coordinate of the immediately preceding pair, and in which $(i, j)$ and $(j, i)$ do not [i]both[/i] appear for any $i$ and $j$. Let $D_n$ be the set of all dominoes whose coordinates are no larger than $n$. Find the length of the longest proper sequence of dominoes that can be formed using the dominoes of $D_n$.

Let $\gamma=\alpha \times \beta$ where \[\alpha=999 \cdots 9\] (2010 '9') and \[\beta=444 \cdots 4\] (2010 '4') Find the sum of digits of $\gamma$.