At Olympic High School, $\frac25$ of the freshmen and $\frac45$ of the sophomores took the AMC-10. Given that the number of freshmen and sophomore contestants was the same, which of the following must be true? $\text{(A)}$ There are five times as many sophomores as freshmen. $\text{(B)}$ There are twice as many sophomores as freshmen. $\text{(C)}$ There are as many freshmen as sophomores. $\text{(D)}$ There are twice as many freshmen as sophomores. $\text{(E)}$ There are five times as many freshmen as sophomores.

A permutation of the integers \(2020, 2021,...,2118, 2119\) is a list \(a_1,a_2,a_3,...,a_{100}\) where each one of the numbers appears exactly once. For each permutation we define the partial sums. $s_1=a_1$ $s_2=a_1+a_2$ $s_3=a_1+a_2+a_3$ $...$ $s_{100}=a_1+a_2+...+a_{100}$ How many of these permutations satisfy that none of the numbers \(s_1,...,s_{100}\) is divisible by $3$?

Convex quadrilateral is divided by diagonals into four triangles with congruent inscribed circles. Prove that this quadrilateral is rhombus.

Consider the regular quadrilateral prism $ABCDA'B'C 'D'$, in which $AB = a,AA' = \frac{a \sqrt {2}}{2}$, and $M$ is the midpoint of $B' C'$. Let $F$ be the foot of the perpendicular from $B$ on line $MC$, Let determine the measure of the angle between the planes $(BDF)$ and $(HBS)$.

It is given positive integer $N$. Let $d_1$, $d_2$,...,$d_n$ be its divisors and let $a_i$ be number of divisors of $d_i$, $i=1,2,...n$. Prove that $$(a_1+a_2+...+a_n)^2={a_1}^3+{a_2}^3+...+{a_n}^3$$

Let $ ABC$ be an acute-angled triangle. $ C_{1}$ and $ C_{2}$ are two circles of diameters $ AB$ and $ AC$, respectively. $ C_{2}$ and $ AB$ intersect again at $ F$, and $ C_{1}$ and $ AC$ intersect again at $ E$. Also, $ BE$ meets $ C_{2}$ at $ P$ and $ CF$ meets $ C_{1}$ at $ Q$. Prove that $ AP=AQ$.

Let $P$ be the set of points in the plane, and let $f : P \to P$ be a function such that the image through $f$ of any triangle is a square (any polygon is considered to be formed by the reunion of the points on its sides). Prove that $f(P)$ is a square.

Prove that there are infinitely many positive integers $n$ such that $\Delta = nr^2$, where $\Delta$ and $r$ are respectively the area and the inradius of a triangle with integer sides.

Determine all possible values of the expression $xy+yz+zx$ with real numbers $x, y, z$ satisfying the conditions $x^2-yz = y^2-zx = z^2-xy = 2$.

A string of letters is called $good$ if it contains a continuous substring $IMONST$ in it. For example, the string $NSIMONSTIM$ is $good$, but the string $IMONNNST$ is not. Find the number of good strings consisting of $12$ letters from $I$, $M$, $O$, $N$, $S$, $T$ only.

A mason has bricks with dimensions $2\times5\times8$ and other bricks with dimensions $2\times3\times7$. She also has a box with dimensions $10\times11\times14$. The bricks and the box are all rectangular parallelepipeds. The mason wants to pack bricks into the box filling its entire volume and with no bricks sticking out. Find all possible values of the total number of bricks that she can pack.

A network is a simple directed graph such that each edge $ e$ has two intger lower and upper capacities $ 0\leq c_l(e)\leq c_u(e)$. A circular flow on this graph is a function such that: 1) For each edge $ e$, $ c_l(e)\leq f(e)\leq c_u(e)$. 2) For each vertex $ v$: \[ \sum_{e\in v^\plus{}}f(e)\equal{}\sum_{e\in v^\minus{}}f(e)\] a) Prove that this graph has a circular flow, if and only if for each partition $ X,Y$ of vertices of the network we have: \[ \sum_{\begin{array}{c}{e\equal{}xy}\\{x\in X,y\in Y}\end{array}} c_l(e)\leq \sum_{\begin{array}{c}{e\equal{}yx}\\{y\in Y,x\in X}\end{array}} c_l(e)\] b) Suppose that $ f$ is a circular flow in this network. Prove that there exists a circular flow $ g$ in this network such that $ g(e)\equal{}\lfloor f(e)\rfloor$ or $ g(e)\equal{}\lceil f(e)\rceil$ for each edge $ e$.

Let $ a$, $ b$, $ c$ be positive reals such that $ a^4 \plus{} b^4 \plus{} c^4 \equal{} 3$. Prove that $ \sum\frac1{4 \minus{} ab}\leq1$, where the $ \sum$ sign stands for cyclic summation. [i]Alternative formulation:[/i] For any positive reals $ a$, $ b$, $ c$ satisfying $ a^4 \plus{} b^4 \plus{} c^4 \equal{} 3$, prove the inequality $ \frac{1}{4\minus{}bc}\plus{}\frac{1}{4\minus{}ca}\plus{}\frac{1}{4\minus{}ab}\leq 1$.

For each positive integer $n$ let $a_n$ be the largest positive integer satisfying \[(a_n)!\left| \prod_{k=1}^n \left\lfloor \frac{n}{k}\right\rfloor\right.\] Show that there are infinitely many positive integers $m$ for which $a_{m+1}<a_m$.

Consider functions $\, f: [0,1] \rightarrow \mathbb{R} \,$ which satisfy (i) $f(x) \geq 0 \,$ for all $\, x \,$ in $\, [0,1],$ (ii) $f(1) = 1,$ (iii) $f(x) + f(y) \leq f(x+y)\,$ whenever $\, x, \, y, \,$ and $\, x + y \,$ are all in $\, [0,1]$. Find, with proof, the smallest constant $\, c \,$ such that \[ f(x) \leq cx \] for every function $\, f \,$ satisfying (i)-(iii) and every $\, x \,$ in $\, [0,1]$.

Suppose that \[x_1+1=x_2+2=x_3+3=\cdots=x_{2008}+2008=x_1+x_2+x_3+\cdots+x_{2008}+2009.\] Find the value of $\left\lfloor|S|\right\rfloor$, where $S=\displaystyle\sum_{n=1}^{2008}x_n$.

The semicircle with centre $O$ and the diameter $AC$ is divided in two arcs $AB$ and $BC$ with ratio $1: 3$. $M$ is the midpoint of the radium $OC$. Let $T$ be the point of arc $BC$ such that the area of the cuadrylateral $OBTM$ is maximum. Find such area in fuction of the radium.

Let $\mathbb{N}_{\ge 1}$ be the set of positive integers. Find all functions $f \colon \mathbb{N}_{\ge 1} \to \mathbb{N}_{\ge 1}$ such that, for all positive integers $m$ and $n$: (a) $n = \left(f(2n)-f(n)\right)\left(2 f(n) - f(2n)\right)$, (b)$f(m)f(n) - f(mn) = \left(f(2m)-f(m)\right)\left(2 f(n) - f(2n)\right) + \left(f(2n)-f(n)\right)\left(2 f(m) - f(2m)\right)$, (c) $m-n$ divides $f(2m)-f(2n)$ if $m$ and $n$ are distinct odd prime numbers.

How many integers between $100$ and $999$, inclusive, have the property that some permutation of its digits is a multiple of $11$ between $100$ and $999$? For example, both $121$ and $211$ have this property. $ \textbf{(A) }226\qquad \textbf{(B) } 243 \qquad \textbf{(C) } 270 \qquad \textbf{(D) }469\qquad \textbf{(E) } 486$

Find the number of subsets of ${1,2,3,...,10}$ that contain exactly one pair of consecutive integers. Examples of such subsets are ${1,2,5}$ and ${1,3,6,7,10}$.

Let $a, b, c$ be positive reals such that $abc=1$. Show that \[\frac{1}{a^5(b+2c)^2} + \frac{1}{b^5(c+2a)^2} + \frac{1}{c^5(a+2b)^2} \ge \frac{1}{3}.\]

Parallelogram $ABCD$ is such that angle $B < 90$ and $AB<BC$. Points E and F are on the circumference of $\omega$ inscribing triangle ABC, such that tangents to $\omega$ in those points pass through D. If $\angle EDA= \angle{FDC}$, find $\angle{ABC}$.

$100$ deputies formed $450$ commissions. Each two commissions has no more than three common deputies, and every $5$ - no more than one. Prove that, that there are $4$ commissions that has exactly one common deputy each.

What is the smallest number ? (A) $3$ (B) $2^{\sqrt2}$ (C) $2^{1+\frac{1}{\sqrt2}}$ (D) $2^{\frac12} + 2^{\frac23}$ (E) $2^{\frac53}$

Circles $A, B$, and $C$ each have radius $r$, and their centers are the vertices of an equilateral triangle of side length $6r$. Two lines are drawn, one tangent to $A$ and $C$ and one tangent to $B$ and $C$, such that $A$ is on the opposite side of each line from $B$ and $C$. Find the sine of the angle between the two lines. [img]http://4.bp.blogspot.com/-IZv8q-3NYZg/XXmrroy2PnI/AAAAAAAAKxg/jSOcOOQ8Kyw0EwHUifXJ1jOd2ENAo1FfACK4BGAYYCw/s200/2008%2Bpumac%2Bb6.png[/img]