The vertices of $\triangle ABC$ are $A = (0,0)$, $B = (0,420)$, and $C = (560,0)$. The six faces of a die are labeled with two $A$'s, two $B$'s, and two $C$'s. Point $P_1 = (k,m)$ is chosen in the interior of $\triangle ABC$, and points $P_2$, $P_3$, $P_4, \dots$ are generated by rolling the die repeatedly and applying the rule: If the die shows label $L$, where $L \in \{A, B, C\}$, and $P_n$ is the most recently obtained point, then $P_{n + 1}$ is the midpoint of $\overline{P_n L}$. Given that $P_7 = (14,92)$, what is $k + m$?

Let $ABC$ be a triangle. The incircle of $ABC$ touches the sides $AB$ and $AC$ at the points $Z$ and $Y$, respectively. Let $G$ be the point where the lines $BY$ and $CZ$ meet, and let $R$ and $S$ be points such that the two quadrilaterals $BCYR$ and $BCSZ$ are parallelogram. Prove that $GR=GS$. [i]Proposed by Hossein Karke Abadi, Iran[/i]

The figure below is used to fold into a pyramid, and consists of four equilateral triangles erected around a square with area nine. What is the length of the dashed path shown? [asy] real r = 1/2 * 3^(1/2); size(45); draw((0,0)--(1,0)--(1,1)--(0,1)--cycle); draw((0,0)--(-r,0.5)--(0,1)--(0.5,1+r)--(1,1)--(1+r,0.5)--(1,0)--(0.5,-r)--cycle,dashed); [/asy] $\textbf{(A) }18\qquad\textbf{(B) }20\qquad\textbf{(C) }21\qquad\textbf{(D) }24\qquad\textbf{(E) }27$

Let $n$ be an integer greater than $1$. Define \[x_1 = n, y_1 = 1, x_{i+1} =\left[ \frac{x_i+y_i}{2}\right] , y_{i+1} = \left[ \frac{n}{x_{i+1}}\right], \qquad \text{for }i = 1, 2, \ldots\ ,\] where $[z]$ denotes the largest integer less than or equal to $z$. Prove that \[ \min \{x_1, x_2, \ldots, x_n \} =[ \sqrt n ]\]

Let $x$, $y$, $z$ be non-zero real numbers, such that $x + y + z = 0$ and the numbers $$ \frac{x}{y} + \frac{y}{z} + \frac{z}{x} \quad \text{and} \quad \frac{x}{z} + \frac{z}{y} + \frac{y}{x} + 1 $$ are equal. Determine their common value.

Ryan has three distinct eggs, one of which is made of rubber and thus cannot break; unfortunately, he doesn't know which egg is the rubber one. Further, in some 100-story building there exists a floor such that all normal eggs dropped from below that floor will not break, while those dropped from at or above that floor will break and cannot be dropped again. What is the minimum number of times Ryan must drop an egg to determine the floor satisfying this property?

If $f:\mathbb{R} \rightarrow \mathbb{R}$ satisfies $f(x^2 +f(y))=y+xf(x)$ for all $x,y \in \mathbb{R}$, find $f(x)$.

Let N be the number of distinct rearrangements of the 34 letters in SUPERCALIFRAGILISTICEXPIALIDOCIOUS. How many positive factors does N have?

Find the complex numbers $ z$ for which the series \[ 1 \plus{} \frac {z}{2!} \plus{} \frac {z(z \plus{} 1)}{3!} \plus{} \frac {z(z \plus{} 1)(z \plus{} 2)}{4!} \plus{} \cdots \plus{} \frac {z(z \plus{} 1)\cdots(z \plus{} n)}{(n \plus{} 2)!} \plus{} \cdots\] converges and find its sum.

Determine all pairs $(a,b)$ of positive integers with the following property: all of the terms of the sequence $(a^n+b^n+1)_{n\geqslant 1}$ have a greatest common divisor $d>1.$

A natural number $n$ is said to be [i]abundant [/i] if the sum of its divisors is greater than $2n$. For example, $18$ is abundant because the sum of its divisors, $1 + 2 + 3 + 6 + 9 + 18$, is greater than $36$. Write every even number greater than $46$ as a sum of two numbers abundant.

In the figure below the triangles $BCD, CAE$ and $ABF$ are equilateral, and the triangle $ABC$ is right-angled with $\angle A = 90^o$. Prove that $|AD| = |EF|$. [img]https://1.bp.blogspot.com/-QMMhRdej1x8/XzP18QbsXOI/AAAAAAAAMUI/n53OsE8rwZcjB_zpKUXWXq6bg3o8GUfSwCLcBGAsYHQ/s0/2019%2Bmohr%2Bp5.png[/img]

$p$ is an odd prime number. Find all $\frac{p-1}2$-tuples $\left(x_1,x_2,\dots,x_{\frac{p-1}2}\right)\in \mathbb{Z}_p^{\frac{p-1}2}$ such that $$\sum_{i = 1}^{\frac{p-1}{2}} x_{i} \equiv \sum_{i = 1}^{\frac{p-1}{2}} x_{i}^{2} \equiv \cdots \equiv \sum_{i = 1}^{\frac{p-1}{2}} x_{i}^{\frac{p - 1}{2}} \pmod p.$$ [i]Proposed by Ali Partofard[/i]

In a triangle $ABC$, the excircle $\omega_a$ opposite $A$ touches $AB$ at $P$ and $AC$ at $Q$, while the excircle $\omega_b$ opposite $B$ touches $BA$ at $M$ and $BC$ at $N$. Let $K$ be the projection of $C$ onto $MN$ and let $L$ be the projection of $C$ onto $PQ$. Show that the quadrilateral $MKLP$ is cyclic. ([i]Bulgaria[/i])

What is the smallest possible area of a rectangle that can completely contain the shape formed by joining six squares of side length $8$ cm as shown below? [asy] size(5cm,0); draw((0,2)--(0,3)); draw((1,1)--(1,3)); draw((2,0)--(2,3)); draw((3,0)--(3,2)); draw((4,0)--(4,1)); draw((2,0)--(4,0)); draw((1,1)--(4,1)); draw((0,2)--(3,2)); draw((0,3)--(2,3)); [/asy] $\text{(A) }384\text{ cm}^2\qquad\text{(B) }576\text{ cm}^2\qquad\text{(C) }672\text{ cm}^2\qquad\text{(D) }768\text{ cm}^2\qquad\text{(E) }832\text{ cm}^2$

The values of $ a$ in the equation: $ \log_{10}(a^2 \minus{} 15a) \equal{} 2$ are: $ \textbf{(A)}\ \frac {15\pm\sqrt {233}}{2} \qquad\textbf{(B)}\ 20, \minus{} 5 \qquad\textbf{(C)}\ \frac {15 \pm \sqrt {305}}{2}$ $ \textbf{(D)}\ \pm20 \qquad\textbf{(E)}\ \text{none of these}$

If the circumradius of a regular $n$-gon is $1$ and the ratio of its perimeter over its area is $\dfrac{4\sqrt 3}{3}$, what is $n$? $ \textbf{(A)}\ 3 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}\ 6 \qquad\textbf{(E)}\ 8 $

Among the coordinates $(x,y)$ $(1\leq x,y\leq 101)$, choose some points such that there does not exist $4$ points which form a isoceles trapezium with its base parallel to either the $x$ or $y$ axis(including rectangles). Find the maximum number of coordinate points that can be chosen.

A convex $n$-side polygon has angles $v_1,v_2,\dots,v_n$ (in degrees), where all $v_k$ ($k = 1,2,\dots,n$) are positive integers divisible by $36$. (a) Determine the largest $n$ for which this is possible. (b) Show that if $n>5$, two of the sides of the $n$-polygon must be parallel.

Define the sequence of positive prime numbers. $p_1,p_2,p_3,...$. Let set $A$ be the infinite set of positive integers whose prime divisor does not exceed $p_n$. How many at least members must be selected from the set $A$ , such that we ensures that there are $2$ numbers whose products are perfect squares? [i](PP-nine)[/i]

Let $A B C$ be an acut-angles triangle of incenter $I$, circumcenter $O$ and inradius $r.$ Let $\omega$ be the inscribed circle of the triangle $A B C$. $A_1$ is the point of $\omega$ such that $A IA_1O$ is a convex trapezoid of bases $A O$ and $IA_1$. Let $\omega_1$ be the circle of radius $r$ which goes through $A_1$, tangent to the line $A B$ and is different from $\omega$ . Let $\omega_2$ be the circle of radius $r$ which goes through $A_1$, is tangent to the line $A C$ and is different from $\omega$ . Circumferences $\omega_1$ and $\omega_2$ they are cut at points $A_1$ and $A_2$. Similarly are defined points $B_2$ and $C_2$. Prove that the lines $A A_2, B B_2$ and $CC2$ they are concurrent.

Let $\left ( a_{n} \right )_{n \geq 1}$ be an increasing sequence of positive integers such that $a_1=1$, and for all positive integers $n$, $a_{n+1}\leq 2n$. Prove that for every positive $n$; there exists positive integers $p$ and $q$ such that $n=a_{p}-a_{q}$.

[b]Problem 3[/b] : Let $x_1,x_2,\cdots,x_{2022}$ be non-negative real numbers such that $$x_k + x_{k+1}+x_{k+2} \leq 2$$ for all $k = 1,2,\cdots,2020$. Prove that $$\sum_{k=1}^{2020}x_kx_{k+2}\leq 1010$$

There are 4 countries: Argentina, Brazil, Peru and Uruguay. Each country consists of 4 islands. There are bridges going back and forth between some of the 16 islands. Carlos noted that whenever he travels between some of the islands using the bridges, without using the same bridge twice, and ending in the island where he started his journey, he will necessarily visit at least one island of each country. Determine the maximum number of bridges there can be.

We wish to place natural numbers around a circle such that the following property is satisfied: the absolute values of the differences of each pair of neighboring numbers are all different. a) Is it possible to place the numbers from 1 to 2009 satisfying this property b) Is it possible to suppress one of the numbers from 1 to 2009 in such a way that the remaining 2008 numbers can be placed satisfying the property