Determine all prime numbers $p, q$ and $r$ with $p + q^2 = r^4$. [i](Karl Czakler)[/i]

Let $E$ be the midpoint of side $[AB]$ of square $ABCD$. Let the circle through $B$ with center $A$ and segment $[EC]$ meet at $F$. What is $|EF|/|FC|$? $ \textbf{(A)}\ 2 \qquad\textbf{(B)}\ \dfrac{3}{2} \qquad\textbf{(C)}\ \sqrt{5}-1 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ \sqrt{3} $

Given a circle, point $A$ on it and point $M$ inside it. We consider the chords $BC$ passing through $M$. Prove that the circles passing through the midpoints of the sides of all the triangles $ABC$ are tangent to a fixed circle.

Raina the frog is playing a game in a circular pond with six lilypads around its perimeter numbered clockwise from $1$ to $6$ (so that pad $1$ is adjacent to pad $6$). She starts at pad $1$, and when she is on pad i, she may jump to one of its two adjacent pads, or any pad labeled with $j$ for which $j - i$ is even. How many jump sequences enable Raina to hop to each pad exactly once?

Determine the sum of the two largest prime factors of the integer $89! + 90!.$

Let triangle $\triangle{ABC}$ be acute. Point $D$ is the foot of the altitude of $\triangle{ABC}$ from $A$ to $\overline{BC},$ and $E$ is the foot of the altitude of $\triangle{ABC}$ from $B$ to $\overline{AC}.$ Let $F$ denote the point of intersection between $\overline{BE}$ and $\overline{AD},$ and let $G$ denote the point of intersection between $\overline{CF}$ and $\overline{DE}.$ The areas of triangles $\triangle{EFG}, \triangle{CDG},$ and $\triangle{CEG}$ are $1,4,$ and $3,$ respectively. Find the area of $\triangle{ABC}.$

All sides and diagonals of a $25$-gon are drawn either red or white. Show that at least $500$ triangles, having all three sides are in same color and having all three vertices from the vertices of the $25$-gon, can be found.

There are $ 6n \plus{} 4$ mathematicians participating in a conference which includes $ 2n \plus{} 1$ meetings. Each meeting has one round table that suits for $ 4$ people and $ n$ round tables that each table suits for $ 6$ people. We have known that two arbitrary people sit next to or have opposite places doesn't exceed one time. 1. Determine whether or not there is the case $ n \equal{} 1$. 2. Determine whether or not there is the case $ n > 1$.

On the side $AB$ of the cyclic quadrilateral $ABCD$ there is a point $X$ such that diagonal $AC$ bisects the segment $DX$, and the diagonal $BD$ bisects the segment $CX$. What is the smallest possible ratio $|AB | : |CD|$ in such a quadrilateral ?

The feet of the altitudes drawn from vertices $A$ and $B$ of an acute triangle $ABC$ are $K$ and $L$, respectively. Prove that if $|BK| = |KL|$ then the triangle $ABC$ is isosceles.

Square $S_{1}$ is $1\times 1.$ For $i\ge 1,$ the lengths of the sides of square $S_{i+1}$ are half the lengths of the sides of square $S_{i},$ two adjacent sides of square $S_{i}$ are perpendicular bisectors of two adjacent sides of square $S_{i+1},$ and the other two sides of square $S_{i+1},$ are the perpendicular bisectors of two adjacent sides of square $S_{i+2}.$ The total area enclosed by at least one of $S_{1}, S_{2}, S_{3}, S_{4}, S_{5}$ can be written in the form $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m-n.$ [asy] size(250); path p=rotate(45)*polygon(4); int i; for(i=0; i<5; i=i+1) { draw(shift(2-(1/2)^(i-1),0)*scale((1/2)^i)*p); } label("$S_1$", (0,-0.75)); label("$S_2$", (1,-0.75)); label("$S_3$", (3/2,-0.75)); label("$\cdots$", (7/4, -3/4)); label("$\cdots$", (2.25, 0));[/asy]

In decimal representation, we call an integer [i]$k$-carboxylic[/i] if and only if it can be represented as a sum of $k$ distinct integers, all of them greater than $9$, whose digits are the same. For instance, $2008$ is [i]$5$-carboxylic[/i] because $2008=1111+666+99+88+44$. Find, with an example, the smallest integer $k$ such that $8002$ is [i]$k$-carboxylic[/i].

On a chessboard (with $64$ squares) there are situated $32$ white and $32$ black pools. We say that two pools form a mixed pair when they are with different colors and they lie on the same row or column. Find the maximum and the minimum of the mixed pairs for all possible situations of the pools. [i]K. Dochev[/i]

Let $ABC$ and $PQR$ be two triangles such that [list] [b](a)[/b] $P$ is the mid-point of $BC$ and $A$ is the midpoint of $QR$. [b](b)[/b] $QR$ bisects $\angle BAC$ and $BC$ bisects $\angle QPR$ [/list] Prove that $AB+AC=PQ+PR$.

A triangle $ABC$ is inscribed in circle $(O)$. $P1, P2$ are two points in the plane of the triangle. $P_1A, P_1B, P_1C$ meet $(O)$ again at $A_1,B_1,C_1$ . $P_2A, P_2B, P_2C$ meet $(O)$ again at $A_2,B_2,C_2$. a) $A_1A_2, B_1B_2, C_1C_2$ intersect $BC,CA,AB$ at $A_3,B_3,C_3$. Prove that three points $A_3,B_3,C_3$ are collinear. b) $P$ is a point on the line $P_1P_2. A_1P,B_1P,C_1P$ meet (O) again at $A_4,B_4,C_4$. Prove that three lines $A_2A_4,B_2B_4,C_2C_4$ are concurrent. Trần Quang Hùng

On a table there is a pile with $ T$ tokens which incrementally shall be converted into piles with three tokens each. Each step is constituted of selecting one pile removing one of its tokens. And then the remaining pile is separated into two piles. Is there a sequence of steps that can accomplish this process? a.) $ T \equal{} 1000$ (Cono Sur) b.) $ T \equal{} 2001$ (BWM)

Non-empty set $M$, that consists of positive integer numbers, has the following property: if for some(not necessarily distinct) positive integers $a_1,\ldots,a_{2024}$ the number $a_1\ldots a_{2024}$ is in $M$, then the number $a_1+a_2+\ldots+a_{2024}$ is also in $M$ Prove that all positive integer numbers, starting from $2049$, are in the $M$ [i]M. Zorka[/i]

Given two positive integers $ x$ and $ y$ with $ x < y$. The percent that $ x$ is less than $ y$ is: $ \textbf{(A)}\ \frac {100(y \minus{} x)}{x} \qquad\textbf{(B)}\ \frac {100(x \minus{} y)}{x} \qquad\textbf{(C)}\ \frac {100(y \minus{} x)}{y} \qquad\textbf{(D)}\ 100(y \minus{} x)$ $ \textbf{(E)}\ 100(x \minus{} y)$

Compute the unique ordered pair $(x, y)$ of real numbers satisfying the system of equations $$\frac{x}{\sqrt{x^2 + y^2}}-\frac{1}{x}= 7 \,\,\, \text{and} \,\,\, \frac{y}{\sqrt{x^2 + y^2}}+\frac{1}{y}=4 $$

Let $ABCDE$ be a convex pentagon. Let $P$ be the intersection of the lines $CE$ and $BD$. Assume that $\angle PAD = \angle ACB$ and $\angle CAP = \angle EDA$. Prove that the circumcentres of the triangles $ABC$ and $ADE$ are collinear with $P$.

Shirley went to the store planning to buy $120$ balloons for $10$ dollars. When she arrived, she was surprised to nd that the balloons were on sale for $20$ percent less than expected. How many balloons could Shirley buy for her $10$ dollars?

Let $S=\{1,2,\cdots ,n\}$ where $n$ is an odd integer. Let $f$ be a function defined on $\{(i,j): i\in S, j \in S\}$ taking values in $S$ such that (i) $f(s,r)=f(r,s)$ for all $r,s \in S$ (ii) $\{f(r,s): s\in S\}=S$ for all $r\in S$ Show that $\{f(r,r): r\in S\}=S$

The number of points in the set $\{(x,y)|\lg(x^3+\frac{1}{3}y^3+\frac{1}{9})=\lg x+\lg y)\}$ is $\text{(A)}0\qquad\text{(B)}1\qquad\text{(C)}2\qquad\text{(D)}$more than $2$

Compute the number of ordered pairs $(a,b)$ of positive integers satisfying $a^b=2^{100}$. [i]Proposed by Nathan Xiong[/i]

Show that there exists a polynomial $P(x)$ with integer coefficients, such that the equation $P(x) = 0$ has no integer solutions, but for each positive integer $n$ there is an $x \in \Bbb{Z}$ such that $n \mid P(x).$