Let $a,b,c,d$ be positive reals such taht $a+b+c+d=1$. Prove that: \[6(a^{3}+b^{3}+c^{3}+d^{3})\geq a^{2}+b^{2}+c^{2}+d^{2}+\frac{1}{8}.\]

Georg has a $4$-sided die with the numbers $2, 3, 4$ and $5$. He rolls the die $17$ times and records the result of each roll on a board, so that eventually $17$ numbers are written on it. Georg notices that the average of the $17$ numbers is an integer. Is it possible that each of the numbers $2, 3, 4$ and $5$ appears at least three times on Georg’s board?

Which convex domains (figures) on a plane can contain an entire straight line? It is assumed that the figure is flat and does not degenerate into a straight line and is closed, that is, it contains all its boundary points.

For any polynomial $P(x)=a_0+a_1x+\ldots+a_kx^k$ with integer coefficients, the number of odd coefficients is denoted by $o(P)$. For $i-0,1,2,\ldots$ let $Q_i(x)=(1+x)^i$. Prove that if $i_1,i_2,\ldots,i_n$ are integers satisfying $0\le i_1<i_2<\ldots<i_n$, then: \[ o(Q_{i_{1}}+Q_{i_{2}}+\ldots+Q_{i_{n}})\ge o(Q_{i_{1}}). \]

Ten points are marked in the plane so that no three of them lie on the same straight line. All points are connected with segments.Each of these segments is painted one of the $k$ colors. For what positive integer $k$ ($1 \le k \le 5$) is it possible to paint the segments so that for any $k$ of the given $10$ points there are $k$ segments with the ends at these $k$ points, all of these segments being painted $k$ different colors ? (E. Barabanov)

The deposition of $1.0$ g of which element from its molten chloride requires the shortest time at a currently of 1 A? ${ \textbf{(A)}\ \text{Na} \qquad\textbf{(B)}\ \text{Mg} \qquad\textbf{(C)}\ \text{Al} \qquad\textbf{(D}}\ \text{Ba} \qquad$

In triangle $ABC$, $D$ is the midpoint of $AB$; $E$ is the midpoint of $DB$; and $F$ is the midpoint of $BC$. If the area of $\triangle ABC$ is $96$, then the area of $\triangle AEF$ is $\textbf{(A) }16\qquad\textbf{(B) }24\qquad\textbf{(C) }32\qquad\textbf{(D) }36\qquad \textbf{(E) }48$

Consider a polynomial with coefficients of real numbers $ \phi(x)\equal{}ax^3\plus{}bx^2\plus{}cx\plus{}d$ with three positive real roots. Assume that $ \phi(0)<0$, prove that \[ 2b^3\plus{}9a^2d\minus{}7abc \le 0.\] [i]Hery Susanto, Malang[/i]

A certain football quarterback can throw a football a maximum range of $80$ meters on level ground. What is the highest point reached by the football if thrown this maximum range? Ignore air friction. (A) $\text{10 m}$ (B) $\text{20 m}$ (C) $\text{30 m}$ (D) $\text{40 m}$ (E) $\text{50 m}$

Find all the positive integers $N$ with an even number of digits with the property that if we multiply the two numbers formed by cutting the number in the middle we get a number that is a divisor of $N$ ( for example $12$ works because $1 \cdot 2$ divides $12$)

Prove that the number $\overline{40...0}9$ (with at least one zero) is not a perfect square. (VA Senderov)

Let $p$ be a prime divisor of $x^3 + x^2 - 4x + 1$. Prove that $p$ is a cubic residue modulo $13$.

Let it \(k\) be a fixed positive integer. Alberto and Beralto play the following game: Given an initial number \(N_0\) and starting with Alberto, they alternately do the following operation: change the number \(n\) for a number \(m\) such that \(m < n\) and \(m\) and \(n\) differ, in its base-2 representation, in exactly \(l\) consecutive digits for some \(l\) such that \(1 \leq l \leq k\). If someone can't play, he loses. We say a non-negative integer \(t\) is a [i]winner[/i] number when the gamer who receives the number \(t\) has a winning strategy, that is, he can choose the next numbers in order to guarrantee his own victory, regardless the options of the other player. Else, we call it [i]loser[/i]. Prove that, for every positive integer \(N\), the total of non-negative loser integers lesser than \(2^N\) is \(2^{N-\lfloor \frac{log(min\{N,k\})}{log 2} \rfloor}\)

Circles $\Omega$ and $\Gamma$ meet at points $A$ and $B$. The line containing their centres intersects $\Omega$ and $\Gamma$ at point $P$ and $Q$, respectively, such that these points lie on same side of the line $AB$ and point $Q$ is closer to $AB$ than point $P$. The circle $\delta$ lies on the same side of the line $AB$ as $P$ and $Q$, touches the segment $AB$ at point $D$ and touches $\Gamma$ at point $T$. The line $PD$ meets $\delta$ and $\Omega$ again at points $K$ and $L$, respectively. Prove that $\angle QTK=\angle DTL$

Let $O$ be the circumcenter of triangle $ABC$: Points $D$ and $E$ are chosen at sides $AB$ and $AC$ respectively such that $\angle ADO = \angle AEO = 60^o$ and $BDEC$ is inscribed quadrangle. Prove or disprove that $ABC$ is isosceles triangle.

2. Let [i]ABCD [/i]be a square and let $\Gamma$ denote the circle with diameter [i]CD[/i]. A tangent line is drawn to the circle $\Gamma$ from [b][i]B[/i][/b], meeting the circle $\Gamma$ at [i]E[/i] and intersecting the segment [i]AD[/i] at [i]K[/i]. Prove that |[i]AD[/i]| = 4 |[i]KD[/i]|.

In triangle $ABC$, point $D$ is on side $BC$ such that $AD$ is the angle bisector of $\angle BAC$. If $AB=12$, $AD=9$, and $AC=15$, compute $\cos\tfrac{\angle BAC}{2}$.

Rachel has the number $1000$ in her hands. When she puts the number $x$ in her left pocket, the number changes to $x+1.$ When she puts the number $x$ in her right pocket, the number changes to $x^{-1}.$ Each minute, she flips a fair coin. If it lands heads, she puts the number into her left pocket, and if it lands tails, she puts it into her right pocket. She then takes the new number out of her pocket. If the expected value of the number in Rachel's hands after eight minutes is $E,$ compute $\left\lfloor\frac{E}{10}\right\rfloor.$

There are $2012$ distinct points in the plane, each of which is to be coloured using one of $n$ colours, so that the numbers of points of each colour are distinct. A set of $n$ points is said to be [i]multi-coloured [/i]if their colours are distinct. Determine $n$ that maximizes the number of multi-coloured sets.

Let $ f(x) \equal{} 1 \plus{} x \plus{} x^2 \plus{} \cdots \plus{} x^{100}$. Find $ f'(1)$.

Let $M$ be a finite sum of numbers, such that among any three of its elements there are two whose sum belongs to $M$. Find the greatest possible number of elements of $M$.

If you were to randomly select an answer to this question, what is the probability it would be correct? $\textbf{(A) }0\%\qquad\textbf{(B) }20\%\qquad\textbf{(C) }40\%\qquad\textbf{(D) }80\%\qquad\textbf{(E) }100\%$

Let n be a nonzero natural number. We say about a function f ∶ R ⟶ R that is n-positive if, for any real numbers $x_1, x_2,...,x_n$ with the property that $x_1+x_2+...+x_n = 0$, the inequality $f(x_1)+f(x_2)+...+f(x_n)=>0$ is true a) Is it true that any 2020-positive function is also 1010-positive? b) Is it true that any 1010-positive function is 2020-positive?

The dimensions of a rectangle $R$ are $a$ and $b$, $a < b$. It is required to obtain a rectangle with dimensions $x$ and $y$, $x < a$, $y < a$, so that its perimeter is one-third that of $R$, and its area is one-third that of $R$. The number of such (different) rectangles is: $\textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1\qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ \text{infinitely many}$

For what positive integers $n$ is it possible to tile an equilateral triangle of side $n$ with trapezoids each of which has sides $1, 1, 1, 2$? (NB Vassiliev)