Prove the following identity for every $ n\in N$: $ \sum_{j\plus{}h\equal{}n,j\geq h}\frac{(\minus{}1)^h2^{j\minus{}h}\binom{j}{h}}{j}\equal{}\frac{2}{n}$

For a pair $ A \equal{} (x_1, y_1)$ and $ B \equal{} (x_2, y_2)$ of points on the coordinate plane, let $ d(A,B) \equal{} |x_1 \minus{} x_2| \plus{} |y_1 \minus{} y_2|$. We call a pair $ (A,B)$ of (unordered) points [i]harmonic[/i] if $ 1 < d(A,B) \leq 2$. Determine the maximum number of harmonic pairs among 100 points in the plane.

Three real numbers $x$, $y$, and $z$ are chosen independently and uniformly at random from the interval $[0,1]$. Compute the probability that $x$, $y$, and $z$ can be the side lengths of a triangle.

Let $E(n)$ denote the sum of the even digits of $n$. For example, $E(5681) = 6+8 = 14$. Find $E(1) + E(2) + E(3) + \cdots + E(100)$. $\text{(A)}\ 200 \qquad \text{(B)}\ 360 \qquad \text{(C)}\ 400 \qquad \text{(D)}\ 900 \qquad \text{(E)}\ 2250$

Let $k\geq 2$ and $n_1, \dots, n_k \in \mathbf{Z}^+$. If $n_2 | (2^{n_1} -1)$, $n_3 | (2^{n_2} -1)$, $\dots$, $n_k | (2^{n_{k-1}} -1)$, $n_1 | (2^{n_k} -1)$, show that $n_1 = \dots = n_k =1$.

Let $ABC$ be a triangle with $\angle BAC> 90 ^ \circ$ and $D$ a point on $BC$. Let $E$ and $F$be the reflections of the point $D$ about $AB$ and $AC$, respectively. Suppose that $BE$ and $CF$ intersect at $P$. Show that $AP$ passes through the circumcenter of triangle $ABC$.

A number $n$ is $\textit{almost prime}$ if any of $n-2$, $n-1$, $n$, $n+1$, or $n+2$ is prime. Compute the smallest positive integer that is not $\textit{almost prime}$.

Find all $n\in \mathbb{N} $ so that 7 divides $5^n + 1$

Consider a function $f:\mathbb{Z}\to \mathbb{Z}$ such that: \[f(m^2+f(n))=f^2(m)+n,\ \forall m,n\in \mathbb{Z}\] Prove that: a)$f(0)=0$; b)$f(1)=1$; c)$f(n)=n,\ \forall n\in \mathbb{Z}$ [i]Lucian Dragomir[/i]

Let $\mathbb{Q}$ denote the set of rational numbers. Determine all functions $f:\mathbb{Q}\longrightarrow\mathbb{Q}$ such that, for all $x, y \in \mathbb{Q}$, $$f(x)f(y+1)=f(xf(y))+f(x)$$ [i]Nicolás López Funes and José Luis Narbona Valiente, Spain[/i]

For real numbers $a$, $b$, $c$ and $d$ holds: $$ a+b+c+d=0$$ $$a^3+b^3+c^3+d^3=0$$ Prove that sum of some two numbers $a$, $b$, $c$ and $d$ is equal to zero

Prove that for any $ 2n$ real numbers $ a_{1}$, $ a_{2}$, ..., $ a_{n}$, $ b_{1}$, $ b_{2}$, ..., $ b_{n}$, we have $ \sum_{i < j}{\left|a_{i}\minus{}a_{j}\right|}\plus{}\sum_{i < j}{\left|b_{i}\minus{}b_{j}\right|}\leq\sum_{i,j\in\left[1,n\right]}{\left|a_{i}\minus{}b_{j}\right|}$.

Let $AD$ be the common chord of two circles $\Gamma_1$ and $\Gamma_2$. A line through $D$ intersects $\Gamma_1$ at $B$ and $\Gamma_2$ at $C$. Let $E$ be a point on the segment $AD$, different from $A$ and $D$. The line $CE$ intersect $\Gamma_1$ at $P$ and $Q$. The line $BE$ intersects $\Gamma_2$ at $M$ and $N$. (i) Prove that $P,Q,M,N$ lie on the circumference of a circle $\Gamma_3$. (ii) If the centre of $\Gamma_3$ is $O$, prove that $OD$ is perpendicular to $BC$.

$n$ is a positive integer, $K$ the set of polynoms of real variables $x_1,x_2,...,x_{n+1}$ and $y_1,y_2,...,y_{n+1}$, function $f:K\rightarrow K$ satisfies \[f(p+q)=f(p)+f(q),\quad f(pq)=f(p)q+pf(q),\quad (\forall)p,q\in K.\] If $f(x_i)=(n-1)x_i+y_i,\quad f(y_i)=2ny_i$ for all $i=1,2,...,n+1$ and \[\prod_{i=1}^{n+1}(tx_i+y_i)=\sum_{i=0}^{n+1}p_it^{n+1-i}\] for any real $t$, prove, that for all $k=1,...,n+1$ \[f(p_{k-1})=kp_k+(n+1)(n+k-2)p_{k-1}\]

Prove that for all odd prime numbers $p$ there exist a natural number $m<p$ and integers $x_1, x_2, x_3$ such that: \[mp=x_1^2+x_2^2+x_3^2.\]

[b]p1.[/b] A [i]multimagic [/i] square is a $3 \times 3$ array of distinct positive integers with the property that the product of the $3$ numbers in each row, each column, and each of the two diagonals of the array is always the same. (a) Prove that the numbers $1, 2, 3, . . . , 9$ cannot be used to form a multimagic square. (b) Give an example of a multimagic square. [b]p2.[/b] A sequence $a_1, a_2, a_3, ... , a_n$ of real numbers is called an arithmetic progression if $$a_1 - a_2 = a_2 - a_3 = ... = a_{n-1} - a_n.$$ Prove that there exist distinct positive integers $n_1, n_2, n_3, ... , n_{2014}$ such that $$\frac{1}{n_1},\frac{1}{n_2}, ... ,\frac{1}{n_{2014}}$$ is an arithmetic progression. [b]p3.[/b] Let $\lfloor x \rfloor$ be the largest integer that is less than or equal to $x$. For example, $\lfloor 3.9 \rfloor = 3$ and $\lfloor 4\rfloor = 4$. Determine (with proof) all real solutions of the equation $$x^2 - 25 \lfloor x\rfloor + 100 = 0.$$ [b]p4.[/b] An army has $10$ cannons and $8$ carts. Each cart can carry at most one cannon. It takes one day for a cart to cross the desert. What is the least number of days that it takes to get the cannons across the desert? (Cannons can be left part way and picked up later during the procedure.) Prove that the amount of time that your solution requires to move the cannons across the desert is the smallest possible. [b]p5.[/b] Let $C$ be a convex polygon with $4031$ sides. Let $p$ be the length of its perimeter and let $d$ be the sum of the lengths of its diagonals. Show that $$\frac{d}{p}> 2014.$$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Suppose that the euro is worth $1.30$ dollars. If Diana has $500$ dollars and Etienne has $400$ euros, by what percent is the value of Etienne's money greater than the value of Diana's money? ${{ \textbf{(A)}\ 2\qquad\textbf{(B)}\ 4\qquad\textbf{(C)}\ 6.5\qquad\textbf{(D)}\ 8}\qquad\textbf{(E)}\ 13} $

Let $a_n$ be a sequence recursively defined by $a_0 = 0, a_1 = 1$ and $a_{n+2} = a_{n+1} + a_n$. Calculate the sum of $a_n\left( \frac 25\right)^n$ for all positive integers $n$. For what value of the base $b$ we get the sum $1$?

The polynomial $f(z)=az^{2018}+bz^{2017}+cz^{2016}$ has real coefficients not exceeding $2019$, and $f(\tfrac{1+\sqrt{3}i}{2})=2015+2019\sqrt{3}i$. Find the remainder when $f(1)$ is divided by $1000$.

Let $ABCD$ be a regular tetrahedron with side length $2$. The plane parallel to edges $AB$ and $CD$ and lying halfway between them cuts $ABCD$ into two pieces. Find the surface area of one of these pieces.

Let $ABC$ be an equilateral triangle. On the plane $(ABC)$ rise the perpendiculars $AA'$ and $BB'$ on the same side of the plane, so that $AA' = AB$ and $BB' =\frac12 AB$. Determine the measure the angle between the planes $(ABC)$ and $(A'B'C')$.

Find all ordered pairs of integers $(a,b)$ such that $3^a + 7^b$ is a perfect square.

In a family there are four children of different ages, each age being a positive integer not less than $2$ and not greater than $16$. A year ago the square of the age of the eldest child was equal to the sum of the squares of the ages of the remaining children. One year from now the sum of the squares of the youngest and the oldest will be equal to the sum of the squares of the other two. How old is each child?

Let $ABCD$ be a trapezium $AB // CD,$ $M$ and $N$ are fixed points on $AB,$ $P$ is a variable point on $CD$. $E = DN \cap AP$, $F = DN \cap MC$, $G = MC \cap PB$, $DP = \lambda \cdot CD$. Find the value of $\lambda$ for which the area of quadrilateral $PEFG$ is maximum.

$10000$ nonzero digits are written in a $100$-by-$100$ table, one digit per cell. From left to right, each row forms a $100$-digit integer. From top to bottom, each column forms a $100$-digit integer. So the rows and columns form $200$ integers (each with $100$ digits), not necessarily distinct. Prove that if at least $199$ of these $200$ numbers are divisible by $2013$, then all of them are divisible by $2013$.