Find all primes $p,q$ and even $n>2$ such that $p^n+p^{n-1}+...+1=q^2+q+1$.

Let $n\geqslant 3$ be an integer. Prove that there exists a set $S$ of $2n$ positive integers satisfying the following property: For every $m=2,3,...,n$ the set $S$ can be partitioned into two subsets with equal sums of elements, with one of subsets of cardinality $m$.

Let $(a_n)_{n\in\mathbb{N}}$ be a sequence of positive numbers such that $$a_n-2a_{n+1}+a_{n+2}\geq 0 \text{ and } \sum_{j=1}^{n} a_j \leq 1, \forall n\in\mathbb{N}.$$ Prove that $0\leq a_{n}-a_{n+1}<\frac{2}{n^2}, \forall n\in\mathbb{N}.$

Consider two equilateral triangles $ABC$ and $MNP$ with the property that $AB \parallel MN, BC \parallel NP$ and $CA \parallel PM$ , so that the surfaces of the triangles intersect after a convex hexagon. The distances between the three pairs of parallel lines are at most equal to $1$. Show that at least one of the two triangles has the side at most equal to $\sqrt {3}$ .

Let $A_1A_2... A_{2n + 1}$ be a convex polygon, $a_1 = A_1A_2$, $a_2 ​​= A_2A_3$, $...$, $a_{2n} = A_{2n}A_{2n + 1}$, $a_{2n + 1} = A_{2n + 1}A_1$. Denote by: $\alpha_i = \angle A_i$, $1 \le i \le 2n + 1$, $\alpha_{k + 2n + 1} = \alpha_k$, $k \ge 1$, $ \beta_i = \alpha_{i + 2} + \alpha_{i + 4} +... + \alpha_{i + 2n}$, $1 \le i \le 2n + 1$. Prove what if $$\frac{\alpha_1}{\sin \beta_1}=\frac{\alpha_2}{\sin \beta_2}=...=\frac{\alpha_{2n+1}}{\sin \beta_{2n+1}}$$ then a circle can be circumscribed around this polygon. Does the inverse statement hold a place?

Let $ ABP, BCQ, CAR$ be three non-overlapping triangles erected outside of acute triangle $ ABC$. Let $ M$ be the midpoint of segment $ AP$. Given that $ \angle PAB \equal{} \angle CQB \equal{} 45^\circ$, $ \angle ABP \equal{} \angle QBC \equal{} 75^\circ$, $ \angle RAC \equal{} 105^\circ$, and $ RQ^2 \equal{} 6CM^2$, compute $ AC^2/AR^2$. [i]Zuming Feng.[/i]

[u]Round 1 [/u] [b]p1.[/b] Ephram was born in May $2005$. How old will he turn in the first year where the product of the digits of the year number is a nonzero perfect square? [b]p2.[/b] Zhao is studying for his upcoming calculus test by reviewing each of the $13$ lectures, numbered Lecture $1$, Lecture $2$, ..., Lecture $13$. For each $n$, he spends $5n$ minutes on Lecture $n$. Which lecture is he reviewing after $4$ hours? [b]p3.[/b] Compute $$\dfrac{3^3 \div 3(3)+3}{\frac{3}{3}}+3!.$$ [u]Round 2 [/u] [b]p4.[/b] At Ingo’s shop, train tickets normally cost $\$2$, but every $5$th ticket costs only $\$1$. At Emmet’s shop, train tickets normally cost $\$3$, but every $5$th ticket is free. Both Ingo and Emmett sold $1000$ tickets. Find the absolute difference between their sales, in dollars. [b]p5.[/b] Ephram paddles his boat in a river with a $4$-mph current. Ephram travels at $10$ mph in still water. He paddles downstream and then turns around and paddles upstream back to his starting position. Find the proportion of time he spends traveling upstream, as a percentage. [b]p6.[/b] The average angle measure of a $13-14-15$ triangle is $m^o$ and the average angle measure of a $5-6-7$ triangle is $n^o$. Find $m-n$. [u]Round 3[/u] [b]p7.[/b] Let $p(x) = x^2 -10x +31$. Find the minimum value of $p(p(x))$ over all real $x$. [b]p8.[/b] Michael H. andMichael Y. are playing a game with $4$ jellybeans. Michael H starts with $3$ of the jellybeans, and Michael Y starts with the remaining $1$. Every minute, a Michael flips a coin, and if heads, Michael H takes a jellybean from Michael Y. If tails, Michael Y takes a jellybean from Michael H. WhicheverMichael gathers all $4$ jellybeans wins. The probability Michael H wins can be written as $\frac{A}{B}$ for relatively prime positive integers $A$ and $B$. Find $1000A+B$. [b]p9.[/b] Define the digit-product of a positive integer to be the product of its non-zero digits. Let $M$ denote the greatest five-digit number with a digit-product of $360$, and let $N$ denote the least five-digit number with a digit-product of $360$. Find the digit-product of $M-N$. [u]Round 4 [/u] [b]p10.[/b] Hannah is attending one of the three IdeaMath classes running at LHS, while Alex decides to randomly visit some combination of classes. He won’t visit all three classes, but he’s equally likely to visit any other combination. The probability Alex visits Hannah’s class can be expressed as $\frac{A}{B}$ for relatively prime positive integers $A$ and $B$. Find $1000A+B$. [b]p11.[/b] In rectangle $ABCD$, let $E$ be the intersection of diagonal $AC$ and the circle centered at $A$ passing through $D$. Angle $\angle ACD = 24^o$. Find the measure of $\angle CED$ in degrees. [b]p12.[/b] During his IdeaMath class, Zach writes the numbers $2, 3, 4, 5, 6, 7$, and $8$ on a whiteboard. Every minute, he chooses two numbers $a$ and $b$ from the board, erases them, and writes the number $ab +a +b$ on the board. He repeats this process until there’s only one number left. Find the sum of all possible remaining numbers. [u]Round 5[/u] [b]p13.[/b] In isosceles right $\vartriangle ABC$ with hypotenuse $AC$, Let $A'$ be the point on the extension of $AB$ past $A$ such that $AA' = 1$. Let $C'$ be the point on the extension of $BC$ past vertex $C$ such that $CC' = 2$. Given that the difference of the areas of triangle $A'BC'$ and $ABC$ is $10$, find the area of $ABC$. [b]p14.[/b] Compute the sumof the greatest and least values of $x$ such that $(x^2 -4x +4)^2 +x^2 -4x \le 16$. [b]p15.[/b] Ephram is starting a fan club. At the fan club’s first meeting, everyone shakes hands with everyone else exactly once, except for Ephram, who is extremely sociable and shakes hands with everyone else twice. Given that a total of $2015$ handshakes took place, how many people attended the club’s first meeting? PS. You should use hide for answers. Rounds 6-9 have been posted [url=https://artofproblemsolving.com/community/c3h3167139p28823346]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let be given points $A,B,C$ on a line, with $C$ between $A$ and $B$. Three semicircles with diameters $AC,BC,AB$ are drawn on the same side of line $ABC$. The perpendicular to $AB$ at $C$ meets the circle with diameter $AB$ at $H$. Given that $CH =\sqrt2$, compute the area of the region bounded by the three semicircles.

Find all pairs of function $f : Q \rightarrow R$ and $g : Q \rightarrow R,$ for which equations \begin{align*} f(x+y) &= f(x) f(y) + g(x) g(y) \\ g(x+y) &= f(x)g(y) + g(x)f(y) + g(x)g(y) \end{align*} holds for all rational numbers $x$ and $y.$ [i]Proposed by Gurgen Asatryan, Armenia [/i]

You are playing a game. A $2 \times 1$ rectangle covers two adjacent squares (oriented either horizontally or vertically) of a $3 \times 3$ grid of squares, but you are not told which two squares are covered. Your goal is to find at least one square that is covered by the rectangle. A "turn" consists of you guessing a square, after which you are told whether that square is covered by the hidden rectangle. What is the minimum number of turns you need to ensue that at least one of your guessed squares is covered by the rectangle? $\textbf{(A)}~3\qquad\textbf{(B)}~5\qquad\textbf{(C)}~4\qquad\textbf{(D)}~8\qquad\textbf{(E)}~6$

Each point of a plane with both coordinates being integers has been colored black or white. Show that there exists an infinite subset of colored points, whose points are in the same color, having a center of symmetry. [EDIT: added condition about being infinite - now it makes sense]

A biologist wants to calculate the number of fish in a lake. On May 1 she catches a random sample of 60 fish, tags them, and releases them. On September 1 she catches a random sample of 70 fish and finds that 3 of them are tagged. To calculate the number of fish in the lake on May 1, she assumes that $25\%$ of these fish are no longer in the lake on September 1 (because of death and emigrations), that $40\%$ of the fish were not in the lake May 1 (because of births and immigrations), and that the number of untagged fish and tagged fish in the September 1 sample are representative of the total population. What does the biologist calculate for the number of fish in the lake on May 1?

Let $p{}$ and $q{}$ be two coprime positive integers. A frog hops along the integer line so that on every hop it moves either $p{}$ units to the right or $q{}$ units to the left. Eventually, the frog returns to the initial point. Prove that for every positive integer $d{}$ with $d < p + q$ there are two numbers visited by the frog which differ just by $d{}$. [i]Nikolay Belukhov[/i]

$ABC$ is an acute-angled triangle. The tangents to the circumcircle at $A$ and $C$ meet the tangent at $B$ at $M$ and $N$. The altitude from $B$ meets $AC$ at $P$. Show that $BP$ bisects the angle $MPN$

Consider an isosceles triangle $ABC$ with $AB=AC$. Point $D$ is on the smaller arc $AB$ of its circumcirle. Point $E$ lies on the continuation of segment $AD$ beyond point $D$ so that both $A$ and $E$ lie in the same half-plane relative to $BC$. The circumcirle of triangle $BDE$ intersects side $AB$ at point $F$. Prove that lines $EF$ and $BC$ are parallel. (Author: R. Zhenodarov)

$n$ assistants start simultaneously from one vertex of a cube-shaped planet with edge length $1$. Each assistant moves along the edges of the cube at a constant speed of $2, 4, 8, \cdots, 2^n$, and can only change their direction at the vertices of the cube. The assistants can pass through each other at the vertices, but if they collide at any point that is not a vertex, they will explode. Determine the maximum possible value of $n$ such that the assistants can move infinitely without any collisions.

There are $n$ holes in a circle. The holes are numbered $1,2,3$ and so on to $n$. In the beginning, there is a peg in every hole except for hole $1$. A peg can jump in either direction over one adjacent peg to an empty hole immediately on the other side. After a peg moves, the peg it jumped over is removed. The puzzle will be solved if all pegs disappear except for one. For example, if $n=4$ the puzzle can be solved in two jumps: peg $3$ jumps peg $4$ to hole $1$, then peg $2$ jumps the peg in $1$ to hole $4$. (See illustration below, in which black circles indicate pegs and white circles are holes.) [center][img]http://i.imgur.com/4ggOa8m.png[/img][/center] [list=a] [*]Can the puzzle be solved for $n=5$? [*]Can the puzzle be solved for $n=2014$? [/list] In each part (a) and (b) either describe a sequence of moves to solve the puzzle or explain why it is impossible to solve the puzzle.

Determine all integers $ n > 1$ such that \[ \frac {2^n \plus{} 1}{n^2} \] is an integer.

Let $d(k)$ denote the number of positive divisors of a positive integer $k$. Prove that there exist infinitely many positive integers $M$ that cannot be written as \[M=\left(\frac{2\sqrt{n}}{d(n)}\right)^2\] for any positive integer $n$.

A natural number is written in each box of the $13 \times 13$ grid area. Prove that you can choose $2$ rows and $4$ columns such that the sum of the numbers written at their $8$ intersections is divisible by $8$.

Find the first two values of $40!(\text{mod }1763)$ [hide=Answer]1311, 3074[\hide]

In a geometric progression whose terms are positive, any term is equal to the sum of the next two following terms. then the common ratio is: $ \textbf{(A)}\ 1 \qquad\textbf{(B)}\ \text{about }\frac{\sqrt{5}}{2} \qquad\textbf{(C)}\ \frac{\sqrt{5}\minus{}1}{2} \qquad\textbf{(D)}\ \frac{1\minus{}\sqrt{5}}{2} \qquad\textbf{(E)}\ \frac{2}{\sqrt{5}}$

Two circles in space are said to be tangent to each other if they have a corni-non tangent at the same point of tangency. Assume that there are three circles in space which are mutually tangent at three distinct points. Prove that they either alI lie in a plane or all lie on a sphere.

Given an acute triangle $ABC$ with $AB <AC$. Points $P$ and $Q$ lie on the angle bisector of $\angle BAC$ so that $BP$ and $CQ$ are perpendicular on that angle bisector. Suppose that point $E, F$ lie respectively at sides $AB$ and $AC$ respectively, in such a way that $AEPF$ is a kite. Prove that the lines $BC, PF$, and $QE$ intersect at one point.

Given a circle with radius 1 and 2 points C, D given on it. Given a constant l with $0<l\le 2$. Moving chord of the circle AB=l and ABCD is a non-degenerated convex quadrilateral. AC and BD intersects at P. Find the loci of the circumcenters of triangles ABP and BCP.