All faces of the tetrahedron $ABCD$ are acute-angled. Take a point $X$ in the interior of the segment $AB$, and similarly $Y$ in $BC, Z$ in $CD$ and $T$ in $AD$. [b]a.)[/b] If $\angle DAB+\angle BCD\ne\angle CDA+\angle ABC$, then prove none of the closed paths $XYZTX$ has minimal length; [b]b.)[/b] If $\angle DAB+\angle BCD=\angle CDA+\angle ABC$, then there are infinitely many shortest paths $XYZTX$, each with length $2AC\sin k$, where $2k=\angle BAC+\angle CAD+\angle DAB$.

A circle $c$ with center $A$ passes through the vertices $B$ and $E$ of a regular pentagon $ABCDE$ . The line $BC$ intersects the circle $c$ for second time at point $F$. The point $G$ on the circle $c$ is chosen such that $| F B | = | FG |$ and $B \ne G$. Prove that the lines $AB, EF$ and $DG$ intersect at one point.

In a village $1998$ persons volunteered to clean up, for a fair, a rectangular field with integer sides and perimeter equla to $3996$ feet. For this purpose, the field was divided into $1998$ equal parts. If each part had an integer area, find the length and breadth of the field.

Find all natural numbers $n < 1978$ with the following property: If $m$ is a natural number, $1 < m < n$, and $(m, n) = 1$ (i.e., $m$ and $n$ are relatively prime), then $m$ is a prime number.

Integers $1$ to $36$ are written in each "Neuro-Millions" bulletin. A bet on "Neuro-Millions" consists of choosing $6$ of these $36$ numbers. Then, $6$ numbers between $1$ and $36$ are drawn, and these constitute the key to "Neuro-Milh˜oes". A bet is awarded if it does not contain any of the key numbers. How many bets, at least, are necessary to guarantee a prize?

$x,y,z\in\left(0,\frac{\pi}{2}\right)$ are given. Prove that: \[ \frac{x\cos x+y\cos y+z\cos z}{x+y+z}\le \frac{\cos x+\cos y+\cos z}{3} \]

Consider a triangle $ABC$ and two congruent triangles $A_1B_1C_1$ and $A_2B_2C_2$ which are respectively similar to $ABC$ and inscribed in it: $A_i,B_i,C_i$ are located on the sides of $ABC$ in such a way that the points $A_i$ are on the side opposite to $A$, the points $B_i$ are on the side opposite to $B$, and the points $C_i$ are on the side opposite to $C$ (and the angle at A are equal to angles at $A_i$ etc.). The circumcircles of $A_1B_1C_1$ and $A_2B_2C_2$ intersect at points $P$ and $Q$. Prove that the line $PQ$ passes through the orthocenter of $ABC$.

Let $(v_1, ..., v_{2^n})$ be the vertices of an $n$-dimensional hypercube. Label each vertex $v_i$ with a real number $x_i$. Label each edge of the hypercube with the product of labels of the two vertices it connects. Let $S$ be the sum of the labels of all the edges. Over all possible labelings, find the minimum possible value of $\frac{S}{x^2_1+ x^2_2+ ...+ x^2_n}$ in terms of $ n$. Note: an $n$ dimensional hypercube is a graph on $2^n$ vertices labeled labeled with the binary strings of length $n$, where two vertices have an edge between them if and only if their labels differ in exactly one place. For instance, the vertices $100$ and $101$ on the $3$ dimensional hypercube are connected, but the vertices $100$ and $111$ are not.

Let $ABC$ be a triangle with $AB=AC$, and let $M$ be the midpoint of $BC$. Let $P$ be a point such that $PB<PC$ and $PA$ is parallel to $BC$. Let $X$ and $Y$ be points on the lines $PB$ and $PC$, respectively, so that $B$ lies on the segment $PX$, $C$ lies on the segment $PY$, and $\angle PXM=\angle PYM$. Prove that the quadrilateral $APXY$ is cyclic.

$n$ people numbered from $1$ to $n$ are arranged in a row. An [i]acceptable movement[/i] consists of each person changing at most once its place with another or remains in its place. For example $\begin{tabular}{|l|l|l|l|l|l|l|l|l|l|l|l|} \hline initial position & 1 & 2 & 3 & 4 & 5 & 6 & ... & n-2 & n-1 & n \\ \hline final position & 2 & 1 & 3 & 6 & 5 & 4 & ... & n & n-1 & n-2 \\ \hline \end{tabular}$ is an a[i]cceptable movement[/i]. Is it possible that starting from the position $\begin{tabular}{|l|l|l|l|l|l|l|l|l|l|l|l|} \hline 1 & 2 & 3 & 4 & 5 & 6 & ... & n-2 & n-1 & n \\ \hline \end{tabular}$ to reach to $\begin{tabular}{|l|l|l|l|l|l|l|l|l|l|l|l|} \hline n & 1 & 2 & 3 & 4 & 5 & 6 & ... & n-2 & n-1 \\ \hline \end{tabular}$ through two [i]acceptable movements[/i]?

How many possible arrangements of bishops are there on a $8 \times 8$ chessboard such that no bishop threatens a square on which another lies and the maximum number of bishops are used? (Note that a bishop threatens any square along a diagonal containing its square.)

If $ f(x) \equal{} ax \plus{} b$ and $ f^{ \minus{} 1}(x) \equal{} bx \plus{} a$ with $ a$ and $ b$ real, what is the value of $ a \plus{} b$? $ \textbf{(A)} \minus{} \!2 \qquad \textbf{(B)} \minus{} \!1 \qquad \textbf{(C)}\ 0 \qquad \textbf{(D)}\ 1 \qquad \textbf{(E)}\ 2$

Given real numbers $x,y,z,a$ satisfying: $$ x+y+z = a$$ $$ \frac{1}{x}+\frac{1}{y}+\frac{1}{z} = \frac{1}{a} $$ Prove that at least one of the numbers $x,y,z$ is equal to $a$.

The integer $n$, between 10000 and 99999, is $abcde$ when written in decimal notation. The digit $a$ is the remainder when $n$ is divided by 2, the digit $b$ is the remainder when $n$ is divided by 3, the digit $c$ is the remainder when $n$ is divided by 4, the digit $d$ is the remainder when $n$ is divied by 5, and the digit $e$ is the reminader when $n$ is divided by 6. Find $n$.

Let $ABCD$ be a circumscribed quadrilateral. Prove that the common point of the diagonals, the incenter of triangle $ABC$ and the centre of excircle of triangle $CDA$ touching the side $AC$ are collinear.

Find all integers $n\geq 1$ such that there exists a permutation $(a_1,a_2,...,a_n)$ of $(1,2,...,n)$ such that $a_1+a_2+...+a_k$ is divisible by $k$ for $k=1,2,...,n$

Let $\mathbb{R}^{+}$ be the set of positive real numbers. Find all functions $f:\mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$ such that for all $x,y>0$ we have $$f(xy+f(x))=yf(x)+x.$$ [i]Proposed by Nikola Velov[/i]

In how many ways rooks can be placed on a $8$ by $8$ chess board such that every row and every column has at least one rook? (Any number of rooks are available,each square can have at most one rook and there is no relation of attacking between them)

Consider $2011^2$ points arranged in the form of a $2011 \times 2011$ grid. What is the maximum number of points that can be chosen among them so that no four of them form the vertices of either an isosceles trapezium or a rectangle whose parallel sides are parallel to the grid lines?

Prove that for every nonnegative integer $n$, the number $7^{7^{n}}+1$ is the product of at least $2n+3$ (not necessarily distinct) primes.

Let $ d$ be the sum of the lengths of all the diagonals of a plane convex polygon with $ n$ vertices (where $ n>3$). Let $ p$ be its perimeter. Prove that: \[ n\minus{}3<{2d\over p}<\Bigl[{n\over2}\Bigr]\cdot\Bigl[{n\plus{}1\over 2}\Bigr]\minus{}2,\] where $ [x]$ denotes the greatest integer not exceeding $ x$.

The incircle of triangle $ABC$ touches its sides at points $A'$, $B'$, $C'$. $I$ is its center. Straight line $B'I$ intersects segment $A'C'$ at point $P$. Prove that straight line $BP$ passes through the midpoint of $AC$.

Solve equation $$x^2-\sqrt{a-x}=a$$ where $x$ is real number and $a$ is real parameter

For each pair of positive integers $m$ and $n$, we define $f_m(n)$ as follows: $$ f_m(n) = \gcd(n, d_1) + \gcd(n, d_2) + \cdots + \gcd(n, d_k), $$ where $1 = d_1 < d_2 < \cdots < d_k = m$ are all the positive divisors of $m$. For example, $f_4(6) = \gcd(6,1) + \gcd(6,2) + \gcd(6,4) = 5$. $a)\:$ Find all positive integers $n$ such that $f_{2017}(n) = f_n(2017)$. $b)\:$ Find all positive integers $n$ such that $f_6(n) = f_n(6)$.

A frog started from the origin of the coordinate plane and made $3$ jumps. Each time, the frog jumped a distance of $5$ units and landed on a point with integer coordinates. How many different position possibilities end of the frog there?