Given a line segment $ PQ$ moving on the parabola $ y \equal{} x^2$ with end points on the parabola. The area of the figure surrounded by $ PQ$ and the parabola is always equal to $ \frac {4}{3}$. Find the equation of the locus of the mid point $ M$ of $ PQ$.

There are seven rays emanating from a point $A$ on a plane, such that the angle between the two consecutive rays is $30 ^{\circ}$. A point $A_1$ is located on the first ray. The projection of $A_1$ onto the second ray is denoted as $A_2$. Similarly, the projection of $A_2$ onto the third ray is $A_3$, and this process continues until the projection of $A_6$ onto the seventh ray is $A_7$. Find the ratio $\frac{A_7A}{A_1A}$. [img]https://i.imgur.com/oxixe5q.png[/img] [i]Proposed by Giorgi Arabidze, Georgia[/i]

Consider functions $f$ satisfying the following four conditions: (1) $f$ is real-valued and defined for all real numbers. (2) For any two real numbers $x$ and $y$ we have $f(xy)=f(x)f(y)$. (3) For any two real numbers $x$ and $y$ we have $f(x+y) \le 2(f(x)+f(y))$. (4) We have $f(2)=4$. Prove that: a) There is a function $f$ with $f(3)=9$ satisfying the four conditions. b) For any function $f$ satisfying the four conditions, we have $f(3) \le 9$.

Let $ABC$ be an equilateral triangle. Let $Q$ be a random point on $BC$, and let $P$ be the meeting point of $AQ$ and the circumscribed circle of $\triangle ABC$. Prove that $\frac{1}{PQ}=\frac{1}{PB}+\frac{1}{PC}$.

(i) $ABC$ is a triangle with a right angle at $A$, and $P$ is a point on the hypotenuse $BC$. The line $AP$ produced beyond $P$ meets the line through $B$ which is perpendicular to $BC$ at $U$. Prove that $BU = BA$ if, and only if, $CP = CA$. (ii) $A$ is a point on the semicircle $CB$, and points $X$ and $Y$ are on the line segment $BC$. The line $AX$, produced beyond $X$, meets the line through $B$ which is perpendicular to $BC$ at $U$. Also the line $AY$, produced beyond $Y$, meets the line through $C$ which is perpendicular to $BC$ at $V$. Given that $BY = BA$ and $CX = CA$, determine the angle $\angle VAU$.

Define the function $f:\mathbb{R} \backslash \{-1,1\} \to \mathbb{R}$ to be \[f(x) = \sum_{a,b=0}^{\infty} \frac{x^{2^a3^b}}{1-x^{2^{a+1}3^{b+1}}} .\] Suppose that $f\left(y\right)-f\left(\tfrac{1}{y}\right)=2016$. Then $y$ can be written in simplest form as $\tfrac{p}{q}$. Find $p+q$. ($\mathbb{R} \backslash \{-1,1\}$ refers to the set of real numbers excluding $-1$ and $1$.)

Let $t_1 < t_2 < t_3 < \cdots < t_{99}$ be real numbers. Consider a function $f: \mathbb{R} \to \mathbb{R}$ given by $f(x)=|x-t_1|+|x-t_2|+...+|x-t_{99}|$ . Show that $f(x)$ will attain minimum value at $x=t_{50}$.

Find all functions \( f : \mathbb{N} \to \mathbb{N} \) that satisfy the following condition: for any positive integers \( m \) and \( n \) such that \( m > n \) and \( m \) is not divisible by \( n \), if we denote by \( r \) the remainder of the division of \( m \) by \( n \), then the remainder of the division of \( f(m) \) by \( n \) is \( f(r) \). [i]Proposed by Mykyta Kharin[/i]

Suppose $a>b>0$, $f(x)=\dfrac{2(a+b)x+2ab}{4x+a+b}$. Show that there exists an unique positive number $x$, such that $f(x)=\left(\dfrac{a^{\frac{1}{3}}+b^{\frac{1}{3}}}{2} \right)^3$.

Determine maximum real $ k$ such that there exist a set $ X$ and its subsets $ Y_{1}$, $ Y_{2}$, $ ...$, $ Y_{31}$ satisfying the following conditions: (1) for every two elements of $ X$ there is an index $ i$ such that $ Y_{i}$ contains neither of these elements; (2) if any non-negative numbers $ \alpha_{i}$ are assigned to the subsets $ Y_{i}$ and $ \alpha_{1}+\dots+\alpha_{31}=1$ then there is an element $ x\in X$ such that the sum of $ \alpha_{i}$ corresponding to all the subsets $ Y_{i}$ that contain $ x$ is at least $ k$.

Let $ a_1\equal{}a_2\equal{}1$ and \[ a_{n\plus{}2}\equal{}\frac{n(n\plus{}1)a_{n\plus{}1}\plus{}n^2a_n\plus{}5}{n\plus{}2}\minus{}2\]for each $ n\in\mathbb N$. Find all $ n$ such that $ a_n\in\mathbb N$.

Given is a half circle with midpoint $O$ and diameter $AB$. Let $Z$ be a random point inside the half circle, and let $X$ be the intersection of $OZ$ and the half circle, and $Y$ the intersection of $AZ$ and the half circle. If $P$ is the intersection of $BY$ with the tangent line in $X$ to the half circle, show that $PZ \perp BX$.

$\{a,b,c\}\subset\left(\frac{1}{\sqrt6},+\infty\right)$ such that $a^{2}+b^{2}+c^{2}=1.$ Prove that $\frac{1+a^{2}}{\sqrt{2a^{2}+3ab-c^{2}}}+\frac{1+b^{2}}{\sqrt{2b^{2}+3bc-a^{2}}}+\frac{1+c^{2}}{\sqrt{2c^{2}+3ca-b^{2}}}\ge2(a+b+c).$

A sequence $\{y_i\}$ is given, where $y_0=-\frac{1}{4},y_1=0$. For every positive integer $n$ the following equality holds: $$y_{n-1}+y_{n+1}=4y_n+1$$ Prove that for every positive integer $n$ the number $2y_{2n}+\frac{3}{2}$ a) is a positive integer b) is a square of a positive integer [i]D. Zmiaikou[/i]

Suppose that $ a,b,$ and $ c$ are positive real numbers. Prove that: $ \frac{a\plus{}b\plus{}c}{3} \le \sqrt{\frac{a^2\plus{}b^2\plus{}c^2}{3}} \le \frac {\frac{ab}{c}\plus{}\frac{bc}{a}\plus{}\frac{ca}{b}}{3}$. For each of the inequalities, find the conditions on $ a,b,$ and $ c$ such that equality holds.

For positive integers $n, k$ with $1 \le k \le n$, define $$L(n, k) = Lcm \,(n, n - 1, n -2, ..., n - k + 1)$$ Let $f(n)$ be the largest value of $k$ such that $L(n, 1) < L(n, 2) < ... < L(n, k)$. Prove that $f(n) < 3\sqrt{n}$ and $f(n) > k$ if $n > k! + k$.

Prove that $n$ ($\ge 4$) points of the plane are vertices of a convex $n$-gon if and only if any $4$ of them are vertices of a convex quadrilateral.

[u]Round 1 [/u] [b]p1.[/b] In the convex quadrilateral $ABCD$ with diagonals $AC$ and $BD$, you know that angle $BAC$ is congruent to angle $CBD$, and that angle $ACD$ is congruent to angle $ADB$. Show that angle $ABC$ is congruent to angle $ADC$. [img]https://cdn.artofproblemsolving.com/attachments/5/d/41cd120813d5541dc73c5d4a6c86cc82747fcc.png[/img] [b]p2.[/b] In how many different ways can you place $12$ chips in the squares of a $4 \times 4$ chessboard so that (a) there is at most one chip in each square, and (b) every row and every column contains exactly three chips. [b]p3.[/b] Students from Hufflepuff and Ravenclaw were split into pairs consisting of one student from each house. The pairs of students were sent to Honeydukes to get candy for Father's Day. For each pair of students, either the Hufflepuff student brought back twice as many pieces of candy as the Ravenclaw student or the Ravenclaw student brought back twice as many pieces of candy as the Hufflepuff student. When they returned, Professor Trelawney determined that the students had brought back a total of $1000$ pieces of candy. Could she have possibly been right? Why or why not? Assume that candy only comes in whole pieces (cannot be divided into parts). [b]p4.[/b] While you are on a hike across Deception Pass, you encounter an evil troll, who will not let you across the bridge until you solve the following puzzle. There are six stones, two colored red, two colored yellow, and two colored green. Aside from their colors, all six stones look and feel exactly the same. Unfortunately, in each colored pair, one stone is slightly heavier than the other. Each of the lighter stones has the same weight, and each of the heavier stones has the same weight. Using a balance scale to make TWO measurements, decide which stone of each color is the lighter one. [b]p5.[/b] Alex, Bob and Chad are playing a table tennis tournament. During each game, two boys are playing each other and one is resting. In the next game the boy who lost a game goes to rest, and the boy who was resting plays the winner. By the end of tournament, Alex played a total of $10$ games, Bob played $15$ games, and Chad played $17$ games. Who lost the second game? [u]Round 2 [/u] [b]p6.[/b] Consider a set of finitely many points on the plane such that if we choose any three points $A,B,C$ from the set, then the area of the triangle $ABC$ is less than $1$. Show that all of these points can be covered by a triangle whose area is less than $4$. [b]p7.[/b] A palindrome is a number that is the same when read forward and backward. For example, $1771$ and $23903030932$ are palindromes. Can the number obtained by writing the numbers from $1$ to $n$ in order be a palindrome for some $n > 1$ ? (For example, if $n = 11$, the number obtained is $1234567891011$, which is not a palindrome.) PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Given an acute triangle $ABC, H$ is the foot of the altitude drawn from the point $A$ on the line $BC, P$ and $K \ne H$ are arbitrary points on the segments $AH$ and$ BC$ respectively. Segments $AC$ and $BP$ intersect at point $B_1$, lines $AB$ and $CP$ at point $C_1$. Let $X$ and $Y$ be the projections of point $H$ on the lines $KB_1$ and $KC_1$, respectively. Prove that points $A, P, X$ and $Y$ lie on one circle.

Given $a_1\ge 1$ and $a_{k+1}\ge a_k+1$ for all $k\ge 1,2,\dots,n$, show that $a_1^3+a_2^3+\dots+a_n^3\ge (a_1+a_2+\dots+a_n)^2$

Find all positive integers $k$ for which the equation: $$ \text{lcm}(m,n)-\text{gcd}(m,n)=k(m-n)$$ has no solution in integers positive $(m,n)$ with $m\neq n$.

A magic $3 \times 5$ board can toggle its cells between black and white. Define a \textit{pattern} to be an assignment of black or white to each of the board's $15$ cells (so there are $2^{15}$ patterns total). Every day after Day 1, at the beginning of the day, the board gets bored with its black-white pattern and makes a new one. However, the board always wants to be unique and will die if any two of its patterns are less than $3$ cells different from each other. Furthermore, the board dies if it becomes all white. If the board begins with all cells black on Day $1$, compute the maximum number of days it can stay alive.

Let $m,n\geq 2$. One needs to cover the table $m \times n$ using only the following tiles: Tile 1 - A square $2 \times 2$. Tile 2 - A L-shaped tile with five cells, in other words, the square $3 \times 3$ [b]without[/b] the upper right square $2 \times 2$. Each tile 1 covers exactly $4$ cells and each tile 2 covers exactly $5$ cells. Rotation is allowed. Determine all pairs $(m,n)$, such that the covering is possible.

An integer $a$ is called friendly if the equation $(m^2+n)(n^2+m)=a(m-n)^3$ has a solution over the positive integers. [b]a)[/b] Prove that there are at least $500$ friendly integers in the set $\{ 1,2,\ldots ,2012\}$. [b]b)[/b] Decide whether $a=2$ is friendly.

A natural number $n$ is said to have the property $P$, if whenever $n$ divides $a^{n}-1$ for some integer $a$, $n^2$ also necessarily divides $a^{n}-1$. [list=a] [*] Show that every prime number $n$ has the property $P$. [*] Show that there are infinitely many composite numbers $n$ that possess the property $P$. [/list]