Ahuiliztli is playing around with some coins (pennies, nickels, dimes, and quarters). She keeps grabbing $k$ coins and calculating the value of her handful. After a while, she begins to notice that if $k$ is even, she more often gets even sums, and if $k$ is odd, she more often gets odd sums. Help her prove this true! Given $k$ coins chosen uniformly and at random, prove that. the probability that the parity of $k$ is the same as the parity of the $k$ coins' value is greater than the probability that the parities are different.

Given a polygon with $n$ sides, we assign the numbers $0,1,...,n-1$ to the vertices, and to each side is assigned the sum of the numbers assigned to its ends. The figure shows an example for $n = 5$. Notice that the numbers assigned to the sides are still in arithmetic progression. [img]https://cdn.artofproblemsolving.com/attachments/c/0/975969e29a7953dcb3e440884461169557f9a7.png[/img] $\bullet$ Make the respective assignment for a $9$-sided polygon, and generalize for odd $n$. $\bullet$ Prove that this is not possible if $n$ is even.

The numbers $$\frac{1}{i+j-1} \,\,\,\,\,\,\, (i = 1,2,...,n; j = 1,2,...,n)$$ are written in an $n$ by $n$ table: the number $1/(i + j - 1)$ stands at the intersection of the $i$-th row and $j$-th column. Chose any $n$ squares of the table so that no two of them stand in the same row and no two of them stand in the same column. Prove that the sum of the numbers in these $n$ squares is not less than $1$. (Sergey Ivanov, St Petersburg)

Prove that a positive integer $n$ is composite if and only if there exist positive integers $a,b,x,y$ such that $a+b = n$ and $\frac{x}{a}+\frac{y}{b}= 1$.

Given a circle and a point $A$ inside the circle, but not at its center. Find points $B$, $C$, $D$ on the circle which maximise the area of the quadrilateral $ABCD$.

For every complex number $z$ different from 0 and 1 we define the following function \[ f(z) := \sum \frac 1 { \log^4 z } \] where the sum is over all branches of the complex logarithm. a) Prove that there are two polynomials $P$ and $Q$ such that $f(z) = \displaystyle \frac {P(z)}{Q(z)} $ for all $z\in\mathbb{C}-\{0,1\}$. b) Prove that for all $z\in \mathbb{C}-\{0,1\}$ we have \[ f(z) = \frac { z^3+4z^2+z}{6(z-1)^4}. \]

Let $ABC$ be a triangle of sides $AB = 15$, $AC = 14$ and $BC = 13$. Let $M$ be the midpoint of side $AB$ and let $I$ be the incenter of triangle $ABC$. The line $MI$ intersects the altitude corresponding to the side $AB$ of triangle $ABC$ at point $P$. Calculate the length of the segment $PC$. Note: The incenter of a triangle is the intersection point of its angle bisectors.

Frieda the frog begins a sequence of hops on a $3 \times 3$ grid of squares, moving one square on each hop and choosing at random the direction of each hop up, down, left, or right. She does not hop diagonally. When the direction of a hop would take Frieda off the grid, she "wraps around'' and jumps to the opposite edge. For example if Frieda begins in the center square and makes two hops "up'', the first hop would place her in the top row middle square, and the second hop would cause Frieda to jump to the opposite edge, landing in the bottom row middle square. Suppose Frieda starts from the center square, makes at most four hops at random, and stops hopping if she lands on a corner square. What is the probability that she reaches a corner square on one of the four hops? $\textbf{(A) }\frac{9}{16}\qquad\textbf{(B) }\frac{5}{8}\qquad\textbf{(C) }\frac{3}{4}\qquad\textbf{(D) }\frac{25}{32}\qquad\textbf{(E) }\frac{13}{16}$

Let all distances between the vertices of a convex $n$-gon ($n > 3$) be different. a) A vertex is called uninteresting if the closest vertex is adjacent to it. What is the minimal possible number of uninteresting vertices (for a given $n$)? b) A vertex is called unusual if the farthest vertex is adjacent to it. What is the maximal possible number of unusual vertices (for a given $n$)? [i](Proposed by B.Frenkin)[/i]

A chord of a circle is perpendicular to a radius at the midpoint of the radius. The ratio of the area of the larger of the two regions into which the chord divides the circle to the smaller can be expressed in the form $\frac{a\pi+b\sqrt{c}}{d\pi-e\sqrt{f}}$, where $a$, $b$, $c$, $d$, $e$, and $f$ are positive integers, $a$ and $e$ are relatively prime, and neither $c$ nor $f$ is divisible by the square of any prime. Find the remainder when the product $abcdef$ is divided by 1000.

Let $ABC$ be any triangle with $\angle BAC \le \angle ACB \le \angle CBA$. Let $D, E$ and $F$ be the midpoints of $BC, CA$ and $AB$, respectively, and let $\epsilon$ be a positive real number. Suppose there is an ant (represented by a point $T$ ) and two spiders (represented by points $P_1$ and $P_2$, respectively) walking on the sides $BC, CA, AB, EF, FD$ and $DE$. The ant and the spiders may vary their speeds, turn at an intersection point, stand still, or turn back at any point; moreover, they are aware of their and the others’ positions at all time. Assume that the ant’s speed does not exceed $1$ mm/s, the first spider’s speed does not exceed $\frac{\sin A}{2 \sin A+\sin B}$ mm/s, and the second spider’s speed does not exceed $\epsilon$ mm/s. Show that the spiders always have a strategy to catch the ant regardless of the starting points of the ant and the spiders. Note: the two spiders can discuss a plan before the hunt starts and after seeing all three starting points, but cannot communicate during the hunt.

Twelve 1-Euro-coins are laid flat on a table, such that their midpoints form a regular $12$-gon. Adjacent coins are tangent to each other. Prove that it is possible to put another seven such coins into the interior of the ring of the twelve coins.

Suppose point $P$ is inside quadrilateral $ABCD$ such that $\angle{PAB} = \angle{PDA}, \angle{PAD} = \angle{PDC}, \angle{PBA} = \angle{PCB}, \angle{PBC} = \angle{PCD}.$ If $PA = 4, PB = 5,$ and $PC = 10$, compute the perimeter of $ABCD$.

For a quadrilateral $ABCD$ is given that $\angle CBD=2\angle ADB$, $\angle ABD=2\angle CDB$, and $AB=CB$. Prove that $AD=CD$.

Find all functions $f:\mathbb R^+\to \mathbb R^+$ such that $$f(x+y)f(f(x))=f(1+yf(x))$$ for all $x,y\in \mathbb R^+.$ [i]Proposed by Ming Hsiao[/i]

Let there be $ n \ge 2$ real numbers such that none of them is greater than the arithmetic mean of the other numbers. Prove that all the numbers are equal.

Nickolas and Peter divide $2n+1$ nuts amongst each other. Both of them want to get as many as possible. Three methods are suggested to them for doing so, each consisting of three stages. The first two stages are the same in all three methods: [i]Stage 1:[/i] Peter divides the nuts into 2 heaps, each containing at least 2 nuts. [i]Stage 2:[/i] Nickolas divides both heaps into 2 heaps, each containing at least 1 nut. Finally, stage 3 varies among the three methods as follows: [i]Method 1:[/i] Nickolas takes the smallest and largest of the heaps. [i]Method 2:[/i] Nickolas takes the two middle size heaps. [i]Method 3:[/i] Nickolas chooses between taking the biggest and the smallest heap or the two middle size heaps, but gives one nut to Peter for the right of choice. Determine the most and the least profitable method for Nickolas.

Let $S=\{1,2,3,\cdots,100\}$. Find the maximum value of integer $k$, such that there exist $k$ different nonempty subsets of $S$ satisfying the condition: for any two of the $k$ subsets, if their intersection is nonemply, then the minimal element of their intersection is not equal to the maximal element of either of the two subsets.

Triangle $ABC$ has sides $AB=9,BC = 5\sqrt{3},$ and $AC=12$. Points $A=P_0, P_1, P_2, \dots, P_{2450} = B$ are on segment $\overline{AB}$ with $P_k$ between $P_{k-1}$ and $P_{k+1}$ for $k=1,2,\dots,2449$, and points $A=Q_0, Q_1, Q_2, \dots ,Q_{2450} = C$ for $k=1,2,\dots,2449$. Furthermore, each segment $\overline{P_kQ_k}, k=1,2,\dots,2449$, is parallel to $\overline{BC}$. The segments cut the triangle into $2450$ regions, consisting of $2449$ trapezoids and $1$ triangle. Each of the $2450$ regions have the same area. Find the number of segments $\overline{P_kQ_k}, k=1,2 ,\dots,2450$, that have rational length.

The number of real triples $(x , y , z )$ that satisfy the equation $x^4 + 4y^4 + z^4 + 4 = 8xyz$ is (A): $0$, (B): $1$, (C): $2$, (D): $8$, (E): None of the above.

Let $S$ be the set of points on the rays forming the sides of a $120^{\circ}$ angle, and let $P$ be a fixed point inside the angle on the angle bisector. Consider all distinct equilateral triangles $PQR$ with $Q$ and $R$ in $S$. (Points $Q$ and $R$ may be on the same ray, and switching the names of $Q$ and $R$ does not create a distinct triangle.) There are [asy] draw((0,0)--(6,10.2), EndArrow); draw((0,0)--(6,-10.2), EndArrow); draw((0,0)--(6,0), dotted); dot((6,0)); label("P", (6,0), S); [/asy] $ \textbf{(A)}\ \text{exactly 2 such triangles} \\ \qquad\textbf{(B)}\ \text{exactly 3 such triangles} \\ \qquad\textbf{(C)}\ \text{exactly 7 such triangles} \\ \qquad\textbf{(D)}\ \text{exactly 15 such triangles} \\ \qquad\textbf{(E)}\ \text{more than 15 such triangles} $

Fifty teams participate in a round robin competition over 50 days. Moreover, all the teams (at least two) that show up in any day must play against each other. Prove that on every pair of consecutive days, there is a team that has to play on those two days.

Let $0 < a < \frac{\pi}{2}$ and $x_1,x_2,...,x_n$ be real numbers such that $\sin x_1 + \sin x_2 +... + \sin x_n \ge n \cdot sin a $. Prove that $\sin (x_1 - a) + \sin (x_2 - a) + ... + \sin (x_n - a) \ge 0$ .

Let $ABC$ be a triangle with area $S$, and let $P$ be a point in the plane. Prove that $AP+BP+CP\geq 2\sqrt[4]{3}\sqrt{S}$.

In a right triangle, the altitude from a vertex to the hypotenuse splits the hypotenuse into two segments of lengths $a$ and $b$. If the right triangle has area $T$ and is inscribed in a circle of area $C$, find $ab$ in terms of $T$ and $C$.