The real numbers $a$, $b$, $c$, $d$ satisfy simultaneously the equations \[abc -d = 1, \ \ \ bcd - a = 2, \ \ \ cda- b = 3, \ \ \ dab - c = -6.\] Prove that $a + b + c + d \not = 0$.

[b]problem 5.[/b] Let $x_1, x_2,\ldots, x_k$ be vectors of $m$-dimensional Euclidean space, such that $x_1+x_2+\ldots + x_k=0$. Show that there exists a permutation $\pi$ of the integers $\{ 1, 2, \ldots, k \}$ such that: $$\left\lVert \sum_{i=1}^n x_{\pi (i)}\right\rVert \leq \left( \sum_{i=1}^k \lVert x_i \rVert ^2\right)^{1/2}$$for each $n=1, 2, \ldots, k$. Note that $\lVert \cdot \rVert$ denotes the Euclidean norm. (18 points).

Let $ABCD$ be a trapezoid with bases $AB$ and $CD$. Bisectors of $AD$ and $BC$ intersect line segments $BC$ and $AD$ respectively in points $P$ and $Q$. Show that $\angle APD = \angle BQC$.

A lattice point in an $xy$-coordinate system is any point $(x,y)$ where both $x$ and $y$ are integers. The graph of $y=mx+2$ passes through no lattice point with $0<x \le 100$ for all $m$ such that $\frac{1}{2}<m<a$. What is the maximum possible value of $a$? $ \textbf{(A)}\ \frac{51}{101} \qquad \textbf{(B)}\ \frac{50}{99} \qquad \textbf{(C)}\ \frac{51}{100} \qquad \textbf{(D)}\ \frac{52}{101} \qquad \textbf{(E)}\ \frac{13}{25} $

Find the area of a square inscribed in an equilateral triangle, with one edge of the square on an edge of the triangle, if the side length of the triangle is $ 2\plus{}\sqrt{3}$.

Show that we cannot form more than $4096$ binary sequences of length $24$ so that any two differ in at least $8$ positions.

Suppose $\triangle ABC$ is such that $AB=13$, $AC=15$, and $BC=14$. It is given that there exists a unique point $D$ on side $\overline{BC}$ such that the Euler lines of $\triangle ABD$ and $\triangle ACD$ are parallel. Determine the value of $\tfrac{BD}{CD}$. (The $\textit{Euler}$ line of a triangle $ABC$ is the line connecting the centroid, circumcenter, and orthocenter of $ABC$.)

An $ (n, k) \minus{}$ tournament is a contest with $ n$ players held in $ k$ rounds such that: $ (i)$ Each player plays in each round, and every two players meet at most once. $ (ii)$ If player $ A$ meets player $ B$ in round $ i$, player $ C$ meets player $ D$ in round $ i$, and player $ A$ meets player $ C$ in round $ j$, then player $ B$ meets player $ D$ in round $ j$. Determine all pairs $ (n, k)$ for which there exists an $ (n, k) \minus{}$ tournament. [i]Proposed by Carlos di Fiore, Argentina[/i]

How many positive integer divisors of $2004^{2004}$ are divisible by exactly $2004$ positive integers?

Let $0\leq x_1\leq x_2\leq \cdots \leq x_n\leq 1 $ $(n\geq 2).$ Prove that $$\sqrt[n]{x_1x_2 \cdots x_n}+ \sqrt[n]{(1-x_1)(1-x_2)\cdots (1-x_n)}\leq \sqrt[n]{1-(x_1- x_n)^2}.$$

Given four weights in geometric progression and an equal arm balance, show how to find the heaviest weight using the balance only twice.

a) Exist $a, b, c, \in N$, such that the numbers $ab+1,bc+1$ and $ca+1$ are simultaneously even perfect squares ? b) Show that there is an infinity of natural numbers (distinct two by two) $a, b, c$ and $d$, so that the numbers $ab+1,bc+1, cd+1$ and $da+1$ are simultaneously perfect squares.

Every evening uncle Chernomor (see Pushkin's tales) appoints either $9$ or $10$ of his 33 "knights" in the "night guard". When it can happen, for the first time, that every knight has been on duty the same number of times?

For a positive integer $k,$ call an integer a $pure$ $k-th$ $power$ if it can be represented as $m^k$ for some integer $m.$ Show that for every positive integer $n,$ there exists $n$ distinct positive integers such that their sum is a pure $2009-$th power and their product is a pure $2010-$th power.

A $20 \times 20$ rectangular grid has been given. It is known that one of the grid's unit squares contains a hidden treasure. To find the treasure, we have been given an opportunity to order several scientific studies at the same time, results of which will be known only after some time. For each study we must choose one $1 \times 4$ rectangle, and the study will tell whether the rectangle contains the treasure. The $1 \times 4$ rectangle can be either horizontal or vertical, and it can extend over a side of the $20 \times 20$ grid, coming back in at the opposite side (you can think of the $20 \times 20$ grid as a torus - the opposite sides are connected). What is the minimal amount of studies that have to ordered for us to precisely determine the unit square containing the treasure?

Let $\gamma_1$ and $\gamma_2$ be circles centered at $O$ and $ P$ respectively, and externally tangent to each other at point $Q$. Draw point $D$ on $\gamma_1$ and point $E$ on $\gamma_2$ such that line $DE$ is tangent to both circles. If the length $OQ = 1$ and the area of the quadrilateral $ODEP$ is $520$, then what is the value of length $PQ$?

A permutation $ \pi$ is selected randomly through all $ n$-permutations. a) if \[ C_a(\pi)\equal{}\mbox{the number of cycles of length }a\mbox{ in }\pi\] then prove that $ E(C_a(\pi))\equal{}\frac1a$ b) Prove that if $ \{a_1,a_2,\dots,a_k\}\subset\{1,2,\dots,n\}$ the probability that $ \pi$ does not have any cycle with lengths $ a_1,\dots,a_k$ is at most $ \frac1{\sum_{i\equal{}1}^ka_i}$

The real numbers $a$, $b$, $c$, $d$ satisfy simultaneously the equations \[abc -d = 1, \ \ \ bcd - a = 2, \ \ \ cda- b = 3, \ \ \ dab - c = -6.\] Prove that $a + b + c + d \not = 0$.

A number is [i]capicua [/i] if it is read equally from left to right as it is from right to the left. For example, $23432$ and $111111$ are capicua numbers. (a) How many $2023$-digit capicua numbers can be formed if you want them to have at least $2022$ equal digits? (b) How many $2023$-digit capicua numbers can be formed if you want them to have at least $2021$ equal digits?

We say that an ordered pair $(a,b)$ of positive integers with $a>b$ is square-ish if both $a+b$ and $a-b$ are perfect squares. For example, $(17,8)$ is square-ish because $17+8=25$ and $17-8=9$ are both perfect squares. How many square-ish pairs $(a,b)$ with $a+b<100$ are there? [i]Proposed by Nathan Xiong[/i]

A positive integer $n$ and a real number $a$ are given. Find all $n$-tuples $(x_1, ... ,x_n)$ of real numbers that satisfy the system of equations $$\sum_{i=1}^{n} x_i^k= a^k \,\,\,\, for \,\,\,\, k = 1,2, ... ,n$$

Five distinct points $A,B,C,D$ and $E$ lie on a line with $|AB|=|BC|=|CD|=|DE|$. The point $F$ lies outside the line. Let $G$ be the circumcentre of the triangle $ADF$ and $H$ the circumcentre of the triangle $BEF$. Show that the lines $GH$ and $FC$ are perpendicular.

Let $k$ be a nonnegative integer. Evaluate \[ \sum_{j=0}^k 2^{k-j} \binom{k+j}{j}. \]

Find the highest point (largest possible $y$-coordinate) on the parabola \[y=-2x^2+28x+418.\]

In the middle of a vast prairie, a firetruck is stationed at the intersection of two perpendicular straight highways. The truck travels at $50$ miles per hour along the highways and at $14$ miles per hour across the prairie. Consider the set of points that can be reached by the firetruck within six minutes. The area of this region is $m/n$ square miles, where $m$ and $n$ are relatively prime positive integers. Find $m+n.$