The Flyfishing Club is choosing officers. There are $23$ members of the club. $14$ of them are boys and $9$ are girls. In how many ways can they choose a President and a Vice President if one of them must be a boy and the other must be a girl (either office can be held by the boy or the girl)?

Let $ABC$ be an acute-angled and non-isosceles triangle with orthocenter $H$. Let $O$ be the center of the circumscribed circle of triangle $ABC$ and let $K$ be center of the circumscribed circle of triangle $AHO$. Prove that the reflection of $K$ wrt $OH$ lies on $BC$.

How many distinct sets are there such that each set contains only non-negative powers of $2$ or $3$ and sum of its elements is $2014$? $ \textbf{(A)}\ 64 \qquad\textbf{(B)}\ 60 \qquad\textbf{(C)}\ 54 \qquad\textbf{(D)}\ 48 \qquad\textbf{(E)}\ \text{None of the preceding} $

Let $ABC$ be a triangle. Suppose that a circle with diameter $BC$ intersects segments $CA$, $AB$ at $E, F$, respectively. Let $D$ be the midpoint of $BC$. Suppose that $AD$ intersects $EF$ at $X$. If $AB =\sqrt9$, $AC =\sqrt{10}$, and $BC =\sqrt{11}$, what is $\frac{EX}{XF}$?

Determine the number of distinct values which appear in the sequence \[\left\lfloor\frac{2025}{1}\right\rfloor,\left\lfloor\frac{2025}{2}\right\rfloor,\left\lfloor\frac{2025}{3}\right\rfloor,\dots,\left\lfloor\frac{2025}{2024}\right\rfloor,\left\lfloor\frac{2025}{2025}\right\rfloor.\]

A block of houses is a square. There is a courtyard there in which a gold medal has fallen. Whoever calculates how long the side of said apple is, knowing that the distances from the medal to three consecutive corners of the apple are, respectively, $40$ m, $60$ m and $80$ m, will win the medal.

Let $ABC$ be a triangle in which $CA>BC>AB$. Let $H$ be its orthocentre and $O$ its circumcentre. Let $D$ and $E$ be respectively the midpoints of the arc $AB$ not containing $C$ and arc $AC$ not containing $B$. Let $D'$ and $E'$ be respectively the reflections of $D$ in $AB$ and $E$ in $AC$. Prove that $O, H, D', E'$ lie on a circle if and only if $A, D', E'$ are collinear.

Given an odd integer $n>3$, let $k$ and $t$ be the smallest positive integers such that both $kn+1$ and $tn$ are squares. Prove that $n$ is prime if and only if both $k$ and $t$ are greater than $\frac{n}{4}$

In $\triangle ABC, \angle BAC$ is a right angle. $BP$ and $CQ$ are bisectors of $\angle B$ and $\angle C$ respectively, which intersect $AC$ and $AB$ at $P$ and $Q$ respectively. Two perpendicular segments $PM$ and $QN$ are drawn on $BC$ from $P$ and $Q$ respectively. Find the value of $\angle MAN$ with proof.

For positive integers $x,y$, find all pairs $(x,y)$ such that $x^2y + x$ is a multiple of $xy^2 + 7$.

Let $n \in N, n\ge 2$, and $a_1, a_2, ..., a_n, b_1, b_2, ..., b_n$ be real positive numbers such that $$\frac{a_1}{b_1} \le \frac{a_2}{b_2} \le ... \le\frac{a_n}{b_n}.$$ Find the largest real $c$ so that $$(a_1-b_1c)x_1+(a_2-b_2c)x_2+...+(a_n-b_nc)x_n \ge 0,$$ for every $x_1, x_2,..., x_n > 0$, with $x_1\le x_2\le ...\le x_n$.

For real numbers $a,b,c,d$ not all equal to $0$ , define a real function $f(x) = a +b\cos{2x} + c\sin{5x} +d \cos{8x}$. Suppose $f(t) = 4a$ for some real $t$. prove that there exist a real number $s$ s.t. $f(s)<0$

Let $p$ be a fixed prime. Determine all the integers $m$, as function of $p$, such that there exist $a_1, a_2, \ldots, a_p \in \mathbb{Z}$ satisfying \[m \mid a_1^p + a_2^p + \cdots + a_p^p - (p+1).\]

Find the sum: \[11 \times \dbinom20 + 10 \times \dbinom31 + 9 \times \dbinom42 + \cdots + 2 \times \dbinom{11}9 + \dbinom{12}{10}\] Where $\tbinom{n}{r}$ is combination function given by $\tfrac{n!}{r!(n-r)!}$ $\textbf{(A) } 351\qquad\textbf{(B) } 841\qquad\textbf{(C) } 901\qquad\textbf{(D) } 991\qquad\textbf{(E) } 1001$

The dimensions of a wooden octahedron are natural numbers. We painted all its surface (the six faces), cut it by planes parallel to the cubed faces of an edge unit and observed that exactly half of the cubes did not have any painted faces. Prove that the number of octahedra with such property is finite. (It may be useful to keep in mind that $\sqrt[3]{\frac{1}{2}}=1,79 ... <1,8$). [hide=original wording] Las dimensiones de un ortoedro de madera son enteras. Pintamos toda su superficie (las seis caras), lo cortamos mediante planos paralelos a las caras en cubos de una unidad de arista y observamos que exactamente la mitad de los cubos no tienen ninguna cara pintada. Probar que el número de ortoedros con tal propiedad es finito[/hide]

Let $X$ be a set of $n$ elements and $k$ be a positive integer. Consider the family $S_k$ of all $k$-tuples $(E_1,...,E_k)$ with $E_i \subseteq X$ for each $i$. Evaluate the sums $\sum_{(E_1,...,E_k) \in S_k }|E_1 \cap ... \cap E_k|$ and $\sum_{(E_1,...,E_k) \in S_k }|E_1 \cup ... \cup E_k|$

Compute the smallest multiple of $63$ with an odd number of ones in its base two representation.

In the spherical shaped planet $X$ there are $2n$ gas stations. Every station is paired with one other station , and every two paired stations are diametrically opposite points on the planet. Each station has a given amount of gas. It is known that : if a car with empty (large enough) tank starting from any station it is always to reach the paired station with the initial station (it can get extra gas during the journey). Find all naturals $n$ such that for any placement of $2n$ stations for wich holds the above condotions, holds: there always a gas station wich the car can start with empty tank and go to all other stations on the planet.(Consider that the car consumes a constant amount of gas per unit length.)

Determine if there exist four polynomials such that the sum of any three of them has a real root while the sum of any two of them does not.

Find all positive integers $N$ with the following properties: (i) $N$ has at least two distinct prime factors, and (ii) if $d_1 < d_2 < d_3 < d_4$ are the four smallest divisors of $N$ then $N =d_1^2 + d_2 ^2+ d_3 ^2+ d_4^2$

Let $ABC$ be a triangle with $AB<AC$ and circuncircle $\omega$. Let $M$ and $N$ be the midpoints of $AC$ and $AB$ respectively and $G$ is the centroid of $ABC$. Let $P$ be the foot of perpendicular of $A$ to the line $BC$, and the point $Q$ is the intersection of $GP$ and $\omega$($Q,P,G$ are collinears in this order). The line $QM$ cuts $\omega$ in $M_1$ and the line $QN$ cuts $\omega$ in $N_1$. If $K$ is the intersection of $BM_1$ and $CN_1$ prove that $P$, $G$ and $K$ are collinears.

Point $Y$ lies on line segment $XZ$ such that $XY = 5$ and $Y Z = 3$. Point $G$ lies on line $XZ$ such that there exists a triangle $ABC$ with centroid $G$ such that $X$ lies on line $BC$, $Y$ lies on line $AC$, and $Z$ lies on line $AB$. Compute the largest possible value of $XG$.

In parallelogram $ABCD$, angle $A$ is acute. Let $X$ be a point, symmetrical to point $C$ wrt to straight line $AD$, $Y$ is a point symmetrical to the point $C$ wrt point $D$, and $M$ is the intersection point of $AC$ and $BD$. It turned out, that the circumcircles of triangles $BMC$ and $AXY$ are tangent internally. Prove that $AM = AB$.

Find all solutions to the equation: \[ \left(\left\lfloor x+\frac{7}{3} \right\rfloor \right)^2-\left\lfloor x-\frac{9}{4} \right\rfloor = 16 \]

Find all positive integers $n$ and sequence of integers $a_0,a_1,\ldots, a_n$ such that the following hold: 1. $a_n\neq 0$; 2. $f(a_{i-1})=a_i$ for all $i=1,\ldots, n$, where $f(x) = a_nx^n+a_{n-1}x^{n-1}+\cdots +a_0$. [i] Proposed by usjl[/i]