A pair of positive integers $(m,n)$ is called [b][i]'steakmaker'[/i][/b] if they maintain the equation 1 + 2$^m$ = n$^2$. For which values of m and n, the pair $(m,n)$ are steakmaker, find the sum of $mn$

In isosceles $\vartriangle ABC, AB = AC, \angle BAC$ is obtuse, and points $E$ and $F$ lie on sides $AB$ and $AC$, respectively, so that $AE = 10, AF = 15$. The area of $\vartriangle AEF$ is $60$, and the area of quadrilateral $BEFC$ is $102$. Find $BC$.

$ABCDEF$ is an equiangular hexagon of perimeter $21$. Given that $AB = 3, CD = 4$, and $EF = 5$, compute the area of hexagon $ABCDEF$.

For $m\in \mathbb{N}$ define $m?$ be the product of first $m$ primes. Determine if there exists positive integers $m,n$ with the following property : \[ m?=n(n+1)(n+2)(n+3) \] [i]Proposed by Matko Ljulj[/i]

A and B plays the following game: they choose randomly $k$ integers from $\{1,2,\dots,100\}$; if their sum is even, A wins, else B wins. For what values of $k$ does A and B have the same chance of winning?

The net of 20 triangles shown below can be folded to form a regular icosahedron. Inside each of the triangular faces, write a number from 1 to 20 with each number used exactly once. Any pair of numbers that are consecutive must be written on faces sharing an edge in the folded icosahedron, and additionally, 1 and 20 must also be on faces sharing an edge. Some numbers have been given to you. (No proof is necessary.) [asy] unitsize(1cm); pair c(int a, int b){return (a-b/2,sqrt(3)*b/2);} draw(c(0,0)--c(0,1)--c(-1,1)--c(1,3)--c(1,1)--c(2,2)--c(3,2)--c(4,3)--c(4,2)--c(3,1)--c(2,1)--c(2,-1)--c(1,-1)--c(1,-2)--c(0,-3)--c(0,-2)--c(-1,-2)--c(1,0)--cycle); draw(c(0,0)--c(1,1)--c(0,1)--c(1,2)--c(0,2)--c(0,1),linetype("4 4")); draw(c(4,2)--c(3,2)--c(3,1),linetype("4 4")); draw(c(3,2)--c(1,0)--c(1,1)--c(2,1)--c(2,2),linetype("4 4")); draw(c(1,-2)--c(0,-2)--c(0,-1)--c(1,-1)--c(1,0)--c(2,0)--c(0,-2),linetype("4 4")); label("2",(c(0,2)+c(1,2))/2,S); label("15",(c(1,1)+c(2,1))/2,S); label("6",(c(0,1)+c(1,1))/2,N); label("14",(c(0,0)+c(1,0))/2,N);[/asy]

Given two monic polynomials $P(x)$ and $Q(x)$ with degrees 2016. $P(x)=Q(x)$ has no real root. [b]Prove that P(x)=Q(x+1) has at least one real root.[/b]

Given points $P$ and $Q$, Jaqueline has a ruler that allows tracing the line $PQ$. Jaqueline also has a special object that allows the construction of a circle of diameter $PQ$. Also, always when two circles (or a circle and a line, or two lines) intersect, she can mark the points of the intersection with a pencil and trace more lines and circles using these dispositives by the points marked. Initially, she has an acute scalene triangle $ABC$. Show that Jaqueline can construct the incenter of $ABC$.

Indra has a bag for bringing flowers for her grandmother. The first day she brings $n$ flowers. From the second day Indra tries to bring three times plus one with respect to the number of flowers of the previous day. However, if this number is greater or equal to $40$, Indra substracts multiples of $40$ until the remainder is less than this number, since her bag cannot containt so many flowers. For which value of $n$ Indra will bring $30$ flowers the day $2016$?

Find all prime numbers such that the square of the prime number can be written as the sum of cubes of two positive integers.

The points in the plane with integer coordinates are numbered as below. [img]https://cdn.artofproblemsolving.com/attachments/0/2/122cb559c6fb4cb8401ffa215528a035346a3d.png[/img] What are the coordinates of the number $2003$?

In triangle $ABC$ , $\angle A = 45^o$ and $M$ is the midpoint of $\overline{BC}$. $\overline{AM}$ intersects the circumcircle of $ABC$ for the second time at $D$, and $AM = 2MD$. Find $cos\angle AOD$, where $O$ is the circumcenter of $ABC$.

In the diagram below, all circles are tangent to each other as shown. The six outer circles are all congruent to each other, and the six inner circles are all congruent to each other. Compute the ratio of the area of one of the outer circles to the area of one of the inner circles. [img]https://cdn.artofproblemsolving.com/attachments/b/6/4cfbc1df86b8d38e082b7ad0a71b9e366548b3.png[/img]

Given a prime number $p\ge 5$. Find the number of distinct remainders modulus $p$ of the product of three consecutive positive integers.

[b]3.[/b] Let $A$ be a subset of n-dimensional space containing at least one inner point and suppose that, for every point pair $x, y \in A$, the subset $A$ contains the mid point of the line segment beteween $x$ and $y$. Show that $A$ consists of a convex open set and of some of its boundary points. [b](St. 1)[/b]

A number is called [i]trilegal[/i] if its digits belong to the set \(\{1, 2, 3\}\) and if it is divisible by \(99\). How many trilegal numbers with \(10\) digits are there?

Let $ x_1,x_2,\ldots,x_n$ be real numbers greater than 1. Show that \[ \frac{x_1x_2}{x_3}\plus{}\frac{x_2x_3}{x_4}\plus{}\cdots\plus{}\frac{x_nx_1}{x_2}\ge4n\] and determine when the equality holds.

Let $ A\in M_2\left( \mathbb{C}\right) $ such that $ \det \left( A^2+A+I_2\right) =\det \left( A^2-A+I_2\right) =3. $ Prove that $ A^2\left( A^2+I_2\right) =2I_2. $

Let $ k$ be a circle and $ k_{1},k_{2},k_{3},k_{4}$ four smaller circles with their centres $ O_{1},O_{2},O_{3},O_{4}$ respectively, on $ k$. For $ i \equal{} 1,2,3,4$ and $ k_{5}\equal{} k_{1}$ the circles $ k_{i}$ and $ k_{i\plus{}1}$ meet at $ A_{i}$ and $ B_{i}$ such that $ A_{i}$ lies on $ k$. The points $ O_{1},A_{1},O_{2},A_{2},O_{3},A_{3},O_{4},A_{4}$ lie in that order on $ k$ and are pairwise different. Prove that $ B_{1}B_{2}B_{3}B_{4}$ is a rectangle.

We have 101 coins and a two-pan scale. In one weighing, we can compare the weights of two coins. What is the smallest number of weighings required in order to decide whether there exist 51 coins which all have the same weight?

Let $M$ and $N$ be the midpoints of the sides $AC$ and $AB$, respectively, of an acute triangle $ABC$, $AB \neq AC$. Let $\omega_B$ be the circle centered at $M$ passing through $B$, and let $\omega_C$ be the circle centered at $N$ passing through $C$. Let the point $D$ be such that $ABCD$ is an isosceles trapezoid with $AD$ parallel to $BC$. Assume that $\omega_B$ and $\omega_C$ intersect in two distinct points $P$ and $Q$. Show that $D$ lies on the line $PQ$.

Let $(a,b)$ be a pair of natural numbers. Henning and Paul play the following game. At the beginning there are two piles of $a$ and $b$ coins respectively. We say that $(a,b)$ is the [i]starting position [/i]of the game. Henning and Paul play with the following rules: $\bullet$ They take turns alternatively where Henning begins. $\bullet$ In every step each player either takes a positive integer number of coins from one of the two piles or takes same natural number of coins from both piles. $\bullet$ The player how take the last coin wins. Let $A$ be the set of all positive integers like $a$ for which there exists a positive integer $b<a$ such that Paul has a wining strategy for the starting position $(a,b)$. Order the elements of $A$ to construct a sequence $a_1<a_2<a_3<\dots$ $(a)$ Prove that $A$ has infinity many elements. $(b)$ Prove that the sequence defined by $m_k:=a_{k+1}-a_{k}$ will never become periodic. (This means the sequence $m_{k_0+k}$ will not be periodic for any choice of $k_0$)

In a triangle $ABC$ with $AB = AC$, let $M$ be the midpoint of $CB$ and let $D$ be a point in $BC$ such that $\angle BAD = \frac{\angle BAC}{6}$. The perpendicular line to $AD$ by $C$ intersects $AD$ in $N$ where $DN = DM$. Find the angles of the triangle $BAC$.

Points $B = B_1 , B_2, \cdots , B_n , B_{n+1} = C$ are chosen on side $BC$ of a triangle $ABC$ in that order. Let $r_j$ be the inradius of triangle $AB_jB_{j+1}$ for $j = 1, \cdots, n$ , and $r$ be the inradius of $\triangle ABC$. Show that there is a constant $\lambda$ independent of $n$ such that : \[(\lambda -r_1)(\lambda -r_2)\cdots (\lambda -r_n) =\lambda^{n-1}(\lambda -r)\]

Prove that if the equation $$x^4 + ax + b = 0$$ has two equal roots, then $$\left( \frac{a}{4} \right)^4 =\left( \frac{b}{3} \right)^3.$$