What is the radius of a circle inscribed in a rhombus with diagonals of length $ 10$ and $ 24$? $ \textbf{(A)}\ 4 \qquad \textbf{(B)}\ 58/13 \qquad \textbf{(C)}\ 60/13 \qquad \textbf{(D)}\ 5 \qquad \textbf{(E)}\ 6$

$ABC$ is an equilateral triangle with side length $ 1$. $BCDE$ is a square. Some point $F$ is equidistant from $A, D$, and $E$. Find the length of $AF$. [img]https://cdn.artofproblemsolving.com/attachments/2/4/194318955f7ed5fed1c58633cb29c33011371a.jpg[/img]

In some company, consisting of $n$ people, any two have at most $k \geq 2$ common friends. Lets call group of people working in the company unsocial if everyone in the group has at most one friend from the group. Prove that there exists an unsocial group consisting of at least $\sqrt{\frac{2n}{k}}$ people [i]M. Zorka[/i]

The numbers from $1$ to $49$ are arranged in a spiral pattern on a square grid, beginning at the center. The first few numbers have been entered into the grid below. Consider the four numbers that will appear in the shaded squares, on the same diagonal as the number $7.$ How many of these four numbers are prime? [asy] size(6cm); fill((4,0)--(5,0)--(5,1)--(4,1)--cycle,mediumgray); fill((3,1)--(4,1)--(4,2)--(3,2)--cycle,mediumgray); fill((1,3)--(1,4)--(2,4)--(2,3)--cycle,mediumgray); fill((0,4)--(0,5)--(1,5)--(1,4)--cycle,mediumgray); label(scale(.9)*"$1$", (3.5,3.5)); label(scale(.9)*"$2$", (4.5,3.5)); label(scale(.9)*"$3$", (4.5,4.5)); label(scale(.9)*"$4$", (3.5,4.5)); label(scale(.9)*"$5$", (2.5,4.5)); label(scale(.9)*"$6$", (2.5,3.5)); label(scale(.9)*"$7$", (2.5,2.5)); draw((1,0)--(1,7)--(2,7)--(2,0)--(3,0)--(3,7)--(4,7)--(4,0)--(5,0)--(5,7)--(6,7)--(6,0)--(7,0)--(7,7),gray); draw((0,1)--(7,1)--(7,2)--(0,2)--(0,3)--(7,3)--(7,4)--(0,4)--(0,5)--(7,5)--(7,6)--(0,6)--(0,7)--(7,7),gray); draw((4,4)--(3,4)--(3,3)--(5,3)--(5,5)--(2,5)--(2,2)--(6,2)--(6,6)--(1,6)--(1,1)--(7,1)--(7,7)--(0,7)--(0,0)--(7,0),linewidth(1.25)); [/asy] $\textbf{(A) } 0\qquad\textbf{(B) } 1\qquad\textbf{(C) } 2\qquad\textbf{(D) } 3\qquad\textbf{(E) } 4$

Let $ n$ be a $ 5$-digit number, and let $ q$ and $ r$ be the quotient and remainder, respectively, when $ n$ is divided by $ 100$. For how many values of $ n$ is $ q \plus{} r$ divisible by $ 11$? $ \textbf{(A)}\ 8180 \qquad \textbf{(B)}\ 8181 \qquad \textbf{(C)}\ 8182 \qquad \textbf{(D)}\ 9000 \qquad \textbf{(E)}\ 9090$

Let $A$ and $B$ be digits between $0$ and $9$, and suppose that the product of the two-digit numbers $\overline{AB}$ and $\overline{BA}$ is equal to $k$. Given that $k+1$ is a multiple of $101$, find $k$. [i]Proposed by Andrew Wu[/i]

Let $ABC$ be a given triangle with circumcenter $O$ and orthocenter $H$. Let $D, E$ and $F$ be the feet of the perpendiculars from $A, B$ and $C$ to the opposite sides, respectively. Let $A'$ be the reflection of $A$ with respect to $EF$. Prove that $HOA'D$ is a cyclic quadrilateral. [i]Proposed by Imad Uddin Ahmad Hasin[/i]

A rectangle $\mathcal{R}$ with odd integer side lengths is divided into small rectangles with integer side lengths. Prove that there is at least one among the small rectangles whose distances from the four sides of $\mathcal{R}$ are either all odd or all even. [i]Proposed by Jeck Lim, Singapore[/i]

Let $n$ be a positive integer. A German set in an $n \times n$ square grid is a set of $n$ cells which contains exactly one cell in each row and column. Given a labelling of thecells with the integers from $1$ to $n^2$ using each integer exactly once, we say that an integer is a German product if it is the product of the labels of the cells in a German set. (a) Let $n=8$. Determine whether there exists a labelling of an $8 \times 8$ grid such that the following condition is fulfilled: The difference of any two German products is alwaysdivisible by $65$. (b) Let $n=10$. Determine whether there exists a labelling of a $10 \times 10$ grid such that the following condition is fulfilled: The difference of any two German products is always divisible by $101$.

Four coplanar regular polygons share a common vertex but have no interior points in common. Each polygon is adjacent to two of the other polygons, and each pair of adjacent polygons has a common side length of $1$. How many possible perimeters are there for all such configurations? $\text{(A) }2\qquad\text{(B) }3\qquad\text{(C) }4\qquad\text{(D) }5\qquad\text{(E) more than }5$

Show that if $a,b,c$ are complex numbers that such that \[ (a+b)(a+c)=b \qquad (b+c)(b+a)=c \qquad (c+a)(c+b)=a\] then $a,b,c$ are real numbers.

A finite number of coins are placed on an infinite row of squares. A sequence of moves is performed as follows: at each stage a square containing more than one coin is chosen. Two coins are taken from this square; one of them is placed on the square immediately to the left while the other is placed on the square immediately to the right of the chosen square. The sequence terminates if at some point there is at most one coin on each square. Given some initial configuration, show that any legal sequence of moves will terminate after the same number of steps and with the same final configuration.

Let $ S$ be the set of all positive integers whose all digits are $ 1$ or $ 2$. Denote $ T_{n}$ as the set of all integers which is divisible by $ n$, then find all positive integers $ n$ such that $ S\cap T_{n}$ is an infinite set.

Triangle $ABC$ has $AB=BC=10$ and $CA=16$. The circle $\Omega$ is drawn with diameter $BC$. $\Omega$ meets $AC$ at points $C$ and $D$. Find the area of triangle $ABD$.

Find all $x,y\in\left(0,\frac{\pi}{2}\right)$ such that \begin{align*} \frac{\cos x}{\cos y}&=2\cos^2 y, \\ \frac{\sin x}{\sin y}&=2\sin^2 y. \end{align*}

Given is a cyclic quadrilateral $ABCD$ with $|AB| = |BC|$. Point $E$ is on the arc $CD$ where $A$ and $B$ are not on. Let $P$ be the intersection point of $BE$ and $CD$ , let $Q$ be the intersection point of $AE$ and $BD$ . Prove that $PQ \parallel AC$.

Given a circle $G$ with center $O$ and radius $r$. Let $AB$ be a fixed diameter of $G$. Let $K$ be a fixed point of segment $AO$. Denote by $t$ the line tangent to at $A$. For any chord $CD$ (other than $AB$) passing through $K$. Let $P$ and $Q$ be the points of intersection of lines $BC$ and $BD$ with $t$. Prove that the product $AP\cdot AQ$ remains costant as the chord $CD$ varies.

The altitudes $AA_1,BB_1,CC_1,DD_1$ of a tetrahedron $ABCD$ intersect in the center $H$ of the sphere inscribed in the tetrahedron $A_1B_1C_1D_1$. Prove that the tetrahedron $ABCD$ is regular. (D. Tereshin)

We call a positive integer $N$ [i]contagious[/i] if there are $1000$ consecutive non-negative integers such that the sum of all their digits is $N$. Find all contagious positive integers.

A word is a sequence of n letters of the alphabet {a, b, c, d}. A word is said to be complicated if it contains two consecutive groups of identic letters. The words caab, baba and cababdc, for example, are complicated words, while bacba and dcbdc are not. A word that is not complicated is a simple word. Prove that the numbers of simple words with n letters is greater than $2^n$, if n is a positive integer.

Prove that the polynomial $P(X) = (X^2-12X +11)^4+23$ can not be written as the product of three non-constant polynomials with integer coefficients. (Le Anh Vinh)

In the questions below: $G$ is a finite group; $H \leq G$ a subgroup of $G; |G : H |$ the index of $H$ in $G; |X |$ the number of elements of $X \subseteq G; Z (G)$ the center of $G; G'$ the commutator subgroup of $G; N_{G}(H )$ the normalizer of $H$ in $G; C_{G}(H )$ the centralizer of $H$ in $G$; and $S_{n}$ the $n$-th symmetric group. Let $a, b \in G.$ Suppose that $ab^{2}= b^{3}a$ and $ba^{2}= a^{3}b.$ Prove that $a = b = 1.$

In a plane a circle with radius $R$ and center $w$ and a line $\Lambda$ are given. The distance between $w$ and $\Lambda$ is $d, d > R$. The points $M$ and $N$ are chosen on $\Lambda$ in such a way that the circle with diameter $MN$ is externally tangent to the given circle. Show that there exists a point $A$ in the plane such that all the segments $MN$ are seen in a constant angle from $A.$

A line drawn from the vertex $A$ of the equilateral triangle $ABC$ meets the side $BC$ at $D$ and the circumcircle of the triangle at point $Q$. Prove that $\frac{1}{QD} = \frac{1}{QB} + \frac{1}{QC}$.

The integers $1, 2, \cdots, n^2$ are placed on the fields of an $n \times n$ chessboard $(n > 2)$ in such a way that any two fields that have a common edge or a vertex are assigned numbers differing by at most $n + 1$. What is the total number of such placements?