The real numbers $a, b, c, d$ are such that $a\geq b\geq c\geq d>0$ and $a+b+c+d=1$. Prove that \[(a+2b+3c+4d)a^ab^bc^cd^d<1\] [i]Proposed by Stijn Cambie, Belgium[/i]

Let $ n > 1$ be an integer. Find all sequences $ a_1, a_2, \ldots a_{n^2 \plus{} n}$ satisfying the following conditions: \[ \text{ (a) } a_i \in \left\{0,1\right\} \text{ for all } 1 \leq i \leq n^2 \plus{} n; \] \[ \text{ (b) } a_{i \plus{} 1} \plus{} a_{i \plus{} 2} \plus{} \ldots \plus{} a_{i \plus{} n} < a_{i \plus{} n \plus{} 1} \plus{} a_{i \plus{} n \plus{} 2} \plus{} \ldots \plus{} a_{i \plus{} 2n} \text{ for all } 0 \leq i \leq n^2 \minus{} n. \] [i]Author: Dusan Dukic, Serbia[/i]

In quadrilateral $ABCD$, $CD = 14$, $\angle BAD = 105^o$, $\angle ACD = 35^o$, and $\angle ACB = 40^o$. Let the midpoint of $CD$ be $M$. Points $P$ and $Q$ lie on $\overrightarrow{AM}$ and $\overrightarrow{BM}$, respectively, such that $\angle AP B = 40^o$ and $\angle AQB = 40^o$ . $P B$ intersects $CD$ at point $R$ and $QA$ intersects $CD$ at point $S$. If $CR = 2$, what is the length of $SM$?

Do there exist distinct natural numbers $x, y, z, t$, all greater than or equal to $2$, such that $x \geq y + 2$, $z \geq t + 2$, and \[ \binom{x}{y} = \binom{z}{t}? \] [i](For natural numbers $n$ and $k$ with $n \geq k$, we define $\binom{n}{k} = \frac{n!}{k!(n-k)!}$.)[/i]

Find all the roots of $(x^2 + 3x + 2)(x^2 - 7x + 12)(x^2- 2x -1) + 24 = 0$.

Let $ACE$ be a triangle with a point $B$ on segment $AC$ and a point $D$ on segment $CE$ such that $BD$ is parallel to $AE$. A point $Y$ is chosen on segment $AE$, and segment $CY$ is drawn. Let $X$ be the intersection of $CY$ and $BD$. If $CX = 5$, $XY = 3$, what is the ratio of the area of trapezoid $ABDE$ to the area of triangle $BCD$? [img]https://cdn.artofproblemsolving.com/attachments/9/2/d6c723e54dbc4c88c12aa6a6ee91ae9e3ea581.png[/img]

Let $ABCD$ be a square and let the points $M$ on $BC$, $N$ on $CD$, $P$ on $DA$, be such that $\angle (AB,AM)=x,\angle (BC,MN)=2x,\angle (CD,NP)=3x$. 1) Show that for any $0\le x\le 22.5$, such a configuration uniquely exists, and that $P$ ranges over the whole segment $DA$; 2) Determine the number of angles $0\le x\le 22.5$ for which$\angle (DA,PB)=4x$. (Dan Schwarz)

The integer $k$ is a [i]good number[/i], if we can divide a square into $k$ squares. How many good numbers not greater than $2006$ are there? $ \textbf{(A)}\ 1003 \qquad\textbf{(B)}\ 1026 \qquad\textbf{(C)}\ 2000 \qquad\textbf{(D)}\ 2003 \qquad\textbf{(E)}\ 2004 $

Let $m, n$ be positive integers. Show that the polynomial $$f(x)=x^m(x^2-100)^n-11$$ cannot be expressed as a product of two non-constant polynomials with integral coefficients.

Let $ABC$ be a triangle with $AC>AB$ , and denote its circumcircle by $\Omega$ and incentre by $I$. Let its incircle meet sides $BC,CA,AB$ at $D,E,F$ respectively. Let $X$ and $Y$ be two points on minor arcs $\widehat{DF}$ and $\widehat{DE}$ of the incircle, respectively, such that $\angle BXD = \angle DYC$. Let line $XY$ meet line $BC$ at $K$. Let $T$ be the point on $\Omega$ such that $KT$ is tangent to $\Omega$ and $T$ is on the same side of line $BC$ as $A$. Prove that lines $TD$ and $AI$ meet on $\Omega$. [right][i]Tommy Walker Mackay, United Kingdom[/i][/right]

Let $\mathbb{R}^+$ denote the set of positive real numbers. Find all functions $f:\mathbb{R}^+\to\mathbb{R}^+$ such that for any $x,y\in\mathbb{R}^+$, we have \[ f(x)f(y)=f(y)f\Big(xf(y)\Big)+\frac{1}{xy}.\]

Let $p{}$ be a fixed prime number. Juku and Miku play the following game. One of the players chooses a natural number $a$ such that $a>1$ and $a$ is not divisible by $p{}$, his opponent chooses any natural number $n{}$ such that $n>1$. Miku wins if the natural number written as $n{}$ "$1$"s in the positional numeral system with base $a$ is divisible by $p{}$, otherwise Juku wins. Which player has a winning strategy if: (a) Juku chooses the number $a$, tells it to Miku and then Miku chooses the number $n{}$; (b) Juku chooses the number $n{}$, tells it to Miku and then Miku chooses the number $a$?

Amelia needs to estimate the quantity $\tfrac ab-c$, where $a$, $b$, and $c$ are large positive integers. She rounds each of the integers so that the calculation will be easier to do mentally. In which of these situations will her answer necessarily be greater than the exact value of $\tfrac ab-c$? $\textbf{(A) }\text{She rounds all three numbers up.}$ $\textbf{(B) }\text{She rounds }a\text{ and }b\text{ up, and she rounds }c\text{ down.}$ $\textbf{(C) }\text{She rounds }a\text{ and }c\text{ up, and she rounds }b\text{ down.}$ $\textbf{(D) }\text{She rounds }a\text{ up, and she rounds }b\text{ and }c\text{ down.}$ $\textbf{(E) }\text{She rounds }c\text{ up, and she rounds }a\text{ and }b\text{ down.}$

The sequence $x_0, x_1, x_2, \ldots$ is defined by the conditions \[ x_0 = a, x_1 = b, x_{n+1} = \frac{x_{n - 1} + (2n - 1) ~x_n}{2n}\] for $n \ge 1,$ where $a$ and $b$ are given numbers. Express $\lim_{n \to \infty} x_n$ concisely in terms of $a$ and $b.$

A real number $a$ is chosen randomly and uniformly from the interval $[-20, 18]$. The probability that the roots of the polynomial \[x^4 + 2ax^3 + (2a-2)x^2 + (-4a+3)x - 2\] are all real can be written in the form $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

We have $p$ players participating in a tournament, each player playing against every other player exactly once. A point is scored for each victory, and there are no draws. A sequence of nonnegative integers $s_1 \le s_2 \le s_3 \le\cdots\le s_p$ is given. Show that it is possible for this sequence to be a set of final scores of the players in the tournament if and only if \[(i)\displaystyle\sum_{i=1}^{p} s_i =\frac{1}{2}p(p-1)\] \[\text{and}\] \[(ii)\text{ for all }k < p,\displaystyle\sum_{i=1}^{k} s_i \ge \frac{1}{2} k(k - 1).\]

Prove that in the interior of an equilateral triangle with side $a$ you can put a finite number of equal circles that do not overlap, with radius $r = \frac{a}{2010}$, so that the sum of their areas is greater than $\frac{17\sqrt3}{80}$ a$^2$.

Three congruent circles of radius $2$ are drawn in the plane so that each circle passes through the centers of the other two circles. The region common to all three circles has a boundary consisting of three congruent circular arcs. Let $K$ be the area of the triangle whose vertices are the midpoints of those arcs. If $K = \sqrt{a} - b$ for positive integers $a, b$, find $100a+b$. [i]Proposed by Michael Tang[/i]

Let \(n>1\) be a natural number, and let \(K\) be the square of side length \(n\) subdivided into \(n^2\) unit squares. Determine for which values of \(n\) it is possible to dissect \(K\) into \(n\) connected regions of equal area using only the diagonals of those unit squares, subject to the condition that from each unit square at most one of its diagonals is used (some unit squares may have neither diagonal).

A deck consists of $2^n$ cards. The deck is shuffled using the following operation: if the cards are initially in the order $a_1,a_2,a_3,a_4,...,a_{2^n-1},a_{2^n}$ then after shuffling the order becomes $a_{2^{n-1}+1},a_1,a_{2^{n-1}+2},a_2,...,a_{2^n},a_{2^{n-1}}$ . Find the smallest number of such operations after which the original order of the cards is restored. (R. Palm)

Let $ p$ be a prime number such that $ p\minus{}1$ is a perfect square. Prove that the equation $ a^{2}\plus{}(p\minus{}1)b^{2}\equal{}pc^{2}$ has infinite many integer solutions $ a$, $ b$ and $ c$ with $ (a,b,c)\equal{}1$

Find all real solutions of the system $\begin{cases} x^3 + y^3 = 1 \\ x^4 + y^4 = 1 \end{cases}$

Let $ \mathcal{H}$ be a family of finite subsets of an infinite set $ X$ such that every finite subset of $ X$ can be represented as the union of two disjoint sets from $ \mathcal{H}$. Prove that for every positive integer $ k$ there is a subset of $ X$ that can be represented in at least $ k$ different ways as the union of two disjoint sets from $ \mathcal{H}$. [i]P. Erdos[/i]

The diagonals of the inscribed quadrangle $ABCD$ intersect at point $K$. Prove that the tangent at point $K$ to the circle circumscribed around the triangle $ABK$ is parallel to $CD$. (A Zaslavsky)

Let there be an integer $n\geq2$. In a chess tournament $n$ players play between each other one game. No game ended in a draw. Show that after the end of the tournament the players can be arranged in a list: $P_1, P_2, P_3,\ldots,P_n$ such that for every $i (1\leq i\leq n-1)$ the player $P_i$ won against player $P_{i+1}$.