Determine the value of \[ \sup_{ 1 \leq \xi \leq 2} [\log E \xi\minus{}E \log \xi],\] where $ \xi$ is a random variable and $ E$ denotes expectation. [i]Z. Daroczy[/i]

Let $ABCD$ be inscribed in a circle $\omega$. Let the line parallel to the tangent to $\omega$ at $A$ and passing $D$ meet $\omega$ at $E$. $F$ is a point on $\omega$ such that lies on the different side of $E$ wrt $CD$. If $AE \cdot AD \cdot CF = BE \cdot BC \cdot DF$ and $\angle CFD = 2\angle AFB$, Show that the tangent to $\omega$ at $A, B$ and line $EF$ concur at one point. ($A$ and $E$ lies on the same side of $CD$)

Find the root that the following three polynomials have in common: \begin{align*} & x^3+41x^2-49x-2009 \\ & x^3 + 5x^2-49x-245 \\ & x^3 + 39x^2 - 117x - 1435\end{align*}

The reals $x, y$ satisfy $x(x-6)\leq y(4-y)+7$. Find the minimal and maximal values of the expression $x+2y$.

Let $ABCD$ be a regular tetrahedron with side length $2$. The plane parallel to edges $AB$ and $CD$ and lying halfway between them cuts $ABCD$ into two pieces. Find the surface area of one of these pieces.

In the figure, $ \angle EAB$ and $ \angle ABC$ are right angles. $ AB \equal{} 4, BC \equal{} 6, AE \equal{} 8$, and $ \overline{AC}$ and $ \overline{BE}$ intersect at $ D$. What is the difference between the areas of $ \triangle ADE$ and $ \triangle BDC$? [asy]unitsize(4mm); defaultpen(linewidth(.8pt)+fontsize(10pt)); pair A=(0,0), B=(4,0), C=(4,6), Ep=(0,8); pair D=extension(A,C,Ep,B); draw(A--C--B--A--Ep--B); label("$A$",A,SW); label("$B$",B,SE); label("$C$",C,N); label("$E$",Ep,N); label("$D$",D,2.5*N); label("$4$",midpoint(A--B),S); label("$6$",midpoint(B--C),E); label("$8$",(0,3),W);[/asy]$ \textbf{(A)}\ 2\qquad \textbf{(B)}\ 4\qquad \textbf{(C)}\ 5\qquad \textbf{(D)}\ 8\qquad \textbf{(E)}\ 9$

Let $\mathbb{N}$ denote the set of positive integers. Find all functions $f:\mathbb{N}\longrightarrow\mathbb{N}$ such that \[n+f(m)\mid f(n)+nf(m)\] for all $m,n\in \mathbb{N}$ [i]Proposed by Dorlir Ahmeti, Albania[/i]

In the $n \times n$ table in every cell there is one child. Every child looks in neigbour cell. So every child sees ear or back of the head of neighbour. What is minimal number children, that see ear ?

Prove that there are infinitely many positive integers $n$ such that $n^2$ written in base $4$ contains only digits $1$ and $2$.

Does exist polynoms of one variable that are irreducible over the field of integers, have degree $ 60 $ and have multiples of the form $ X^n-1? $ If so, how many of them?

Mientka Publishing Company prices its bestseller [i]Where's Walter?[/i] as follows: \[C(n) \equal{} \begin{cases} 12n, &\text{if } 1 \le n \le 24\\ 11n, &\text{if } 25 \le n \le 48\\ 10n, &\text{if } 49 \le n \end{cases}\] where $ n$ is the number of books ordered, and $ C(n)$ is the cost in dollars of $ n$ books. Notice that $ 25$ books cost less than $ 24$ books. For how many values of $ n$ is it cheaper to buy more than $ n$ books than to buy exactly $ n$ books? $ \textbf{(A)}\ 3\qquad \textbf{(B)}\ 4\qquad \textbf{(C)}\ 5\qquad \textbf{(D)}\ 6\qquad \textbf{(E)}\ 8$

Points $A$, $B$, and $C$ lie on a circle $\Omega$ such that $A$ and $C$ are diametrically opposite each other. A line $\ell$ tangent to the incircle of $\triangle ABC$ at $T$ intersects $\Omega$ at points $X$ and $Y$. Suppose that $AB=30$, $BC=40$, and $XY=48$. Compute $TX\cdot TY$.

Let $\mathbb{R}$ be the set of real numbers. Determine all functions $f: \mathbb{R}\longrightarrow \mathbb{R}$ so that the equality $$f(x+yf(x+y)) +xf(x)= f(xf(x+y+1))+y^2$$ is true for any real numbers $x,y$.

Each of the six boxes $B_1$, $B_2$, $B_3$, $B_4$, $B_5$, $B_6$ initially contains one coin. The following operations are allowed Type 1) Choose a non-empty box $B_j$, $1\leq j \leq 5$, remove one coin from $B_j$ and add two coins to $B_{j+1}$; Type 2) Choose a non-empty box $B_k$, $1\leq k \leq 4$, remove one coin from $B_k$ and swap the contents (maybe empty) of the boxes $B_{k+1}$ and $B_{k+2}$. Determine if there exists a finite sequence of operations of the allowed types, such that the five boxes $B_1$, $B_2$, $B_3$, $B_4$, $B_5$ become empty, while box $B_6$ contains exactly $2010^{2010^{2010}}$ coins. [i]Proposed by Hans Zantema, Netherlands[/i]

Let $ S $ be a set of $ n $ concentric circles in the plane. Prove that if a function $ f: S\to S $ satisfies the property \[ d( f(A),f(B)) \geq d(A,B) \] for all $ A,B \in S $, then $ d(f(A),f(B)) = d(A,B) $, where $ d $ is the euclidean distance function.

Given a triangle $\vartriangle ABC$,construct squares $BAQP$ and $ACRS$ outside the triangle $ABC$ (with vertices in that listed in counterclockwise order).Show that the line from $A$ perpendicular to $BC$ passes through the midpoint of the segment $QS$.

In the land of Binary , the unit of currency is called Ben and currency notes are available in denominations $1,2,2^2,2^3,..$ Bens. The rules of the Government of Binary stipulate that one can not use more than two notes of any one denomination in any transaction. For example, one can give change for $2$ Bens in two ways : $2$ one Ben notes or $1$ two Ben note. For $5$ Ben one can given $1$ one Ben and $1$ four Ben note or $1$ Ben note and $2$ two Ben notes. Using $5$ one Ben notes or $3$ one Ben notes and $1$ two Ben notes for a $5$ Ben transaction is prohibited. Find the number of ways in which one can give a change $100$ Bens following the rules of the Government.

Given are positive integers $ n>1$ and $ a$ so that $ a>n^2$, and among the integers $ a\plus{}1, a\plus{}2, \ldots, a\plus{}n$ one can find a multiple of each of the numbers $ n^2\plus{}1, n^2\plus{}2, \ldots, n^2\plus{}n$. Prove that $ a>n^4\minus{}n^3$.

What is the least positive integer $k$ satisfying that $n+k\in S$ for every $n\in S$ where $S=\{n : n3^n + (2n+1)5^n \equiv 0 \pmod 7\}$? $ \textbf{(A)}\ 6 \qquad\textbf{(B)}\ 7 \qquad\textbf{(C)}\ 14 \qquad\textbf{(D)}\ 21 \qquad\textbf{(E)}\ 42 $

Let $ABC$ be a scalene triangle and its incentre $I$. Denote by $F_A$ the intersection of the line $BC$ and the perpendicular to the angle bisector at $A$ through $I$. Let us define points $F_B$ and $F_C$ in a similar manner. Prove that points $F_A, F_B$ and $F_C$ are collinear.

There are $2006$ points marked on the surface of a sphere. Prove that the surface can be cut into $2006$ congruent pieces so that each piece contains exactly one of these points inside it.

Let be a nonnegative integer $ n. $ Find all continuous functions $ f:\mathbb{R}_{\ge 0}\longrightarrow\mathbb{R} $ for which the following equation holds: $$ (1+n)\int_0^x f(t) dt =nxf(x) ,\quad\forall x>0. $$

Let $ABCD$ be a convex quadrilateral such that $\angle ABC = \angle ADC = 135^{\circ}$ and \[AC^2\cdot BD^2 = 2\cdot AB\cdot BC\cdot CD\cdot DA.\] Prove that the diagonals of the quadrilateral $ABCD$ are perpendicular.

How many real numbers $a$ are there for which both solutions to the equation \[x^2 + (a - 2024)x + a = 0\] are integers? \[\mathrm a. ~15\qquad \mathrm b. ~16 \qquad \mathrm c. ~18 \qquad\mathrm d. ~20\qquad\mathrm e. ~24\qquad\]

For each rational number $r$ consider the statement: If $x$ is a real number such that $x^2-rx$ and $x^3-rx$ are both rational, then $x$ is also rational. [list](a) Prove the claim for $r \ge \frac43$ and $r \le 0$. (b) Let $p,q$ be different odd primes such that $3p <4q$. Prove that the claim for $r=\frac{p}q$ does not hold. [/list]