Let $S \subset \mathbb{R}^d$ be a convex compact body with nonempty interior. Show that there is an $\alpha > 0$ such that if $S = \cap_{i \in I} H_i$, where $I$ is an index set and $(H_i)_{i \in I}$ are halfspaces, then for any $P \in \mathbb{R}^d$, there is an $i \in I$ for which $\mathrm{dist}(P, H_i) \ge \alpha \, \mathrm{dist}(P, S)$.

Let rational numbers $a, b, c$ satisfy the conditions $a + b + c = a^2 + b^2 + c^2 \in Z$. Prove that there exist two relative prime numbers $m, n$ such that $abc =\frac{m^2}{n^3}$ .

A table has $m$ rows and $n$ columns. The following permutations of its $mn$ elements are permitted: an arbitrary permutation leaving each element in the same row (a“horizontal move”) and an arbitrary permutation leaving each element in the same column (a “vertical move”). Find the number $k$ such that any permutation of $mn$ elements can be obtained by $k$ permitted moves but there exists a permutation that cannot be achieved in less than $k$ moves. (A Andjans, Riga0

It is known that for quadratic polynomials $P(x)=x^2+ax+b$ and $Q(x)=x^2+cx+d$ the equation $P(Q(x))=Q(P(x))$ does not have real roots. Prove that $b \neq d$.

Determine all pairs $ (n,p)$ of positive integers, where $ p$ is prime, such that $ 3^p\minus{}np\equal{}n\plus{}p$.

Four fair six-sided dice are rolled. What is the probability that at least three of the four dice show the same value? $ \textbf{(A) } \frac{1}{36} \qquad\textbf{(B) } \frac{7}{72} \qquad\textbf{(C) } \frac{1}{9} \qquad\textbf{(D) }\frac{5}{36}\qquad\textbf{(E) }\frac{1}{6} \qquad $

We are given $3n$ points $A_1,A_2, \ldots , A_{3n}$ in the plane, no three of them collinear. Prove that one can construct $n$ disjoint triangles with vertices at the points $A_i.$

Let $ABC$ be an acute triangle. Points $B, D, E$, and $C$ lie on a line in this order and satisfy $BD = DE = EC$. Let $M$ and $N$ be the midpoints of $AD$ and $AE$, respectively. Suppose triangle $ADE$ is acute, and let $H$ be its orthocentre. Points $P$ and $Q$ lie on lines $BM$ and $CN$, respectively, such that $D, H, M,$ and $P$ are concyclic and pairwise different, and $E, H, N,$ and $Q$ are concyclic and pairwise different. Prove that $P, Q, N,$ and $M$ are concyclic.

Let $ABC$ be an equilateral triangle and let $D$ be an inner point of the side $BC$. A circle is tangent to $BC$ at $D$ and intersects the sides $AB$ and $AC$ in the inner points $M, N$ and $P, Q$ respectively. Prove that $|BD| + |AM| + |AN| = |CD| + |AP| + |AQ|$.

A radio program has a quiz consisting of $3$ multiple-choice questions, each with $3$ choices. A contestant wins if he or she gets $2$ or more of the questions right. The contestant answers randomly to each question. What is the probability of winning? $\textbf{(A) } \frac{1}{27}\qquad \textbf{(B) } \frac{1}{9}\qquad \textbf{(C) } \frac{2}{9}\qquad \textbf{(D) } \frac{7}{27}\qquad \textbf{(E) } \frac{1}{2}$

Among all ellipses with center at the origin, exactly one such ellipse is tangent to the graph of the curve $x^3 - 6x^2y + 3xy^2 + y^3 + 9x^2 - 9xy + 9y^2 = 0$ at three distinct points. The area of this ellipse is $\frac{k\pi\sqrt{m}}{n}$, where $k,m,$ and $n$ are positive integers such that $\gcd(k,n)=1$ and $m$ is not divisible by the square of any prime. Compute $100k+10m+n$. [i]Proposed by Jaedon Whyte[/i]

Show that the product of every $k$ consecutive members of the Fibonacci sequence is divisible by $f_1f_2\ldots f_k$ (where $f_0=0$ and $f_1=1$).

Let $a,\ b$ and $c$ be positive real numbers such that $a b+b c+c a=1$. Show that $$ \frac{a^3}{a^2+3 b^2+3 a b+2 b c}+\frac{b^3}{b^2+3 c^2+3 b c+2 c a}+\frac{c^3}{c^2+3 a^2+3 c a+2 a b}>\frac{1}{6\left(a^2+b^2+c^2\right)^2} . $$

In $\Delta ABC$ with $\angle ACB=135^\circ$, are chosen points $M$ and $N$ on side $AB$, so that $\angle MCN=90^\circ$. Segments $MD$ and $NQ$ are angle bisectors of $\Delta AMC$ and $\Delta NBC$ respectively. Prove that the reflection of $C$ in line $PQ$ lies on the line $AB$.

Points $K, L$ on sides $AC, CB$ respectively of a triangle $ABC$ are the points of contact of the excircles with the corresponding sides . Prove that the straight line through the midpoints of $KL$ and $AB$ (a) divides the perimeter of triangle $ABC$ in half, (b) is parallel to the bisector of angle $ACB$. ( L Emelianov)

a)Let $A,B\in \mathcal{M}_3(\mathbb{R})$ such that $\text{rank}\ A>\text{rank}\ B$. Prove that $\text{rank}\ A^2\ge \text{rank}\ B^2$. b)Find the non-constant polynomials $f\in \mathbb{R}[X]$ such that $(\forall)A,B\in \mathcal{M}_4(\mathbb{R})$ with $\text{rank}\ A>\text{rank}\ B$, we have $\text{rank}\ f(A)>\text{rank}\ f(B)$.

Teams $\text{A}$ and $\text{B}$ are playing a series of games. If the odds for either to win any game are even and Team $\text{A}$ must win two or Team $\text{B}$ three games to win the series, then the odds favoring Team $\text{A}$ to win the series are $\textbf{(A) }11\text{ to }5\qquad\textbf{(B) }5\text{ to }2\qquad\textbf{(C) }8\text{ to }3\qquad\textbf{(D) }3\text{ to }2\qquad \textbf{(E) }13\text{ to }6$

A certain $4$ digit prime number has all prime digits. When any one of the digits is removed, the remaning three digits form a composite number in their initial order (i.e. if $1234$ were the answer, then $123$, $234$, $134$, and $124$ would have to be composite.) What is the largest possible value of this number?

Let $s(n)$ denote the sum of the digits (in base ten) of a positive integer $n.$ Compute the number of positive integers $n$ at most $10^4$ that satisfy $$s(11n)=2s(n).$$

Prove that for every natural number $n$ we have $$\sum_{s=n}^{2n} 2^{2n-s}{s \choose n}= 2^{2n}.$$

Given triangle $ABC$ the point $J$ is the centre of the excircle opposite the vertex $A.$ This excircle is tangent to the side $BC$ at $M$, and to the lines $AB$ and $AC$ at $K$ and $L$, respectively. The lines $LM$ and $BJ$ meet at $F$, and the lines $KM$ and $CJ$ meet at $G.$ Let $S$ be the point of intersection of the lines $AF$ and $BC$, and let $T$ be the point of intersection of the lines $AG$ and $BC.$ Prove that $M$ is the midpoint of $ST.$ (The [i]excircle[/i] of $ABC$ opposite the vertex $A$ is the circle that is tangent to the line segment $BC$, to the ray $AB$ beyond $B$, and to the ray $AC$ beyond $C$.) [i]Proposed by Evangelos Psychas, Greece[/i]

Rick has $7$ books on his shelf: three identical red books, two identical blue books, a yellow book, and a green book. Dave accidentally knocks over the shelf and has to put the books back on in the same order. He knows that none of the red books were next to each other and that the yellow book was one of the first four books on the shelf, counting from the left. If Dave puts back the books according to the rules, but otherwise randomly, what is the probability that he puts the books back correctly?

Let $n$ be a positive integer and let $T$ denote the collection of points $(x_1, x_2, \ldots, x_n) \in \mathbb R^n$ for which there exists a permutation $\sigma$ of $1, 2, \ldots , n$ such that $x_{\sigma(i)} - x_{\sigma(i+1) } \geq 1$ for each $i=1, 2, \ldots , n.$ Prove that there is a real number $d$ satisfying the following condition: For every $(a_1, a_2, \ldots, a_n) \in \mathbb R^n$ there exist points $(b_1, \ldots, b_n)$ and $(c_1,\ldots, c_n)$ in $T$ such that, for each $i = 1, . . . , n,$ \[a_i=\frac 12 (b_i+c_i) , \quad |a_i - b_i|  \leq d, \quad \text{and} \quad |a_i - c_i| \leq d.\]

Let $ABC$ be a triangle and $D$ be a point on segment $AC$. The circumscribed circle of the triangle $BDC$ cuts $AB$ again at $E$ and the circumference circle of the triangle $ABD$ cuts $BC$ again at $F$. Prove that $AE = CF$ if and only if $BD$ is the interior bisector of $\angle ABC$.

A simple pendulum experiment is constructed from a point mass $m$ attached to a pivot by a massless rod of length $L$ in a constant gravitational field. The rod is released from an angle $\theta_0 < \frac{\pi}{2}$ at rest and the period of motion is found to be $T_0$. Ignore air resistance and friction. The experiment is repeated with a new pendulum with a rod of length $4L$, using the same angle $\theta_0$, and the period of motion is found to be $T$. Which of the following statements is correct? $\textbf{(A) } T = 2T_0 \text{ regardless of the value of } \theta_0\\ \textbf{(B) } T > 2T_0 \text{ with } T \approx 2T_0 \text{ if } \theta_0 \ll 1\\ \textbf{(C) } T < 2T_0 \text{ with } T \approx 2T_0 \text{ if } \theta_0 \ll 1\\ \textbf{(D) } T < 2T_0 \text{ with some values of } \theta_0 \text{ and } T > 2T_0 \text{ for other values of } \theta_0\\ \textbf{(E) } T \text{ and } T_0 \text{ are not defined because the motion is not periodic unless } \theta_0 \ll 1$