Prove that if a cyclic polygon with an odd number of sides has all angles equal, then this polygon is regular.

Let $\Omega$ be the circumcircle of an acute-angled triangle $ABC$. A point $D$ is chosen on the internal bisector of $\angle ACB$ so that the points $D$ and $C$ are separated by $AB$. A circle $\omega$ centered at $D$ is tangent to the segment $AB$ at $E$. The tangents to $\omega$ through $C$ meet the segment $AB$ at $K$ and $L$, where $K$ lies on the segment $AL$. A circle $\Omega_1$ is tangent to the segments $AL, CL$, and also to $\Omega$ at point $M$. Similarly, a circle $\Omega_2$ is tangent to the segments $BK, CK$, and also to $\Omega$ at point $N$. The lines $LM$ and $KN$ meet at $P$. Prove that $\angle KCE = \angle LCP$. Poland

[color=blue][size=150]PERU TST IMO - 2006[/size] Saturday, may 20.[/color] [b]Question 03[/b] In each square of a board drawn into squares of $2^n$ rows and $n$ columns $(n\geq 1)$ are written a 1 or a -1, in such a way that the rows of the board constitute all the possible sequences of length $n$ that they are possible to be formed with numbers 1 and -1. Next, some of the numbers are replaced by zeros. Prove that it is possible to choose some of the rows of the board (It could be a row only) so that in the chosen rows, is fulfilled that the sum of the numbers in each column is zero. ---- [url=http://www.mathlinks.ro/Forum/viewtopic.php?t=88511]Spanish version[/url] $\text{\LaTeX}{}$ed by carlosbr

Given a square \(ABCD\) with a side length of 4 cm and a point \(E\) on side \(BC\), a square \(AEFG\) is constructed with side \(AE\), as shown in the figure. It is known that triangle \(DFG\) has an area of 1 cm\(^2\). Determine the area of square \(AEFG\).

Let $p>2$ be a prime number and \[1+\frac{1}{2^3}+\frac{1}{3^3}+\ldots +\frac{1}{(p-1)^3}=\frac{m}{n}\] where $m$ and $n$ are relatively prime. Show that $m$ is a multiple of $p$.

The altitudes of a triangles are $12$, $15$, and $20$. The largest angle in this triangle is $\textbf{(A) }72^\circ\qquad\textbf{(B) }75^\circ\qquad\textbf{(C) }90^\circ\qquad\textbf{(D) }108^\circ\qquad\textbf{(E) }120^\circ$

Let $a, b,c$ be positive integers with no common factor and satisfy the conditions $\frac1a +\frac1b=\frac1c$ Prove that $a + b$ is a square.

For integer $a$, $a \neq 0$, $v_2(a)$ is greatest nonnegative integer $k$ such that $2^k | a$. For given $n \in \mathbb{N}$ determine highest possible cardinality of subset $A$ of set $ \{1,2,3,...,2^n \} $ with following property: For all $x, y \in A$, $x \neq y$, number $v_2(x-y)$ is even.

Suppose that $x,y,z$ are nonnegative real numbers satisfying the equation $$\sqrt{xyz}-\sqrt{(1-x)(1-y)z} - \sqrt{(1-x)y(1-z)}-\sqrt{x(1-y)(1-z)} = -\frac{1}{2}.$$ The largest possible value of $\sqrt{xy}$ equals $\tfrac{a+\sqrt{b}}{c}.$ where $a,b,$ and $c$ are positive integers such that $b$ is not divisible by the square of any prime. Find $a^2+b^2+c^2.$

Let $ a,b> 0$ such that $ a\ne b$. Prove that: $ \sqrt {ab} < \dfrac{a \minus{} b}{\ln a \minus{} \ln b} < \dfrac{a \plus{} b}{2}$

(a) Alenka has two jars, each with $6$ marbles labeled with numbers $1$ through $6$. She draws one marble from each jar at random. Denote by $p_n$ the probability that the sum of the labels of the two drawn marbles is $n$. Compute pn for each $n \in N$. (b) Barbara has two jars, each with $6$ marbles which are labeled with unknown numbers. The sets of labels in the two jars may differ and two marbles in the same jar can have the same label. If she draws one marble from each jar at random, the probability that the sum of the labels of the drawn marbles is $n$ equals the probability $p_n$ in Alenka’s case. Determine the labels of the marbles. Find all solution

Prove that an integer $n > 1$ is a prime number if and only if, for every integer $k$ with $1\le k \le n-1$, the binomial coefficient $n \choose k$ is divisible by $n$.

Let $XYZ$ be a triangle with $ \angle X = 60^\circ $ and $ \angle Y = 45^\circ $. A circle with center $P$ passes through points $A$ and $B$ on side $XY$, $C$ and $D$ on side $YZ$, and $E$ and $F$ on side $ZX$. Suppose $AB=CD=EF$. Find $ \angle XPY $ in degrees.

A quadrilateral $ABCD$ is circumscribed around a circle $\omega$ centered at $I$. Lines $AC$ and $BD$ meet at point $P$, lines $AB$ and $CD$ meet at point $£$, lines $AD$ and $BC$ meet at point $F$. Point $K$ on the circumcircle of triangle $E1F$ is such that $\angle IKP = 90^o$. The ray $PK$ meets $\omega$ at point $Q$. Prove that the circumcircle of triangle $EQF$ touches $\omega$.

Let $X$ be a $5\times 5$ matrix with each entry be $0$ or $1$. Let $x_{i,j}$ be the $(i,j)$-th entry of $X$ ($i,j=1,2,\hdots,5$). Consider all the $24$ ordered sequence in the rows, columns and diagonals of $X$ in the following: \begin{align*} &(x_{i,1}, x_{i,2},\hdots,x_{i,5}),\ (x_{i,5},x_{i,4},\hdots,x_{i,1}),\ (i=1,2,\hdots,5) \\ &(x_{1,j}, x_{2,j},\hdots,x_{5,j}),\ (x_{5,j},x_{4,j},\hdots,x_{1,j}),\ (j=1,2,\hdots,5) \\ &(x_{1,1},x_{2,2},\hdots,x_{5,5}),\ (x_{5,5},x_{4,4},\hdots,x_{1,1}) \\ &(x_{1,5},x_{2,4},\hdots,x_{5,1}),\ (x_{5,1},x_{4,2},\hdots,x_{1,5}) \end{align*} Suppose that all of the sequences are different. Find all the possible values of the sum of all entries in $X$.

Given two distinct numbers $ b_1$ and $ b_2$, their product can be formed in two ways: $ b_1 \times b_2$ and $ b_2 \times b_1.$ Given three distinct numbers, $ b_1, b_2, b_3,$ their product can be formed in twelve ways: $ b_1\times(b_2 \times b_3);$ $ (b_1 \times b_2) \times b_3;$ $ b_1 \times (b_3 \times b_2);$ $ (b_1 \times b_3) \times b_2;$ $ b_2 \times (b_1 \times b_3);$ $ (b_2 \times b_1) \times b_3;$ $ b_2 \times(b_3 \times b_1);$ $ (b_2 \times b_3)\times b_1;$ $ b_3 \times(b_1 \times b_2);$ $ (b_3 \times b_1)\times b_2;$ $ b_3 \times(b_2 \times b_1);$ $ (b_3 \times b_2) \times b_1.$ In how many ways can the product of $ n$ distinct letters be formed?

Find all pairs $(m, n)$ of integers for which $$\sqrt{m^2-6}<2\sqrt{n}-m<\sqrt{m^2-2}.$$

Let $G$ be the set of $2\times 2$ matrices that such $$ G = \left\{ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \mid\, a,b,c,d \in \mathbb{Z}, ad-bc = 1, c \text{ is a multiple of } 3 \right\} $$ and two matrices in $G$: $$ A = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}\;\;\; B = \begin{pmatrix} -1 & 1 \\ -3 & 2 \end{pmatrix} $$ Show that any matrix in $G$ can be written as a product $M_1M_2\cdots M_r$ such that $M_i \in \{A, A^{-1}, B, B^{-1}\}, \forall i \leq r$

Let $k$ be a circle with diameter $AB$. Let $C$ be a point on the straight line $AB$, so that $B$ between $A$ and $C$ lies. Let $T$ be a point on $k$ such that $CT$ is a tangent to $k$. Let $l$ be the parallel to $CT$ through $A$ and $D$ the intersection of $l$ and the perpendicular to $AB$ through $T$. Show that the line $DB$ bisects segment $CT$.

Kermit writes down the numbers $1$, $2$, $3$, $4$, $5$. He then erases one number, and discovers that the sum of the remaining numbers is $13$. Which number was erased? $\textbf{(A) }1\qquad\textbf{(B) }2\qquad\textbf{(C) }3\qquad\textbf{(D) }4\qquad\textbf{(E) }5$

A man walked a certain distance at a constant rate. If he had gone $\textstyle\frac{1}{2}$ mile per hour faster, he would have walked the distance in four-fifths of the time; if he had gone $\textstyle\frac{1}{2}$ mile per hour slower, he would have been $2\textstyle\frac{1}{2}$ hours longer on the road. The distance in miles he walked was $\textbf{(A) }13\textstyle\frac{1}{2}\qquad\textbf{(B) }15\qquad\textbf{(C) }17\frac{1}{2}\qquad\textbf{(D) }20\qquad \textbf{(E) }25$

There are $n$ boys and $n$ girls in a school class, where $n$ is a positive integer. The heights of all the children in this class are distinct. Every girl determines the number of boys that are taller than her, subtracts the number of girls that are taller than her, and writes the result on a piece of paper. Every boy determines the number of girls that are shorter than him, subtracts the number of boys that are shorter than him, and writes the result on a piece of paper. Prove that the numbers written down by the girls are the same as the numbers written down by the boys (up to a permutation). [i]Proposed by Stephan Wagner, Austria[/i]

The tangent lines from a point $P$ meet a circle $k$ at $A$ and $B$. Let $X$ be an arbitrary point on the shorter arc $AB$, and $C$ and $D$ be the orthogonal projections of $P$ onto the lines $AX$ and $BX$, respectively. Prove that the line $CD$ passes through a fixed point $Y$ as $X$ moves along the arc $AB$.

Given two points $P,\ Q$ on the parabola $C: y=x^2-x-2$ in the $xy$ plane. Note that the $x$ coodinate of $P$ is less than that of $Q$. (a) If the origin $O$ is the midpoint of the lines　egment $PQ$, then find the equation of the line $PQ$. (b) If the origin $O$ divides internally the line segment $PQ$ by 2:1, then find the equation of $PQ$. (c) If the origin $O$ divides internally the line segment $PQ$ by 2:1, find the area of the figure bounded by the parabola $C$ and the line $PQ$.

Is there an infinite sequence of positive integers where no two terms are relatively prime, no term divides any other term, and there is no integer larger than $1$ that divides every term of the sequence?