Find all quadruplets of positive integers $(m,n,k,l)$ such that $3^m = 2^k + 7^n$ and $m^k = 1 + k + k^2 + k^3 + \ldots + k^l$.

A convex quadrilateral has area $30$ and side lengths $5, 6, 9,$ and $7,$ in that order. Denote by $\theta$ the measure of the acute angle formed by the diagonals of the quadrilateral. Then $\tan \theta$ can be written in the form $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

In a $2^n \times 2^n$ square with $n$ positive integer is covered with at least two non-overlapping rectangle pieces with integer dimensions and a power of two as surface. Prove that two rectangles of the covering have the same dimensions (Two rectangles have the same dimensions as they have the same width and the same height, wherein they, not allowed to be rotated.)

In a sequence, $x_1=\frac{1}{2}$ and $x_{n+1}=1-x_1x_2x_3...x_n$ for $n\ge 1$. Prove that $0.99<x_{100}<0.991$. Fresh translation. This problem may be similar to one of the 9th grade problems.

Consider a quadrilateral with $\angle DAB=60^{\circ}$, $\angle ABC=90^{\circ}$ and $\angle BCD=120^{\circ}$. The diagonals $AC$ and $BD$ intersect at $M$. If $MB=1$ and $MD=2$, find the area of the quadrilateral $ABCD$.

$3k$ points are marked on the circumference. They divide it onto $3k$ arcs. Some $k$ of them have length $1$, other $k$ of them have length $2$, the rest $k$ of them have length $3$. Prove that some two of the marked points are the ends of one diameter.

In a triangle $ABC$ denote by $D,E,F$ the points where the angle bisectors of $\angle CAB,\angle ABC,\angle BCA$ respectively meet it's circumcircle. a) Prove that the orthocenter of triangle $DEF$ coincides with the incentre of triangle $ABC$. b) Prove that if $\overrightarrow{AD}+\overrightarrow{BE}+\overrightarrow{CF}=0$, then the triangle $ABC$ is equilateral. [i]Marin Ionescu[/i]

Let $ a$, $ b$, $ c$, $ d$ be positive real numbers such that $ abcd \equal{} 1$ and $ a \plus{} b \plus{} c \plus{} d > \dfrac{a}{b} \plus{} \dfrac{b}{c} \plus{} \dfrac{c}{d} \plus{} \dfrac{d}{a}$. Prove that \[ a \plus{} b \plus{} c \plus{} d < \dfrac{b}{a} \plus{} \dfrac{c}{b} \plus{} \dfrac{d}{c} \plus{} \dfrac{a}{d}\] [i]Proposed by Pavel Novotný, Slovakia[/i]

Let $n\ge 2$ be a given integer. Show that the number of strings of length $n$ consisting of $0'$s and $1'$s such that there are equal number of $00$ and $11$ blocks in each string is equal to \[2\binom{n-2}{\left \lfloor \frac{n-2}{2}\right \rfloor}\]

A convex quadrilateral $ABCD$ is given in which the bisectors of the interior angles $\angle ABC$ and $\angle ADC$ have a common point on the diagonal $AC$. Prove that the bisectors of the interior angles $\angle BAD$ and $\angle BCD$ have a common point on the diagonal $BD$.

$ABCD$ is a convex quadrilateral with \[\angle BAC = 30^\circ \]\[\angle CAD = 20^\circ\]\[\angle ABD = 50^\circ\]\[\angle DBC = 30^\circ\] If the diagonals intersect at $P$, show that $PC = PD$.

Let $ABC$ be an acute triangle and suppose points $A, A_b, B_a, B, B_c, C_b, C, C_a,$ and $A_c$ lie on its perimeter in this order. Let $A_1 \neq A$ be the second intersection point of the circumcircles of triangles $AA_bC_a$ and $AA_cB_a$. Analogously, $B_1 \neq B$ is the second intersection point of the circumcircles of triangles $BB_cA_b$ and $BB_aC_b$, and $C_1 \neq C$ is the second intersection point of the circumcircles of triangles $CC_aB_c$ and $CC_bA_c$. Suppose that the points $A_1, B_1,$ and $C_1$ are all distinct, lie inside the triangle $ABC$, and do not lie on a single line. Prove that lines $AA_1, BB_1, CC_1,$ and the circumcircle of triangle $A_1B_1C_1$ all pass through a common point. [i]Josef Tkadlec (Czech Republic), Patrik Bak (Slovakia)[/i]

Find the integer closest to \[\frac{1}{\sqrt[4]{5^4+1}-\sqrt[4]{5^4-1}}\]

Let $M$ be a point inside square $ABCD$ and $A_1,B_1,C_1,D_1$ be the second intersection points of $AM$, $BM$, $CM$, $DM$ with the circumcircle of the square. Prove that $A_1B_1\cdot C_1D_1=A_1D_1\cdot B_1C_1$.

On the horizontal line from left to right are the points $P, \, \, Q, \, \, R, \, \, S$. Construct a square $ABCD$, for which on the line $AD$ lies lies the point $P$, on the line $BC$ lies the point $Q$, on the line $AB$ lies the point $R$, on the line $CD$ lies the point $S $.

Find all triplets of primes in the form $ (p, 2p\plus{}1, 4p\plus{}1)$.

Let $n \ge 3$ be an integer. Let $\mathcal{P}$ denote the set of vertices of a regular $n$-gon on the plane. A polynomial $f(x, y)$ of two variables with real coefficients is called $\textit{regular}$ if $$\mathcal{P} = \{(u, v) \in \mathbb{R}^2 \, | \, f(u, v) = 0 \}.$$ Find the smallest possible value of the degree of a regular polynomial. [i]Proposed by Navid Safaei[/i]

Find the smallest positive integer $ n$ with the property that the polynomial $ x^4 \minus{} nx \plus{} 63$ can be written as a product of two nonconstant polynomials with integer coefficients.

A paper square is bent along the line $\ell$, which passes through its center, so that a non-convex hexagon is formed. Investigate the question of the circle of largest radius that can be placed in such a hexagon.

Suppose that $A$ be the set of all positive integer that can write in form $a^2+2b^2$ (where $a,b\in\mathbb {Z}$ and $b$ is not equal to $0$). Show that if $p$ be a prime number and $p^2\in A$ then $p\in A$. [i]Marcel Tena[/i]

Prove that the polynomial $P_n(x)=1+x+\frac{x^2}{2!}+\cdots +\frac{x^n}{n!}$ has no real zeros if $n$ is even and has exatly one real zero if $n$ is odd

Prove that $2^{147} - 1$ is divisible by $343.$

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 $E$ be intersection of the diagonals of convex quadrilateral $ABCD$. It is given that $m(\widehat{EDC}) = m(\widehat{DEC})=m(\widehat{BAD})$. If $F$ is a point on $[BC]$ such that $m(\widehat{BAF}) + m(\widehat{EBF})=m(\widehat{BFE})$, show that $A$, $B$, $F$, $D$ are concyclic.

On the team round, an LMT team of six students wishes to divide itself into two distinct groups of three, one group to work on part $1,$ and one group to work on part $2.$ In addition, a captain of each group is designated. In how many ways can this be done?