Two circles have radius 5 and 26. The smaller circle passes through center of the larger one. What is the difference between the lengths of the longest and shortest chords of the larger circle that are tangent to the smaller circle? [i]Ray Li.[/i]

Let $n$ and $k$ be relatively prime positive integers with $k<n$. Each number in the set $M=\{1,2,3,\ldots,n-1\}$ is colored either blue or white. For each $i$ in $M$, both $i$ and $n-i$ have the same color. For each $i\ne k$ in $M$ both $i$ and $|i-k|$ have the same color. Prove that all numbers in $M$ must have the same color.

Nine positive integers $a_1,a_2,...,a_9$ have their last $2$-digit part equal to $11,12,13,14,15,16,17,18$ and $19$ respectively. Find the last $2$-digit part of the sum of their squares.

A triangle $ABC$ and a positive number $k$ are given. Find the locus of a point $M$ inside the triangle such that the projections of $M$ on the sides of $\Delta ABC$ form a triangle of area $k$.

A Haiku is a Japanese poem of seventeen syllables, in three lines of five, seven, and five. Using the four words “Hi”, “hey”, “hello”, and “haiku”, How many haikus Can somebody make? (Repetition is allowed, Order does matter.) [i]Proposed by Jeff Lin[/i]

Find all sets of four real numbers $x_1, x_2, x_3, x_4$ such that the sum of any one and the product of the other three is equal to 2.

Let $n$ be an integer of the form $a^2 + b^2$, where $a$ and $b$ are relatively prime integers and such that if $p$ is a prime, $p \leq \sqrt{n}$, then $p$ divides $ab$. Determine all such $n$.

Consider the polynomial $ P(x) \equal{} x^3 \plus{} 153x^2 \minus{} 111x \plus{} 38$. (a) Prove that there are at least nine integers $ a$ in the interval $ [1, 3^{2000}]$ for which $ P(a)$ is divisible by $ 3^{2000}$. (b) Find the number of integers $ a$ in $ [1, 3^{2000}]$ with the property from (a).

Let $a$, $x$, $y$, $z$ be real numbers such that $ax-y+z=3a-1$ ve $x-ay+z=a^2-1$, which of the followings cannot be equal to $x^2+y^2+z^2$? $ \textbf{(A)}\ \sqrt 2 \qquad\textbf{(B)}\ \sqrt 3 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ \sqrt[3]{4} \qquad\textbf{(E)}\ \text{None of the preceding} $

We are given two point sets $A$ and $B$ which are both composed of finite disjoint arcs on the unit circle. Moreover, the length of each arc in $B$ is equal to $\dfrac{\pi}{m}$ ($m \in \mathbb{N}$). We denote by $A^j$ the set obtained by a counterclockwise rotation of $A$ about the center of the unit circle for $\dfrac{j\pi}{m}$ ($j=1,2,3,\dots$). Show that there exists a natural number $k$ such that $l(A^k\cap B)\ge \dfrac{1}{2\pi}l(A)l(B)$.(Here $l(X)$ denotes the sum of lengths of all disjoint arcs in the point set $X$)

$1994$ points from the plane are given so that any $100$ of them can be selected $98$ that can be rounded (some points may be at the border of the circle) with a diameter of $1$. Determine the smallest number of circles with radius $1$, sufficient to cover all $1994$

For any real number $t$, denote by $[t]$ the greatest integer which is less than or equal to $t$. For example: $[8] = 8$, $[\pi] = 3$, and $[-5/2] = -3$. Show that the equation \[[x] + [2x] + [4x] + [8x] + [16x] + [32x] = 12345\] has no real solution.

All points in the plane that have both integer coordinates are painted, using the colors red, green, and yellow. If the points are painted so that there is at least one point of each color. Prove that there are always three points $X$, $Y$ and $Z$ of different colors, such that $\angle XYZ = 45^{\circ} $

Let $D:=\{(x, y)\mid x,y\in\mathbb R_+,x \ne y,x^y=y^x\}$. (Obvious that $x\ne1$ and $y\ne1$). And let $\alpha\le\beta$ be positive real numbers. Find $$\inf_{(x,y)\in D}x^\alpha y^\beta.$$ [i]Proposed by Arkady Alt[/i]

$ a,b,c,d,e,f $ are real numbers. It is true that: $ a+b+c+d+e+f=20 $ $ (a-2)^2+(b-2)^2+...+(f-2)^2=24 $ Find the maximum value of d.

Let $p$ be a prime number. Find all positive integers $a,b,c\ge 1$ such that: \[a^p+b^p=p^c.\]

In a conference there are $n \geq 10$ people. It is known that: [b]I.[/b] Each person knows at least $\left[\frac{n+2}{3}\right]$ other people. [b]II.[/b] For each pair of person $A$ and $B$ who don't know each other, there exist some people $A(1), A(2), \ldots, A(k)$ such that $A$ knows $A(1)$, $A(i)$ knows $A(i+1)$ and $A(k)$ knows $B$. [b]III.[/b] There doesn't exist a Hamilton path. Prove that: We can divide those people into 2 groups: $A$ group has a Hamilton cycle, and the other contains of people who don't know each other.

The function $f$ satisfies $f(x)+3f(1-x)=x^2$ for every real $x$. If $S=\{x \mid f(x)=0 \}$, which one is true? $\textbf{(A)}$ $S$ is an infinite set. $\textbf{(B)}$ $\{0,1\} \subset S$ $\textbf{(C)}$ $S=\phi$ $\textbf{(D)}$ $S = \{(3+\sqrt 3)/2, (3-\sqrt 3)/2\}$ $\textbf{(E)}$ None of above

For how many ordered pairs of positive integers $(x, y)$ is the least common multiple of $x$ and $y$ equal to $1{,}003{,}003{,}001$?

Prove that in any triangle, at most one side can be shorter than the altitude from the opposite vertex.

Find all permutations of the numbers $1,2,\ldots,9$ in which no two adjacent numbers have a sum divisible by $7$ or $13$.

Three dice with faces numbered 1 through 6 are stacked as shown. Seven of the eighteen faces are visible, leaving eleven faces hidden (back, bottom, between). The total number of dots NOT visible in this view is [asy] draw((0,0)--(2,0)--(3,1)--(3,7)--(1,7)--(0,6)--cycle); draw((3,7)--(2,6)--(0,6)); draw((3,5)--(2,4)--(0,4)); draw((3,3)--(2,2)--(0,2)); draw((2,0)--(2,6)); dot((1,1)); dot((.5,.5)); dot((1.5,.5)); dot((1.5,1.5)); dot((.5,1.5)); dot((2.5,1.5)); dot((.5,2.5)); dot((1.5,2.5)); dot((1.5,3.5)); dot((.5,3.5)); dot((2.25,2.75)); dot((2.5,3)); dot((2.75,3.25)); dot((2.25,3.75)); dot((2.5,4)); dot((2.75,4.25)); dot((.5,5.5)); dot((1.5,4.5)); dot((2.25,4.75)); dot((2.5,5.5)); dot((2.75,6.25)); dot((1.5,6.5)); [/asy] $\text{(A)}\ 21 \qquad \text{(B)}\ 22 \qquad \text{(C)}\ 31 \qquad \text{(D)}\ 41 \qquad \text{(E)}\ 53$

In any triangle $ABC$ prove that the following relationship holds: $$\begin{vmatrix}(b+c)^2&a^2&a^2\\b^2&(c+a)^2&b^2\\c^2&c^2&(a+b)^2\end{vmatrix}\ge93312r^6$$ [i]Proposed by D.M. Bătinețu-Giurgiu and Daniel Sitaru[/i]

Let $n\ge 1$ be an integer and consider the sum $$x=\sum_{k\ge 0} \dbinom{n}{2k} 2^{n-2k}3^k=\dbinom{n}{0}2^n+\dbinom{n}{2}2^{n-2}\cdot{}3+\dbinom{n}{4}2^{n-k}\cdot{}3^2 + \cdots{}.$$ Show that $2x-1,2x,2x+1$ form the sides of a triangle whose area and inradius are also integers.

We are given two mutually tangent circles in the plane, with radii $r_1, r_2$. A line intersects these circles in four points, determining three segments of equal length. Find this length as a function of $r_1$ and $r_2$ and the condition for the solvability of the problem.