(a) Show that if $f:[-1,1]\to \mathbb{R}$ is a convex and $C^2$ function such that $f(1),f(-1)\geq 0$, then: \[\min_{x\in[-1,1]} \{f(x)\} \geq - \int_{-1}^1 f''\] (b) Let $B\subset \mathbb{R}^2$ the closed ball with center $0$ and radius $1$. Show that if $f: B \to \mathbb{R}$ is a convex and $C^2$ function and $f\geq 0$ in $\partial B$, then: \[f(0)\geq -\frac{1}{\sqrt{\pi}} \left( \int_{B} (f_{xx}f_{yy}-f_{xy}^2) \right)^{1/2}\]

The classrooms at MIT are each identified with a positive integer (with no leading zeroes). One day, as President Reif walks down the Infinite Corridor, he notices that a digit zero on a room sign has fallen off. Let $N$ be the original number of the room, and let $M$ be the room number as shown on the sign. The smallest interval containing all possible values of $\frac{M}{N}$ can be expressed as $[\frac{a}{b}, \frac{c}{d} )$ where $a,b,c,d$ are positive integers with $\gcd(a,b) = \gcd(c,d) = 1$. Compute $1000a+100b+10c+d$.

Given are four circles $\Gamma_1, \Gamma_2, \Gamma_3, \Gamma_4$. Circles $\Gamma_1$ and $\Gamma_2$ are externally tangent at point $A$. Circles $\Gamma_2$ and $\Gamma_3$ are externally tangent at point $B$. Circles $\Gamma_3$ and $\Gamma_4$ are externally tangent at point $C$. Circles $\Gamma_4$ and $\Gamma_1$ are externally tangent at point $D$. Prove that $ABCD$ is cyclic.

Show that a square may be partitioned into $n$ smaller squares for sufficiently large $n$. Show that for some constant $N(d)$, a $d$-dimensional cube can be partitioned into $n$ smaller cubes if $n \geq N(d) $.

1.14. Let P and Q be interior points of triangle ABC such that \ACP = \BCQ and \CAP = \BAQ. Denote by D;E and F the feet of the perpendiculars from P to the lines BC, CA and AB, respectively. Prove that if \DEF = 90, then Q is the orthocenter of triangle BDF.

For a triangle $ABC$ and a point $M$ inside the triangle we consider the lines $AM, BM,CM$ which intersect the sides $BC, CA, AB$ in $A_1, B_1, C_1$ respectively. Take $A', B', C'$ to be the intersection points between the lines $AA_1, BB_1, CC_1$ and $B_1C_1, C_1A_1, A_1B_1$ respectively. a) Prove that the lines $BC', CB'$ and $AA'$ intersect in a point $A_2$; b) Define similarly points $B_2, C_2$. Find the loci of $M$ such that the triangle $A_1B_1C_1$ is similar with the triangle $A_2B_2C_2$.

In triangle $ABC,\angle B > \angle C, T$ is the midpoint of arc $BAC$ of the circumcicle of $ABC$, and $I$ is the incentre of $ABC$. Let $E$ be point such that $\angle AEI = 90^0$ and $AE$ is parallel to $BC$. If $TE$ intersects the circumcircle of $ABC$ at $P(\ne T)$ and $\angle B = \angle IPB$, determine $\angle A$.

In an acute angled triangle $ABC$, let $H$ be the orthocenter, and let $D,E,F$ be the feet of altitudes from $A,B,C$ to the opposite sides, respectively. Let $L,M,N$ be the midpoints of the segments $AH, EF, BC$ respectively. Let $X,Y$ be the feet of altitudes from $L,N$ on to the line $DF$ respectively. Prove that $XM$ is perpendicular to $MY$.

Prove that the numbers $${{2^n-1} \choose {i}}, i = 0, 1, . . ., 2^{n-1} - 1,$$ have pairwise different residues modulo $2^n$

The vertices of 100-gon (i.e., polygon with 100 sides) are colored alternately white or black. One of the vertices contains a checker. Two players in turn do two things: move the checker into other vertice along the side of 100-gon and then erase some side. The game ends when it is impossible to move the checker. At the end of the game if the checker is in the white vertice then the first player wins. Otherwise the second player wins. Does any of the players have winning strategy? If yes, then who? [i]Remark.[/i] The answer may depend on initial position of the checker.

There are $2000$ apples , contained in several baskets. One can remove baskets and /or remove apples from the baskets. Prove that it is possible to then have an equal number of apples in each of the remaining baskets, with the total number of apples being not less than $100$ . (A. Razborov)

How many pairs of positive integers $(a,b)$ with $a\leq b$ satisfy $\dfrac{1}{a}+\dfrac{1}{b}=\dfrac{1}{6}$?

Inside a quadrilateral $ABCD$ with $AB=BC=CD$, the points $P$ and $Q$ are chosen so that $AP=PB=CQ=QD$. The line through the point $P$ parallel to the diagonal $AC$ intersects the line through the point $Q$ parallel to the diagonal $BD$ at the point $T$. Prove that $BT=CT$. [i]Proposed by Mykhailo Shtandenko[/i]

Is it possible to dissect a regular decagon along some of its diagonals so that the resulting parts can form two regular polygons? by N.Beluhov

Assume we are given a set of weights, $x_1$ of which have mass $d_1$, $x_2$ have mass $d_2$, etc, $x_k$ have mass $d_k$, where $x_i,d_i$ are positive integers and $1\le d_1<d_2<\ldots<d_k$. Let us denote their total sum by $n=x_1d_1+\ldots+x_kd_k$. We call such a set of weights [i]perfect[/i] if each mass $0,1,\ldots,n$ can be uniquely obtained using these weights. (a) Write down all sets of weights of total mass $5$. Which of them are perfect? (b) Show that a perfect set of weights satisfies $$(1+x_1)(1+x_2)\cdots(1+x_k)=n+1.$$ (c) Conversely, if $(1+x_1)(1+x_2)\cdots(1+x_k)=n+1$, prove that one can uniquely choose the corresponding masses $d_1,d_2,\ldots,d_k$ with $1\le d_1<\ldots<d_k$ in order for the obtained set of weights is perfect. (d) Determine all perfect sets of weights of total mass $1993$.

The lengths of a rectangle’s sides and of its diagonal are integers. Prove that the area of the rectangle is an integer multiple of $12$.

In a non-equilateral triangle $ABC$ point $I$ is the incenter and point $O$ is the circumcenter. A line $s$ through $I$ is perpendicular to $IO$. Line $\ell$ symmetric to like $BC$ with respect to $s$ meets the segments $AB$ and $AC$ at points $K$ and $L$, respectively ($K$ and $L$ are different from $A$). Prove that the circumcenter of triangle $AKL$ lies on the line $IO$. [i]Dušan Djukić[/i]

The figure of equation $|x-y^2|=1-|x|$ is [img]https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvNi80LzQ4YjgxN2YxMjc0YTBkNzZiZjJiMTRhMjBiNDExN2I5OGZhZGY3LnBuZw==&rn=MjAwMDAwMDAwMDAwMC5wbmc=[/img]

A circle of radius 1 is placed in a corner of a room (i.e., it touches the horizontal floor and two vertical walls perpendicular to each other). Find the locus of the center of the band for all of its possible positions. [b]Note.[/b] For the solution of this problem, it is useful to know the following Monge theorem: The locus of all points $P$, such that the two tangents from $P$ to the ellipse with equation $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ are perpendicular to each other, is a circle − a so-called Monge circle − with equation $x^2 + y^2 = a^2 + b^2$.

How many zeroes appear at the end of $209$ factorial?

For each integer $n\ge 1,$ compute the smallest possible value of \[\sum_{k=1}^{n}\left\lfloor\frac{a_k}{k}\right\rfloor\] over all permutations $(a_1,\dots,a_n)$ of $\{1,\dots,n\}.$ [i]Proposed by Shahjalal Shohag, Bangladesh[/i]

Let be two distinct natural numbers $ k_1 $ and $ k_2 $ and a sequence $ \left( x_n \right)_{n\ge 0} $ which satisfies $$ x_nx_m +k_1k_2\le k_1x_n +k_2x_m,\quad\forall m,n\in\{ 0\}\cup\mathbb{N}. $$ Calculate $ \lim_{n\to\infty}\frac{n!\cdot (-1)^{1+n}\cdot x_n^2}{n^n} . $

Let $\mathcal{S}$ be a set consisting of $n \ge 3$ positive integers, none of which is a sum of two other distinct members of $\mathcal{S}$. Prove that the elements of $\mathcal{S}$ may be ordered as $a_1, a_2, \dots, a_n$ so that $a_i$ does not divide $a_{i - 1} + a_{i + 1}$ for all $i = 2, 3, \dots, n - 1$.

On the board there are three real numbers $(a,b,c)$. During a $procedure$ the numbers are erased and in their place another three numbers a written, either $(c,b,a)$ or every time a nonzero real number $ d $ is chosen and the numbers $(a, 2ad+b, ad^2+bd+c)$ are written. 1) If we start with $(1,-2,-1)$ written on the board, can we have the numbers $(2,0,-1)$ on the board after a finite number of procedures? 2) If we start with $(1,-2,-1)$ written on the board, can we have the numbers $(2,-1,-1)$ on the board after a finite number of procedures?

Cyclic quadrilateral $ABCD$ is inscribed in circle $k$ with center $O$. The angle bisector of $ABD$ intersects $AD$ and $k$ in $K,M$ respectively, and the angle bisector of $CBD$ intersects $CD$ and $k$ in $L,N$ respectively. If $KL\parallel MN$ prove that the circumcircle of triangle $MON$ bisects segment $BD$.