Integers $a, b$ and $c$ satisfy the condition $ab + bc + ca = 1$. Is it true that the number $(1+a^2)(1+b^2)(1+c^2)$ is a perfect square? Why?

In a test, $3n$ students participate, who are located in three rows of $n$ students in each. The students leave the test room one by one. If $N_1(t), N_2(t), N_3(t)$ denote the numbers of students in the first, second, and third row respectively at time $t$, find the probability that for each t during the test, \[|N_i(t) - N_j(t)| < 2, i \neq j, i, j = 1, 2, \dots .\]

Suppose that $f$ is a differentiable function such that $f(0) = 20$ and $|f'(x)| \leq 4$ for all real numbers $x$. Let $a$ and $b$ be real numbers such that [i]every[/i] such function $f$ satisfies $a \leq f(22) \leq b$. Find the smallest possible value of $|a| + |b|$.

Let $\Gamma$ be the circumcircle of a fixed triangle $ABC$, and let $M$ and $N$ be the midpoints of the arcs $BC$ and $CA$, respectively. For any point $X$ on the arc $AB$, let $O_1$ and $O_2$ be the incenters of $\vartriangle XAC$ and $\vartriangle XBC$, and let the circumcircle of $\vartriangle XO_1O_2$ intersect $\Gamma$ at $X$ and $Q$. Prove that triangles $QNO_1$ and $QMO_2$ are similar, and find all possible locations of point $Q$.

$p=3k+1$ is a prime number. For each $m \in \mathbb Z_p$, define function $L$ as follow: $L(m) = \sum_{x \in \mathbb{Z}_p}^{ } \left ( \frac{x(x^3 + m)}{p} \right )$ [i]a)[/i] For every $m \in \mathbb Z_p$ and $t \in {\mathbb Z_p}^{*}$ prove that $L(m) = L(mt^3)$. (5 points) [i]b)[/i] Prove that there is a partition of ${\mathbb Z_p}^{*} = A \cup B \cup C$ such that $|A| = |B| = |C| = \frac{p-1}{3}$ and $L$ on each set is constant. Equivalently there are $a,b,c$ for which $L(x) = \left\{\begin{matrix} a & & &x \in A \\ b& & &x \in B \\ c& & & x \in C \end{matrix}\right.$ . (7 points) [i]c)[/i] Prove that $a+b+c = -3$. (4 points) [i]d)[/i] Prove that $a^2 + b^2 + c^2 = 6p+3$. (12 points) [i]e)[/i] Let $X= \frac{2a+b+3}{3},Y= \frac{b-a}{3}$, show that $X,Y \in \mathbb Z$ and also show that :$p= X^2 + XY +Y^2$. (2 points) (${\mathbb Z_p}^{*} = \mathbb Z_p \setminus \{0\}$)

A convex hexagon $A_1B_1A_2B_2A_3B_3$ it is inscribed in a circumference $\Omega$ with radius $R$. The diagonals $A_1B_2$, $A_2B_3$, $A_3B_1$ are concurrent in $X$. For each $i=1,2,3$ let $\omega_i$ tangent to the segments $XA_i$ and $XB_i$ and tangent to the arc $A_iB_i$ of $\Omega$ that does not contain the other vertices of the hexagon; let $r_i$ the radius of $\omega_i$. $(a)$ Prove that $R\geq r_1+r_2+r_3$ $(b)$ If $R= r_1+r_2+r_3$, prove that the six points of tangency of the circumferences $\omega_i$ with the diagonals $A_1B_2$, $A_2B_3$, $A_3B_1$ are concyclic

A rectangular garden 50 feet long and 10 feet wide is enclosed by a fence. To make the garden larger, while using the same fence, its shape is changed to a square. By how many square feet does this enlarge the garden? $ \text{(A)}\ 100\qquad\text{(B)}\ 200\qquad\text{(C)}\ 300\qquad\text{(D)}\ 400\qquad\text{(E)}\ 500 $

Given a set $n$ of points in the plane that are not contained in a single straight line. Prove that there exists a circle passing through at least three of these points, inside which there are none of the remaining points of the set.

There are $100$ white points on a circle. Asya and Borya play the following game: they alternate, starting with Asya, coloring a white point in green or blue. Asya wants to obtain as much as possible pairs of adjacent points of distinct colors, while Borya wants these pairs to be as less as possible. What is the maximal number of such pairs Asya can guarantee to obtain, no matter how Borya plays.

Georg has a set of sticks. From these sticks he must create a closed figure with the property that each stick makes right angles with its neighbouring sticks. All the sticks must be used. If the sticks have the lengths $1, 1, 2, 2, 2, 3, 3$ and $4$, the figure might for example look like this: [img]https://cdn.artofproblemsolving.com/attachments/9/7/c16a3143a52ec6f442208c63b41f2df1ae735c.png[/img] (a) Prove that he can create such a figure if the sticks have the lengths $1, 1, 1, 2, 2, 3, 4$ and $4$. (b) Prove that it cannot be done if the sticks have the lengths $1, 2, 2, 3, 3, 3, 4, 4$ and $4$. (c) Determine whether it is doable if the sticks have the lengths $1, 2, 2, 2, 3, 3, 3, 4, 4$ and $5$.

Evaluate $ \int_0^1 \frac{(1\minus{}2x)e^{x}\plus{}(1\plus{}2x)e^{\minus{}x}}{(e^x\plus{}e^{\minus{}x})^3}\ dx.$

Consider an arbitrary “figure” $F$ (non convex polygon). A chord of $F$ is defined to be a segment which lies entirely within $ F$ and whose ends are on its boundary. (a) Does there always exist a chord of $F$ that divides its area in half? (b) Prove that for any $F$ there exists a chord such that the area of each of the two parts of $F$ is not less than $ 1/3$ of the area of $F$. (c) Can the number $1/3$ in (b) be changed to a greater one? (V Proizvolov)

Does there exist a number $N$ so that there are $N - 1$ infinite arithmetic progressions with differences $2 , 3 , 4 ,..., N$ , and every natural number belongs to at least one of these progressions?

Let $\Delta A_1A_2A_3, \Delta B_1B_2B_3, \Delta C_1C_2C_3$ be three equilateral triangles. The vertices in each triangle are numbered [u]clockwise[/u]. It is given that $A_3=B_3=C_3$. Let $M$ be the center of mass of $\Delta A_1B_1C_1$, and let $N$ be the center of mass of $\Delta A_2B_2C_2$. Prove that $\Delta A_3MN$ is an equilateral triangle.

\[\int_0^{\pi/2} \frac{\sin(x)}{2-\sin(x)\cos(x)}\,\mathrm dx\] [i]Proposed by Connor Gordon[/i]

Can be colored the positive integers with 2009 colors if we know that each color paints infinitive integers and that we can not find three numbers colored by three different colors for which the product of two numbers equal to the third one?

Let $a$, $b$, $c$ be real numbers greater than or equal to $1$. Prove that \[ \min \left(\frac{10a^2-5a+1}{b^2-5b+10},\frac{10b^2-5b+1}{c^2-5c+10},\frac{10c^2-5c+1}{a^2-5a+10}\right )\leq abc. \]

Alberto wants to invite Ximena to his house. Since Alberto knows that Ximena is amateur to mathematics, instead of pointing out exactly which Transantiago buses serve him, he tells him: [i]the numbers of the buses that take me to my house have three digits, where the leftmost digit is not null, furthermore, these numbers are multiples of $13$, and the second digit of them is the average of the other two.[/i] What are the bus lines that go to Alberto's house?

An ice skater can rotate about a vertical axis with an angular velocity $\omega_\text{0}$ by holding her arms straight out. She can then pull in her arms close to her body so that her angular velocity changes to $2\omega_\text{0}$, without the application of any external torque. What is the ratio of her final rotational kinetic energy to her initial rotational kinetic energy? (A)$\sqrt{2}$ (B) $2$ (C) $2\sqrt{2}$ (D) $4$ (E) $8$

Consider a horizontal line $L$ with $n\ge 4$ different points $P_1, P_2, ..., P_n$. For each pair of points $P_i$ ,$P_j $a circle is drawn such that the segment $P_iP_j$ is a diameter. Determine the maximum number of intersections between circles that can occur, considering only those that occur strictly above $L$. [hide=original wording]Considere una recta horizontal $L$ con $n\ge 4$ puntos $P_1, P_2, ..., P_n$ distintos en ella. Para cada par de puntos $P_i,P_j$ se traza un circulo de manera tal que el segmento $P_iP_j$ es un diametro. Determine la cantidad maxima de intersecciones entre circulos que pueden ocurrir, considerando solo aquellas que ocurren estrictamente arriba de $L$.[/hide]

a) Find all the integers $m$ and $n$ such that \[9m^2+3n=n^2+8\] b) Let $a,b\in \mathbb{N}^*$ . If $x=a^{a+b}+(a+b)^a$ and $y=a^a+(a+b)^{a+b}$ which one is bigger? [i]Florin Nicoara, Valer Pop[/i]

Let $a, b>0$. Prove that:$$ (a^3+b^3+a^3b^3)(\frac{1}{a^3} + \frac{1}{b^3} + \frac{1}{a^3b^3} ) +27 \ge 6(a+b+\frac{1}{a} +\frac{1}{b} +\frac{a}{b} +\frac{b}{a}) $$

A panel contains $100$ light bulbs, arranged in a $10$ by $10$ square array. Some of them are on, the others are off. The electrical system is such that when the switch corresponding to a light bulb is pressed, all the light bulbs that are on the same row or column of it (including the bulb linked to the pressed switch) change their state (that is they are turned on or off). [list] [*] From which starting configurations, pressing the right sequence of switches, is it possible to achieve that all bulbs are on at the same time? [*] What is the answer to the previous question if the bulbs are $81$, arranged in a $9$ by $9$ panel?[/list]

For any set $A = \{x_1, x_2, x_3, x_4, x_5\}$ of five distinct positive integers denote by $S_A$ the sum of its elements, and denote by $T_A$ the number of triples $(i, j, k)$ with $1 \le i < j < k \le 5$ for which $x_i + x_j + x_k$ divides $S_A$. Find the largest possible value of $T_A$.

Do there exist four points $P_i = (x_i, y_i) \in \mathbb{R}^2\ (1\leq i \leq 4)$ on the plane such that: [list] [*] for all $i = 1,2,3,4$, the inequality $x_i^4 + y_i^4 \le x_i^3+ y_i^3$ holds, and [*] for all $i \neq j$, the distance between $P_i$ and $P_j$ is greater than $1$? [/list] [i]Pakawut Jiradilok[/i]