Let $f = f(x,y,z)$ be a polynomial in three variables $x$, $y$, $z$ such that $f(w,w,w) = 0$ for all $w \in \mathbb{R}$. Show that there exist three polynomials $A$, $B$, $C$ in these same three variables such that $A + B + C = 0$ and \[ f(x,y,z) = A(x,y,z) \cdot (x-y) + B(x,y,z) \cdot (y-z) + C(x,y,z) \cdot (z-x). \] Is there any polynomial $f$ for which these $A$, $B$, $C$ are uniquely determined?

[b]p1.[/b] A matrix is a rectangular array of numbers. For example, $\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$ and $\begin{pmatrix} 1 & 3 \\ 2 & 4 \end{pmatrix}$ are $2 \times 2$ matrices. A [i]saddle [/i] point in a matrix is an entry which is simultaneously the smallest number in its row and the largest number in its column. a. Write down a $2 \times 2$ matrix which has a saddle point, and indicate which entry is the saddle point. b. Write down a $2 \times 2$ matrix which has no saddle point c. Prove that a matrix of any size, all of whose entries are distinct, can have at most one saddle point. [b]p2.[/b] a. Find four different pairs of positive integers satisfying the equation $\frac{7}{m}+\frac{11}{n}=1$. b. Prove that the solutions you have found in part (a) are all possible pairs of positive integers satisfying the equation $\frac{7}{m}+\frac{11}{n}=1$. [b]p3.[/b] Let $ABCD$ be a quadrilateral, and let points $M, N, O, P$ be the respective midpoints of sides $AB$, $BC$, $CD$, $DA$. a. Show, by example, that it is possible that $ABCD$ is not a parallelogram, but $MNOP$ is a square. Be sure to prove that your construction satisfies all given conditions. b. Suppose that $MO$ is perpendicular to $NP$. Prove that $AC = BD$. [b]p4.[/b] A [i]Pythagorean triple[/i] is an ordered collection of three positive integers $(a, b, c)$ satisfying the relation $a^2 + b^2 = c^2$. We say that $(a, b, c)$ is a [i]primitive [/i] Pythagorean triple if $1$ is the only common factor of $a, b$, and $c$. a. Decide, with proof, if there are infinitely many Pythagorean triples. b. Decide, with proof, if there are infinitely many primitive Pythagorean triples of the form $(a, b, c)$ where $c = b + 2$. c. Decide, with proof, if there are infinitely many primitive Pythagorean triples of the form $(a, b, c)$ where $c = b + 3$. [b]p5.[/b] Let $x$ and $y$ be positive real numbers and let $s$ be the smallest among the numbers $\frac{3x}{2}$,$\frac{y}{x}+\frac{1}{x}$ and $\frac{3}{y}$. a. Find an example giving $s > 1$. b. Prove that for any positive $x$ and $y,s <2$. c. Find, with proof, the largest possible value of $s$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $ABC$ be a triangle with $BC=13, CA=11, AB=10$. Let $A_1$ be the midpoint of $BC$. A variable line $\ell$ passes through $A_1$ and meets $AC,AB$ at $B_1,C_1$. Let $B_2,C_2$ be points with $B_2B=B_2C, B_2C_1\perp AB, C_2B=C_2C, C_2B_1 \perp AC$, and define $P=BB_2\cap CC_2$. Suppose the circles of diameters $BB_2, CC_2$ meet at a point $Q\neq A_1$. Given that $Q$ lies on the same side of line $BC$ as $A$, the minimum possible value of $\dfrac{PB}{PC}+\dfrac{QB}{QC}$ can be expressed in the form $\dfrac{a\sqrt{b}}{c}$ for positive integers $a,b,c$ with $\gcd (a,c)=1$ and $b$ squarefree. Determine $a+b+c$. [i]Proposed by Vincent Huang[/i]

A segment through the focus $F$ of a parabola with vertex $V$ is perpendicular to $\overline{FV}$ and intersects the parabola in points $A$ and $B$. What is $\cos(\angle AVB)$? $ \textbf{(A)}\ -\frac{3\sqrt{5}}{7} \qquad \textbf{(B)}\ -\frac{2\sqrt{5}}{5} \qquad \textbf{(C)}\ -\frac{4}{5} \qquad \textbf{(D)}\ -\frac{3}{5} \qquad \textbf{(E)}\ -\frac{1}{2} $

Let $a_1, a_2, a_3, \dots$ be a sequence of integers such that $\text{(i)}$ $a_1=0$ $\text{(ii)}$ for all $i\geq 1$, $a_{i+1}=a_i+1$ or $-a_i-1$. Prove that $\frac{a_1+a_2+\cdots+a_n}{n}\geq-\frac{1}{2}$ for all $n\geq 1$.

Let $ABCD$ be a unit square. A circle with radius $\frac{32}{49}$ passes through point $D$ and is tangent to side $AB$ at point $E$. Then $DE = \frac{m}{n}$ , where $m$, $n$ are positive integers and gcd $(m, n) = 1$. Find $100m + n$.

Let $CD$ be a diameter of circle $K$. Let $AB$ be a chord that is parallel to $CD$. The line segment $AE$, with $E$ on $K$, is parallel to $CB$; $F$ is the point of intersection of line segments $AB$ and $DE$. The line segment $FG$, with $G$ on $DC$, extended is parallel to $CB$. Is $GA$ tangent to $K$ at point $A \?$

Let $S = f\{0.1. 2.3,...\}$ be the set of the non-negative integers. Find all strictly increasing functions $f : S \to S$ such that $n + f(f(n)) \le 2f(n)$ for every $n$ in $S$

Let $a_{1},a_{2},...,a_{n+1}(n>1)$ are positive real numbers such that $a_{2}-a_{1}=a_{3}-a_{2}=...=a_{n+1}-a_{n}$. Prove that $\sum_{k=2}^{n}\frac{1}{a_{k}^{2}}\leq\frac{n-1}{2}.\frac{a_{1}a_{n}+a_{2}a_{n+1}}{a_{1}a_{2}a_{n}a_{n+1}}$

Let $a,b,c$ be positive real numbers where $ab+bc+ca = 3$. Prove that $$\dfrac{1}{a^2+1}+\dfrac{1}{b^2+1}+\dfrac{1}{c^2+1}\geq\dfrac{3} {2}.$$ [i](dektep)[/i]

Ms. Math's kindergarten class has $16$ registered students. The classroom has a very large number, $N$, of play blocks which satisfies the conditions: (a) If $16$, $15$, or $14$ students are present, then in each case all the blocks can be distributed in equal numbers to each student, and (b) There are three integers $0 < x < y < z < 14$ such that when $x$, $y$, or $z$ students are present and the blocks are distributed in equal numbers to each student, there are exactly three blocks left over. Find the sum of the distinct prime divisors of the least possible value of $N$ satisfying the above conditions.

Find the minimum value of\[ (u-v)^2+\left(\sqrt{2-u^2}-\frac{9}{v}\right)^2 \]for $0<u<\sqrt{2}$ and $v>0$

Rectangle $ABCD$ has $AB=3$ and $BC=4.$ Point $E$ is the foot of the perpendicular from $B$ to diagonal $\overline{AC}.$ What is the area of $\triangle ADE?$ $\textbf{(A)} \text{ 1} \qquad \textbf{(B)} \text{ }\frac{42}{25} \qquad \textbf{(C)} \text{ }\frac{28}{15} \qquad \textbf{(D)} \text{ 2} \qquad \textbf{(E)} \text{ }\frac{54}{25}$

An integer sequence $a_1,a_2,\ldots,a_n$ has $a_1=0$, $a_n\leq 10$ and $a_{i+1}-a_i\geq 2$ for $1\leq i<n$. How many possibilities are there for this sequence? The sequence may be of any length.

Prove that there are exactly three right-angled triangles whose sides are integers while the area is numerically equal to twice the perimeter.

Maria has a board of size $n \times n$, initially with all the houses painted white. Maria decides to paint black some houses on the board, forming a mosaic, as shown in the figure below, as follows: she paints black all the houses from the edge of the board, and then leaves white the houses that have not yet been painted. Then she paints the houses on the edge of the next remaining board again black, and so on. a) Determine a value of $n$ so that the number of black houses equals $200$. b) Determine the smallest value of $n$ so that the number of black houses is greater than $2012$.

Find all pairs of positive integers $(m,n)$, for which $$(m^n-n)^m=n!+m$$ [i]D. Volkovets[/i]

100 people from 50 countries, two from each countries, stay on a circle. Prove that one may partition them onto 2 groups in such way that neither no two countrymen, nor three consecutive people on a circle, are in the same group.

The 9 members of a baseball team went to an ice-cream parlor after their game. Each player had a singlescoop cone of chocolate, vanilla, or strawberry ice cream. At least one player chose each flavor, and the number of players who chose chocolate was greater than the number of players who chose vanilla, which was greater than the number of players who chose strawberry. Let $N$ be the number of different assignments of flavors to players that meet these conditions. Find the remainder when $N$ is divided by $1000.$

Let $\{a_n\}_{n\geq 1}$ be a sequence with $a_1=1$, $a_2=4$ and for all $n>1$, \[ a_{n} = \sqrt{ a_{n-1}a_{n+1} + 1 } . \] a) Prove that all the terms of the sequence are positive integers. b) Prove that $2a_na_{n+1}+1$ is a perfect square for all positive integers $n$. [i]Valentin Vornicu[/i]

A round table has radius $ 4$. Six rectangular place mats are placed on the table. Each place mat has width $ 1$ and length $ x$ as shown. They are positioned so that each mat has two corners on the edge of the table, these two corners being end points of the same side of length $ x$. Further, the mats are positioned so that the inner corners each touch an inner corner of an adjacent mat. What is $ x$? [asy]unitsize(4mm); defaultpen(linewidth(.8)+fontsize(8)); draw(Circle((0,0),4)); path mat=(-2.687,-1.5513)--(-2.687,1.5513)--(-3.687,1.5513)--(-3.687,-1.5513)--cycle; draw(mat); draw(rotate(60)*mat); draw(rotate(120)*mat); draw(rotate(180)*mat); draw(rotate(240)*mat); draw(rotate(300)*mat); label("$x$",(-2.687,0),E); label("$1$",(-3.187,1.5513),S);[/asy]$ \textbf{(A)}\ 2\sqrt {5} \minus{} \sqrt {3} \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ \frac {3\sqrt {7} \minus{} \sqrt {3}}{2} \qquad \textbf{(D)}\ 2\sqrt {3} \qquad \textbf{(E)}\ \frac {5 \plus{} 2\sqrt {3}}{2}$

Let ABC be an acute-angle triangle with BC > CA. Let O, H and F be the circumcenter, orthocentre and the foot of its altitude CH, respectively. Suppose that the perpendicular to OF at F meet the side CA at P. Prove FHP = BAC.

Let $n$ be a fixed positive integer. Compute over $\mathbb{R}$ the minimum of the following polynomial: \[f(x)=\sum_{t=0}^{2n}(2n+1-t)x^t.\]

Archipelago consists of $ n$ islands : $ I_1,I_2,...,I_n$ and $ a_1,a_2,...,a_n$ - number of the roads on each island. $ a_1 \equal{} 55$, $ a_k \equal{} a_{k \minus{} 1} \plus{} (k \minus{} 1)$, ($ k \equal{} 2,3,...,n$) a) Does there exist an island with 2008 roads? b) Calculate $ a_1 \plus{} a_2 \plus{} ... \plus{} a_n.$

Natural numbers from $1$ to $200$ were divided into $50$ sets. Prove that one of them contains three numbers that are the lengths of the sides of some triangle