The real numbers $x, y$ and $z$ are such that $x^2 + y^2 + z^2 = 1$. a) Determine the smallest and the largest possible values of $xy + yz - xz$. b) Prove that there does not exist a triple $(x, y, z)$ of rational numbers, which attains any of the two values in a).

Find all functions $f:\mathbb{N} \to \mathbb{N}$ such that for each natural integer $n>1$ and for all $x,y \in \mathbb{N}$ the following holds: $$f(x+y) = f(x) + f(y) + \sum_{k=1}^{n-1} \binom{n}{k}x^{n-k}y^k$$

Let $AA_1$, $BB_1$, $CC_1$ be the altitudes of an acute-angled, nonisosceles triangle $ABC$, and $A_2$, $B_2$, $C_2$ be the touching points of sides $BC$, $CA$, $AB$ with the correspondent excircles. It is known that line $B_1C_1$ touches the incircle of $ABC$. Prove that $A_1$ lies on the circumcircle of $A_2B_2C_2$.

For any natural numbers $n$, ( $n \geq 3$), let $f(n)$ denote the number of congruent integer-sided triangles with perimeter $n$. Show that (i) $f(1999) > f (1996)$; (ii) $f(2000) = f(1997)$.

$n$ points are given on the surface of a sphere. Show that the surface can be divided into $n$ congruent regions such that each of them contains exactly one of the given points.

Tyrone had $ 97$ marbles and Eric had $ 11$ marbles. Tyrone then gave some of his marbles to Eric so that Tyrone ended with twice as many marbles as Eric. How many marbles did Tyrone give to Eric? $ \textbf{(A)}\ 3 \qquad \textbf{(B)}\ 13 \qquad \textbf{(C)}\ 18 \qquad \textbf{(D)}\ 25 \qquad \textbf{(E)}\ 29$

What is the largest integer $n$ such that $n$ is divisible by every integer less than $\sqrt[3]{n}$?

If $\log_2(\log_{\frac{1}{2}}(\log_2x))=\log_3(\log_{\frac{1}{3}}(\log_3y))=\log_5(\log_{\frac{1}{5}}(\log_5z))=0$, then $\text{(A)}z<x<y\qquad\text{(B)}x<y<z\qquad\text{(C)}y<z<x\qquad\text{(D)}z<y<x$

$x, y, z$ are positive reals such that $x \leq 1$. Prove that $$xy+y+2z \geq 4 \sqrt{xyz}$$

Let G, H be two finite groups, and let $\varphi, \psi: G \to H$ be two surjective but non-injective homomorphisms. Prove that there exists an element of G that is not the identity element of G but whose images under $\varphi$ and $\psi$ are the same.

Let $A(x,y)$ be the number of points $(m,n)$ in the plane with integer coordinates $m$ and $n$ satisfying $m^2+n^2\le x^2+y^2$. Let $g=\sum_{k=1}^\infty e^{-k^2}$. Express $$\int^\infty_{-\infty}\int^\infty_{-\infty}A(x,y)e^{-x^2-y^2}dxdy$$ as a polynomial in $g$.

Every integer is painted white or black, so that if $m$ is white then $m + 20$ is also white, and if $k$ is black then $k + 35$ is also black. For which $n$ can exactly $n$ of the numbers $1, 2, ..., 50$ be white?

In triangle $ABC$ shown in the adjoining figure, $M$ is the midpoint of side $BC$, $AB=12$ and $AC=16$. Points $E$ and $F$ are taken on $AC$ and $AB$, respectively, and lines $EF$ and $AM$ intersect at $G$. If $AE=2AF$ then $\frac{EG}{GF}$ equals [asy] draw((0,0)--(12,0)--(14,7.75)--(0,0)); draw((0,0)--(13,3.875)); draw((5,0)--(8.75,4.84)); label("A", (0,0), S); label("B", (12,0), S); label("C", (14,7.75), E); label("E", (8.75,4.84), N); label("F", (5,0), S); label("M", (13,3.875), E); label("G", (7,1)); [/asy] $ \textbf{(A)}\ \frac{3}{2} \qquad\textbf{(B)}\ \frac{4}{3} \qquad\textbf{(C)}\ \frac{5}{4} \qquad\textbf{(D)}\ \frac{6}{5} \\ \qquad\textbf{(E)}\ \text{not enough information to solve the problem} $

Let $ABC$ be an acute triangle. Find the locus of the centers of the rectangles which have their vertices on the sides of $ABC$.

We are given a square $ ABCD$. Let $ P$ be a point not equal to a corner of the square or to its center $ M$. For any such $ P$, we let $ E$ denote the common point of the lines $ PD$ and $ AC$, if such a point exists. Furthermore, we let $ F$ denote the common point of the lines $ PC$ and $ BD$, if such a point exists. All such points $ P$, for which $ E$ and $ F$ exist are called acceptable points. Determine the set of all acceptable points, for which the line $ EF$ is parallel to $ AD$.

An event occurs many years ago. It occurs periodically in $x$ consecutive years, then there is a break of $y$ consecutive years. We know that the event occured in $1964$, $1986$, $1996$, $2008$ and it didn't occur in $1976$, $1993$, $2006$, $2013$. What is the first year in that the event will occur again?

What is the sum of all the solutions of $ x \equal{} |2x \minus{} |60\minus{}2x\parallel{}$? $ \textbf{(A)}\ 32\qquad\textbf{(B)}\ 60\qquad\textbf{(C)}\ 92\qquad\textbf{(D)}\ 120\qquad\textbf{(E)}\ 124$

Each point on the plane is colored either red or blue. Show that there are three points of the same color that form a triangle with side lengths $1, 2,\sqrt3$.

In triangle $ABC$, angle $B$ is obtuse and equal to $a$. The bisectors of angles $A$ and $C$ intersect opposite sides at points $P$ and $M$, respectively. On the side $AC$, points $K$ and $L$ are taken so that $\angle ABK = \angle CBL = 2a - 180^o$. What is the angle between straight lines $KP$ and $LM$?

On a 50-question multiple choice math contest, students receive 4 points for a correct answer, 0 points for an answer left blank, and -1 point for an incorrect answer. Jesse's total score on the contest was 99. What is the maximum number of questions that Jesse could have answered correctly? $ \textbf{(A)}\ 25\qquad\textbf{(B)}\ 27\qquad\textbf{(C)}\ 29\qquad\textbf{(D)}\ 31\qquad\textbf{(E)}\ 33$

Do there exist 2021 points with integer coordinates on the plane such that the pairwise distances between them are pairwise distinct consecutive integers?

Let $S = \{1,2,3,\dots,2014\}$. What is the largest subset of $S$ that contains no two elements with a difference of $4$ or $7$?

In a tetrahedron, the six triangles determined by an edge of the tetrahedron and the midpoint of the opposite edge all have equal area. Prove that the tetrahedron is regular.

Find the locus of the affix $M$, of the complex number $z$, so that it is aligned with the affixes of $i$ and $iz$ .

In a convex quadrilateral (in the plane) with the area of $32 \text{ cm}^{2}$ the sum of two opposite sides and a diagonal is $16 \text{ cm}$. Determine all the possible values that the other diagonal can have.