Let \( AL \) be the bisector of triangle \( ABC \), \( O \) the center of its circumcircle, and \( D \) and \( E \) the midpoints of \( BL \) and \( CL \), respectively. Points \( P \) and \( Q \) are chosen on segments \( AD \) and \( AE \) such that \( APLQ \) is a parallelogram. Prove that \( PQ \perp AO \). [i]Proposed by Mykhailo Plotnikov[/i]

Determine all functions $f: \mathbb R \to \mathbb R$ with the property \[f(f(x)+y)=2x+f(f(y)-x), \quad \forall x,y \in \mathbb R.\]

Let $f$ be a function on non-negative integers defined as follows $$f(2n)=f(f(n))~~~\text{and}~~~f(2n+1)=f(2n)+1$$ [b](a)[/b] If $f(0)=0$ , find $f(n)$ for every $n$. [b](b)[/b] Show that $f(0)$ cannot equal $1$. [b](c)[/b] For what non-negative integers $k$ (if any) can $f(0)$ equal $2^k$ ?

Shrek throws $5$ balls into $5$ empty bins, where each ball’s target is chosen uniformly at random. After Shrek throws the balls, the probability that there is exactly one empty bin can be written in the form $m/n$ , where $m$ and $n$ are relatively prime positive integers. Compute $m + n$.

Sasha wants to bake $6$ cookies in his $8$ inch $\times$ $8$ inch square baking sheet. With a cookie cutter, he cuts out from the dough six circular shapes, each exactly $3$ inches in diameter. Can he place these six dough shapes on the baking sheet without the shapes touching each other? If yes, show us how. If no, explain why not. (Assume that the dough does not expand during baking.)

The areas of two right-angled triangles have ratio equal to the squares of their hypotenuses. Show that these triangles are similar.

Let $A$ be a set of $N$ residues $\pmod{N^{2}}$. Prove that there exists a set $B$ of of $N$ residues $\pmod{N^{2}}$ such that $A + B = \{a+b|a \in A, b \in B\}$ contains at least half of all the residues $\pmod{N^{2}}$.

Let $p$ be a fixed prime. Find all pairs $(x,y)$ of positive numbers satisfying $p(x-y) = xy$.

An integer between $1000$ and $9999$, inclusive, is chosen at random. What is the probability that it is an odd integer whose digits are all distinct? $\textbf{(A) }\frac{14}{75}\qquad\textbf{(B) }\frac{56}{225}\qquad\textbf{(C) }\frac{107}{400}\qquad\textbf{(D) }\frac{7}{25}\qquad\textbf{(E) }\frac{9}{25}$

The base of isosceles $\triangle{ABC}$ is $24$ and its area is $60$. What is the length of one of the congruent sides? $\textbf{(A)}\ 5 \qquad \textbf{(B)}\ 8 \qquad \textbf{(C)}\ 13 \qquad \textbf{(D)}\ 14 \qquad \textbf{(E)}\ 18$

A group of women working together at the same rate can build a wall in $45$ hours. When the work started, all the women did not start working together. They joined the worked over a period of time, one by one, at equal intervals. Once at work, each one stayed till the work was complete. If the first woman worked 5 times as many hours as the last woman, for how many hours did the first woman work?

Let $a,b,c$ be positive numbers. Prove that a triangle with sides $a,b,c$ exists if and only if the system of equations $$\begin{cases}\dfrac{y}{z}+\dfrac{z}{y}=\dfrac{a}{x} \\ \\ \dfrac{z}{x}+\dfrac{x}{z}=\dfrac{b}{y} \\ \\ \dfrac{x}{y}+\dfrac{y}{x}=\dfrac{c}{z}\end{cases}$$ has a real solution.

For a positive real number $ a > 1$, prove the following inequality. $ \frac {1}{a \minus{} 1}\left(1 \minus{} \frac {\ln a}{a\minus{}1}\right) < \int_0^1 \frac {x}{a^x}\ dx < \frac {1}{\ln a}\left\{1 \minus{} \frac {\ln (\ln a \plus{} 1)}{\ln a}\right\}$

Find all ordered triples of positive integers $(x,y,z)$ such that \[3^{x}+11^{y}=z^{2}\]

An $n\times n$ square table is divided into $n^2$ unit cells. Some unit segments of the obtained grid (i.e. the side of any unit cell) is colored black so that any unit cell of the given square has exactly one black side. Find [b]a)[/b] the smallest [b]b)[/b] the greatest possible number of black unit segments.

Let $S$ be a finite set of size $s\geq 1$ defined with a uniform probability $\mathbb{P}$( i.e. for any subset $X\subset S$ of size $x$, $\mathbb{P}(x)=\frac{x}{s}$). Suppose $A$ and $B$ are subsets of $S$. They are said to be independent iff $\mathbb{P}(A)\mathbb{P}(B)=\mathbb{P}(A\cap B)$. Which if these is sufficient for independence? (A)$|A\cup B|=|A|+|B|$ (B)$|A\cap B|=|A|+|B|$ (C)$|A\cup B|=|A|\cdot |B|$ (D)$|A\cap B|=|A|\cdot |B|$

Let $f(x) = ax^7+bx^3+cx-5$, where $a,b$ and $c$ are constants. If $f(-7) = 7$, the $f(7)$ equals $\textbf {(A) } -17 \qquad \textbf {(B) } -7 \qquad \textbf {(C) } 14 \qquad \textbf {(D) } 21\qquad \textbf {(E) } \text{not uniquely determined}$

Given $6$ points in a plane, assume that each two of them are connected by a segment. Let $D$ be the length of the longest, and $d$ the length of the shortest of these segments. Prove that $\frac Dd\ge\sqrt3$.

Find all pairs $(x,y)$ with $x,y$ real, satisfying the equations $$\sin\bigg(\frac{x+y}{2}\bigg)=0~,~\vert x\vert+\vert y\vert=1$$

In the Gregorian calendar: (i) years not divisible by $4$ are common years; (ii) years divisible by $4$ but not by $100$ are leap years; (iii) years divisible by $100$ but not by $400$ are common years; (iv) years divisible by $400$ are leap years; (v) a leap year contains $366$ days; a common year $365$ days. Prove that the probability that Christmas falls on a Wednesday is not $1/7.$

If $f(x)=\dfrac{x^4+x^2}{x+1}$, then $f(i)$, where $i=\sqrt{-1}$, is equal to: $\textbf{(A) }1+i\qquad\textbf{(B) }1\qquad\textbf{(C) }-1\qquad\textbf{(D) }0\qquad \textbf{(E) }-1-i$

A given convex polyhedron has no vertex which belongs to exactly 3 edges. Prove that the number of faces of the polyhedron that are triangles, is at least 8.

The eyes of a magician are blindfolded while a person $A$ from the audience arranges $n$ identical coins in a row, some are heads and the others are tails. The assistant of the magician asks $A$ to write an integer between $1$ and $n$ inclusive and to show it to the audience. Having seen the number, the assistant chooses a coin and turns it to the other side (so if it was heads it becomes tails and vice versa) and does not touch anything else. Afterwards, the bandages are removed from the magician, he sees the sequence and guesses the written number by $A$. For which $n$ is this possible? [hide=Spoiler hint] The original formulation asks: a) Show that if $n$ is possible, so is $2n$; b) Show that only powers of $2$ are possible; I have omitted this from the above formulation, for the reader's interest. [/hide]

Find all four-digit natural numbers $\overline{xyzw}$ with the property that their sum plus the sum of their digits equals $2003$.

Given three tetrahedrons $A_iB_i C_i D_i$ ($i=1,2,3$), planes $\alpha _i,\beta _i,\gamma _i$ ($i=1,2,3$) are drawn through $B_i ,C_i ,D_i$ respectively, and they are perpendicular to edges $A_i B_i, A_i C_i, A_i D_i$ ($i=1,2,3$) respectively. Suppose that all nine planes $\alpha _i,\beta _i,\gamma _i$ ($i=1,2,3$) meet at a point $E$, and points $A_1,A_2,A_3$ lie on line $l$. Determine the intersection (shape and position) of the circumscribed spheres of the three tetrahedrons.