Real numbers $a, b, c, d, e$, have the property $$|a - b| = 2|b -c| = 3|c - d| = 4|d- e| = 5|e - a|.$$ Prove they are all equal.

We define $A, B, C$ as a [i]partition[/i] of $\mathbb{N}$ if $A,B,C$ satisfies the following. (i) $A, B, C \not= \phi$ (ii) $A \cap B = B \cap C = C \cap A = \phi$ (iii) $A \cup B \cup C = \mathbb{N}$. Prove that the partition of $\mathbb{N}$ satisfying the following does not exist. (i) $\forall$ $a \in A, b \in B$, we have $a+b+2008 \in C$ (ii) $\forall$ $b \in B, c \in C$, we have $b+c+2008 \in A$ (iii) $\forall$ $c \in C, a \in A$, we have $c+a+2008 \in B$

If $a_{1}$, $a_{2}$, $\ldots$, $a_{n}\geq 0$ are such that \[a_{1}^{2}+\cdots+a_{n}^{2}=1,\] then find the maximum value of the product $(1-a_{1})\cdots (1-a_{n})$.

We say that a positive integer $n$ is circular if it is possible to place the numbers $1, 2, \cdots , n$ in a circumference so that there are no three adjacent numbers whose sum is a multiple of 3. a) Show that 9 is not circular b) Show that any integer greater than 9 is circular.

Let $ABCDEF$ be a convex hexagon. In the hexagon there is a point $K$, such that $ABCK,DEFK$ are both parallelograms. Prove that the three lines connecting $A,B,C$ to the midpoints of segments $CE,DF,EA$ meet at one point.

The sequence $ (a_n)$ satisfies $ a_0 \equal{} 0$ and $ \displaystyle a_{n \plus{} 1} \equal{} \frac85a_n \plus{} \frac65\sqrt {4^n \minus{} a_n^2}$ for $ n\ge0$. Find the greatest integer less than or equal to $ a_{10}$.

For a given natural number $n > 3$, the real numbers $x_1, x_2, \ldots, x_n, x_{n + 1}, x_{n + 2}$ satisfy the conditions $0 < x_1 < x_2 < \cdots < x_n < x_{n + 1} < x_{n + 2}$. Find the minimum possible value of \[\frac{(\sum _{i=1}^n \frac{x_{i + 1}}{x_i})(\sum _{j=1}^n \frac{x_{j + 2}}{x_{j + 1}})}{(\sum _{k=1}^n \frac{x_{k + 1} x_{k + 2}}{x_{k + 1}^2 + x_k x_{k + 2}})(\sum _{l=1}^n \frac{x_{l + 1}^2 + x_l x_{l + 2}}{x_l x_{l + 1}})}\] and find all $(n + 2)$-tuplets of real numbers $(x_1, x_2, \ldots, x_n, x_{n + 1}, x_{n + 2})$ which gives this value.

In every cell of a board $9 \times 9$ is written an integer. For any $k$ numbers in the same row (column), their sum is also in the same row (column). Find the smallest possible number of zeroes in the board for $a)$ $k=5;$ $b)$ $k=8.$

Find all pairs $(p, q)$ of prime numbers satisfying \[ p^3+7q=q^9+5p^2+18p. \]

We define the bulldozer of triangle $ABC$ as the segment between points $P$ and $Q$, distinct points in the plane of $ABC$ such that $PA\cdot BC=PB\cdot CA=PC\cdot AB$ and $QA\cdot BC=QB\cdot CA=QC\cdot AB$. Let $XY$ be a segment of unit length in a plane $\mathcal{P}$, and let $\mathcal{S}$ be the region of $\mathcal P$ that the bulldozer of $XYZ$ sweeps through as $Z$ varies across the points in $\mathcal{P}$ satisfying $XZ=2YZ$. Find the greatest integer that is less than $100$ times the area of $\mathcal S$. [i]Proposed by Michael Ren[/i]

Let $f$ be a function defined on the set of positive rational numbers with the property that $f(a\cdot b)=f(a)+f(b)$ for all positive rational numbers $a$ and $b$. Suppose that $f$ also has the property that $f(p)=p$ for every prime number $p$. For which of the following numbers $x$ is $f(x)<0?$ $\textbf{(A) } \frac{17}{32} \qquad \textbf{(B) } \frac{11}{16} \qquad \textbf{(C) } \frac{7}{9} \qquad \textbf{(D) } \frac{7}{6} \qquad \textbf{(E) } \frac{25}{11}$

Let $n>1$ be an integer. Prove that there exists an integer $n-1 \ge m \ge \left \lfloor \frac{n}{2} \right \rfloor$ such that the following equation has integer solutions with $a_m>0:$ $$\frac{a_{m}}{m+1}+\frac{a_{m+1}}{m+2}+ \cdots + \frac{a_{n-1}}{n}=\frac{1}{\textrm{lcm}\left ( 1,2, \cdots , n \right )}$$ [i]Proposed by Navid Safaei[/i]

We say that a rational number is [i]spheric[/i] if it is the sum of three squares of rational numbers (not necessarily distinct). Prove that: [b]a)[/b] $ 7 $ is not spheric. [b]b)[/b] a rational spheric number raised to the power of any natural number greater than $ 1 $ is spheric.

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)