Let $a \in \mathbb{R}.$ Show that for $n \geq 2$ every non-real root $z$ of polynomial $X^{n+1}-X^2+aX+1$ satisfies the condition $|z| > \frac{1}{\sqrt[n]{n}}.$

Let $S_n$ be the set of all permutations of $(1,2,\ldots,n)$. Then find : \[ \max_{\sigma \in S_n} \left(\sum_{i=1}^{n} \sigma(i)\sigma(i+1)\right) \] where $\sigma(n+1)=\sigma(1)$.

For each positive integer $x$, let $\varphi(x)$ be the number of integers $1 \leq k \leq x$ that do not have prime factors in common with $x$. Determine all positive integers $n$ such that there are distinct positive integers $a_1,a_2, \ldots, a_n$ so that the set: $$S = \{a_1, a_2, \ldots, a_n, \varphi(a_1), \varphi(a_2), \ldots, \varphi(a_n)\}$$ Have exactly $2n$ consecutive integers (in some order).

Points $M$ and $N$ are marked on the straight line containing the side $AC$ of triangle $ABC$ so that $MA = AB$ and $NC = CB$ (the order of the points on the line: $M, A, C, N$). Prove that the center of the circle inscribed in triangle $ABC$ lies on the common chord of the circles circumscribed around triangles $MCB$ and $NAB$ .

In the triangle $ABC$, the median $AM$ is extended to the intersection with the circumscribed circle at point $D$. It is known that $AB = 2AM$ and $AD = 4AM$. Find the angles of the triangle $ABC$. (Gryhoriy Filippovskyi)

Determine the locus of points, for whom the ratio of the distances to two given points has a constant value.

For what values of the velocity $c$ does the equation $u_t = u -u^2 + u_{xx}$ have a solution in the form of a traveling wave $u = \varphi(x-ct)$, $\varphi(-\infty) = 1$, $\varphi(\infty) = 0$, $0 \le u \le 1$?

In quadrilateral $ABCD$ with diagonals $\overline{AC}$ and $\overline{BD}$ intersecting at $O$, $\overline{BO}=4$, $\overline{AO}=8$, $\overline{OC}=3$, and $\overline{AB}=6$. The length of $\overline{AD}$ is: $\textbf{(A)}\ 9\qquad \textbf{(B)}\ 10\qquad \textbf{(C)}\ 6\sqrt{3}\qquad \textbf{(D)}\ 8\sqrt{2}\qquad \textbf{(E)}\ \sqrt{166}$

Let $P_0P_5Q_5Q_0$ be a rectangular chocolate bar, one half dark chocolate and one half white chocolate, as shown in the diagram below. We randomly select $4$ points on the segment $P_0P_5$, and immediately after selecting those points, we label those $4$ selected points $P_1, P_2, P_3, P_4$ from left to right. Similarly, we randomly select $4$ points on the segment $Q_0Q_5$, and immediately after selecting those points, we label those $4$ points $Q_1, Q_2, Q_3, Q_4$ from left to right. The segments $P_1Q_1, P_2Q_2, P_3Q_3, P_4Q_4$ divide the rectangular chocolate bar into $5$ smaller trapezoidal pieces of chocolate. The probability that exactly $3$ pieces of chocolate contain both dark and white chocolate can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$. [Diagram in the individuals file for this exam on the Chmmc website]

Show that in any set of eleven integers there are six whose sum is divisible by $6$.

Let $ P$ be a convex $ n$ polygon each of which sides and diagnoals is colored with one of $ n$ distinct colors. For which $ n$ does: there exists a coloring method such that for any three of $ n$ colors, we can always find one triangle whose vertices is of $ P$' and whose sides is colored by the three colors respectively.

Ralph went to the store and bought 12 pairs of socks for a total of \$24. Some of the socks he bought cost \$1 a pair, some of the socks he bought cost \$3 a pair, and some of the socks he bought cost \$4 a pair. If he bought at least one pair of each type, how many pairs of \$1 socks did Ralph buy? $ \textbf{(A) } 4 \qquad \textbf{(B) } 5 \qquad \textbf{(C) } 6 \qquad \textbf{(D) } 7 \qquad \textbf{(E) } 8 $

Prove that for all $ n\ge 2 $ natural numbers there exist $ a_n\in\mathbb{Q} $ such that $$ X^{2n}+a_nX^n+1\Huge\vdots X^2+\frac{1}{2}X+1, $$ and that there isn´t any $ a_n\in\mathbb{R}\setminus\mathbb{Q} $ with this property.

Find all positive values of $a$, for which there is a number $b$ such that the parabola $y=ax^2-b$ intersects the unit circle at four distinct points. Also prove that for every such a there exists $b$ such that the parabola $y=ax^2-b$ intersects the unit circle at four distinct points whose $x$-coordinates form an arithmetic progression.

What are the signs of $\triangle{H}$ and $\triangle{S}$ for a reaction that is spontaneous only at low temperatures? $ \textbf{(A)}\ \triangle{H} \text{ is positive}, \triangle{S} \text{ is positive} \qquad\textbf{(B)}\ \triangle{H}\text{ is positive}, \triangle{S} \text{ is negative} \qquad$ $\textbf{(C)}\ \triangle{H} \text{ is negative}, \triangle{S} \text{ is negative} \qquad\textbf{(D)}\ \triangle{H} \text{ is negative}, \triangle{S} \text{ is positive} \qquad $

For every $n\in\mathbb{N}$, define the [i]power sum[/i] of $n$ as follows. For every prime divisor $p$ of $n$, consider the largest positive integer $k$ for which $p^k\le n$, and sum up all the $p^k$'s. (For instance, the power sum of $100$ is $2^6+5^2=89$.) Prove that the [i]power sum[/i] of $n$ is larger than $n$ for infinitely many positive integers $n$.

Let $1=d_1<d_2<....<d_k=n$ be all different divisors of positive integer n written in ascending order. Determine all n such that: \[d_6^{2} +d_7^{2} - 1=n\]

Let $G$ be a simple $k$ edge-connected graph on $n$ vertices and let $u$ and $v$ be different vertices of $G$. Prove that there are $k$ edge-disjoint paths from $u$ to $v$ each having at most $\frac{20n}{k}$ edges.

Consider a $ n\times n $ square grid which is divided into $ n^2 $ unit squares(think of a chess-board). The set of all unit squares intersecting the main diagonal of the square or lying under it is called an $n$-staircase. Find the number of ways in which an $n$-stair case can be partitioned into several rectangles, with sides along the grid lines, having mutually distinct areas.

Prove that the sequence $(a_n)$, where $$a_n=\sum_{k=1}^n\left\{\frac{\left\lfloor2^{k-\frac12}\right\rfloor}2\right\}2^{1-k},$$converges, and determine its limit as $n\to\infty$.

Let $0\le a\le b\le c$ be real numbers. Prove that \[(a+3b)(b+4c)(c+2a)\ge 60abc \]

On the sides $ AB $ and $ AC $ of the triangle $ ABC $ consider the points $ D, $ respectively, $ E, $ such that $$ \overrightarrow{DA} +\overrightarrow{DB} +\overrightarrow{EA} +\overrightarrow{EC} =\overrightarrow{O} . $$ If $ T $ is the intersection of $ DC $ and $ BE, $ determine the real number $ \alpha $ so that: $$ \overrightarrow{TB} +\overrightarrow{TC} =\alpha\cdot\overrightarrow{TA} . $$

Let $\mathbb{Z}_{>0}$ denote the set of positive integers. Consider a function $f: \mathbb{Z}_{>0} \to \mathbb{Z}_{>0}$. For any $m, n \in \mathbb{Z}_{>0}$ we write $f^n(m) = \underbrace{f(f(\ldots f}_{n}(m)\ldots))$. Suppose that $f$ has the following two properties: (i) if $m, n \in \mathbb{Z}_{>0}$, then $\frac{f^n(m) - m}{n} \in \mathbb{Z}_{>0}$; (ii) The set $\mathbb{Z}_{>0} \setminus \{f(n) \mid n\in \mathbb{Z}_{>0}\}$ is finite. Prove that the sequence $f(1) - 1, f(2) - 2, f(3) - 3, \ldots$ is periodic. [i]Proposed by Ang Jie Jun, Singapore[/i]

Suppose that $ D,E,F$ are points on sides $ BC,CA,AB$ of a triangle $ ABC$ respectively such that $ BD\equal{}CE\equal{}AF$ and $ \angle BAD\equal{}\angle CBE\equal{}\angle ACF$.Prove that the triangle $ ABC$ is equilateral.

From a point in space, $n$ rays are issuing, whereas the angle among any two of these rays is at least $30^{\circ}$. Prove that $n < 59$.