Find all triples $(a,b,k)$, $k \geq 2$, of positive integers such that $(a^k+b)(b^k+a)$ is a power of two.

Susie pays for $4$ muffins and $3$ bananas. Calvin spends twice as much paying for $2$ muffins and $16$ bananas. A muffin is how many times as expensive as a banana? $ \textbf {(A) } \frac{3}{2} \qquad \textbf {(B) } \frac{5}{3} \qquad \textbf {(C) } \frac{7}{4} \qquad \textbf {(D) } 2 \qquad \textbf {(E) } \frac{13}{4}$

Let $\triangle ABC$ and $\triangle C'B'A$ be two congruent triangles ( with this order and orient. ). Define point $M$ as the midpoint of segment $AB$ and suppose that the extension of $CB'$ from $B'$ passes trough $M$ , if $F$ be a point on the smaller arc $MC$ of circumcircle of triangle $\triangle BMC$ such that $\angle FB'A=90$ and $\angle C'CB' \neq 90$ , then prove that $\angle B'C'C=\angle CAF$. [i]Proposed by Alireza Dadgarnia[/i]

Let $ABCD$ be a rectangle with $AB = 20$ and $AD = 23.$ Let $M$ be the midpoint of $CD,$ and let $X$ be the reflection of $M$ across point $A.$ Compute the area of triangle $XBD.$

We define a relation between subsets of $\mathbb R ^n$. $A \sim B\Longleftrightarrow$ we can partition $A,B$ in sets $A_1,\dots,A_n$ and $B_1,\dots,B_n$(i.e $\displaystyle A=\bigcup_{i=1} ^n A_i,\ B=\bigcup_{i=1} ^n B_i, A_i\cap A_j=\emptyset,\ B_i\cap B_j=\emptyset$) and $A_i\simeq B_i$. Say the the following sets have the relation $\sim$ or not ? a) Natural numbers and composite numbers. b) Rational numbers and rational numbers with finite digits in base 10. c) $\{x\in\mathbb Q|x<\sqrt 2\}$ and $\{x\in\mathbb Q|x<\sqrt 3\}$ d) $A=\{(x,y)\in\mathbb R^2|x^2+y^2<1\}$ and $A\setminus \{(0,0)\}$

Let $p,q \in \mathbb N^{\ast}$, $p,q \geq 2$. We say that a set $X$ has the property $\left( \mathcal S \right)$ if no matter how we choose $p$ subsets $B_i \subset X$, $i = \overline{1,n}$, not necessarily distinct, each with $q$ elements, there is a subset $Y \subset X$ with $p$ elements s.t. the intersection of $Y$ with each of the $B_i$'s has an element at most, $i=\overline{1,p}$. Prove that: (a) if $p=4,q=3$ then any set composed of $9$ elements doesn't have $\left( \mathcal S \right)$; (b) any set $X$ composed of $pq-q$ elements doesn't have the property $\left( \mathcal S \right)$; (c) any set $X$ composed of $pq-q+1$ elements has the property $\left( \mathcal S \right)$. [i]Dan Schwarz[/i]

Let $ABC$ be a triangle such that the altitude $CD$ is equal to $AB$. The squares $DBEF$ and $ADGH$ are constructed with $F,G$ on $CD$. Show that the segments $CD,AE$ and $BH$ are concurrent.

It is given that $a_{11}, a_{22}$ are real numbers, that $x_1, x_2, a_{12}, b_1, b_2$ are complex numbers, and that $a_{11}a_{22}=a_{12}\overline{a_{12}}$ (Where $\overline{a_{12}}$ is he conjugate of $a_{12}$). We consider the following system in $x_1, x_2$: \[\overline{x_1}(a_{11}x_1 + a_{12}x_2) = b_1,\]\[\overline{x_2}(a_{12}x_1 + a_{22}x_2) = b_2.\] [b](a) [/b]Give one condition to make the system consistent. [b](b) [/b]Give one condition to make $\arg x_1 - \arg x_2 = 98^{\circ}.$

Let $ABC$ be a scalene acute triangle, where $AL$ $(L \in BC)$ is the internal bisector of $\angle BAC$ and $M$ is the midpoint of $BC$. Let the internal bisectors of $\angle AMB$ and $\angle CMA$ intersect $AB$ and $AC$ in $P$ and $Q$, respectively. Prove that the circumcircle of $APQ$ is tangent to $BC$ if and only if $L$ belongs to it.

Consider the infinite grid of lattice points in $\mathbb{Z}^3$. Little D and Big Z play a game, where Little D first loses a shoe on an unmunched point in the grid. Then, Big Z munches a shoe-free plane perpendicular to one of the coordinate axes. They continue to alternate turns in this fashion, with Little D's goal to lose a shoe on each of $n$ consecutive lattice points on a line parallel to one of the coordinate axes. Determine all $n$ for which Little D can accomplish his goal. [i]David Yang.[/i]

Let $a, b, c, d$ be positive real numbers satisfying $a+b+c+d =1$. Show that \[\frac{2}{(a+b)(c+d)} \leq \frac{1}{\sqrt{ab}}+ \frac{1}{\sqrt{cd}}\mbox{.}\] [i](Swiss Mathematical Olympiad 2011, Final round, problem 6)[/i]

We are numbering the rows and columns of a $29 \text{x} 29$ chess table with numbers $1, 2, ..., 29$ in order (Top row is numbered with $1$ and first columns is numbered with $1$ as well). We choose some of the squares in this chess table and for every selected square, we know that there exist at most one square having a row number greater than or equal to this selected square's row number and a column number greater than or equal to this selected square's column number. How many squares can we choose at most?

Let $s=a+b+c$, where $a$, $b$, and $c$ are integers that are lengths of the sides of a box. The volume of the box is numerically equal to the sum of the lengths of the twelve edges of the box plus its surface area. Find the sum of the possible values of $s$.

Let $n \ge 2$ be an integer. There are $n$ beads numbered $1, 2, \ldots, n$. Two necklaces made out of some of these beads are considered the same if we can get one by rotating the other (with no flipping allowed). For example, with $n \ge 5$, the necklace with four beads $1, 5, 3, 2$ in the clockwise order is same as the one with $5, 3, 2, 1$ in the clockwise order, but is different from the one with $1, 2, 3, 5$ in the clockwise order. We denote by $D_0(n)$ (respectively $D_1(n)$) the number of ways in which we can use all the beads to make an even number (resp. an odd number) of necklaces each of length at least $3$. Prove that $n - 1$ divides $D_1(n) - D_0(n)$.

Let a circle $ (O)$ with diameter $ AB$. A point $ M$ move inside $ (O)$. Internal bisector of $ \widehat{AMB}$ cut $ (O)$ at $ N$, external bisector of $ \widehat{AMB}$ cut $ NA,NB$ at $ P,Q$. $ AM,BM$ cut circle with diameter $ NQ,NP$ at $ R,S$. Prove that: median from $ N$ of triangle $ NRS$ pass over a fix point.

Find all integers $b$ and $c$ for which the equation $x^2-bx+c=0$ has two real roots $x_{1}$ and $x_{2}$ satisfying $x_{1}^2+x_{2}^2=5$.

Let $a, b, c $ be distinct positive integers. a) Prove that $a^2b^2 + a^2c^2 + b^2c^2 \ge 9$. b) if, moreover, $ab + ac + bc +3 = abc > 0,$ show that $$(a -1)(b -1)+(a -1)(c -1)+(b -1)(c -1) \ge 6.$$

In a tournament, every two of the $n$ players played exactly one match with each other (no draws). Prove that it is possible either (i) to partition the league in two groups $A$ and $B$ such that everybody in $A$ defeated everybody in $B$; or (ii) to arrange all the players in a chain $x_1, x_2, . . . , x_n, x_1$ in such a way that each player defeated his successor.

Consider the $2015$ integers $n$, from $ 1$ to $2015$. Determine for how many values ​​of $n$ it is verified that the number $n^3 + 3^n$ is a multiple of $5$.

Let $F_1=1, F_2=1,$ and $F_{n+2}=F_{n+1}+F_n.$ Then, $$S = \sum_{n=1}^{\infty} \arctan\left(\frac{1}{F_n}\right)\arctan\left(\frac{1}{F_{n+1}}\right)$$ Find $\lfloor 80S \rfloor.$ (Hint: it may be useful to note that $\arctan(\tfrac{1}{1}) = \arctan(\tfrac{1}{2})+\arctan(\tfrac{1}{3}).$)

Let $ABCD$ be a convex quadrilateral with $AD = BC, CD \nparallel AB, AD \nparallel BC$. Points $M$ and $N$ are the midpoints of the sides $CD$ and $AB$, respectively. a) If $E$ and $F$ are points, such that $MCBF$ and $ADME$ are parallelograms, prove that $\vartriangle BF N \equiv \vartriangle AEN$. b) Let $P = MN \cap BC$, $Q = AD \cap MN$, $R = AD \cap BC$. Prove that the triangle $PQR$ is iscosceles.

Let $C$ be a coloring of all edges and diagonals of a convex $n$−gon in red and blue (in Romanian, rosu and albastru). Denote by $q_r(C)$ (resp. $q_a(C)$) the number of quadrilaterals having all its edges and diagonals red (resp. blue). Prove: $ \underset{C}{min} (q_r(C)+q_a(C)) \le \frac{1}{32} {n \choose 4}$

Let $ a, b \in \mathbb{Z}$ which are not perfect squares. Prove that if \[ x^2 \minus{} ay^2 \minus{} bz^2 \plus{} abw^2 \equal{} 0\] has a nontrivial solution in integers, then so does \[ x^2 \minus{} ay^2 \minus{} bz^2 \equal{} 0.\]

For a positive integer $n$, let $S(n), P(n)$ denote the sum and the product of all the digits of $n$ respectively. 1) Find all values of n such that $n = P(n)$: 2) Determine all values of n such that $n = S(n) + P(n)$.

How many four-digit integers $abcd$, with $a \neq 0$, have the property that the three two-digit integers $ab<bc<cd$ form an increasing arithmetic sequence? One such number is $4692$, where $a=4$, $b=6$, $c=9$, and $d=2$. $\textbf{(A)}\ 9\qquad\textbf{(B)}\ 15\qquad\textbf{(C)}\ 16\qquad\textbf{(D)}\ 17\qquad\textbf{(E)}\ 20$