Let $p> 5$ be a prime such that none of its digits is divisible by $3$ or $7$. Prove that the equation $x^4 + p = 3y^4$ does not have integer solutions.

A triangle is formed by joining three points whose coordinates are integers. If the $ x$-coordinate and the $ y$-coordinate each have a value of $ 1$, then the area of the triangle, in square units: $ \textbf{(A)}\ \text{must be an integer}\qquad \textbf{(B)}\ \text{may be irrational}\qquad \textbf{(C)}\ \text{must be irrational}\qquad \textbf{(D)}\ \text{must be rational}\qquad \\ \textbf{(E)}\ \text{will be an integer only if the triangle is equilateral.}$

In a triangle $ABC$ let $D$ be the intersection of the angle bisector of $\gamma$, angle at $C$, with the side $AB.$ And let $F$ be the area of the triangle $ABC.$ Prove the following inequality: \[2 \cdot \ F \cdot \left( \frac{1}{AD} -\frac{1}{BD} \right) \leq AB.\]

Let be given a real number $\alpha\ne0$. Show that there is a unique point $P$ in the coordinate plane, such that for every line through $P$ which intersects the parabola $y=\alpha x^2$ in two distinct points $A$ and $B$, segments $OA$ and $OB$ are perpendicular (where $O$ is the origin).

We call a positive integer [i]alternating[/i] if every two consecutive digits in its decimal representation are of different parity. Find all positive integers $n$ such that $n$ has a multiple which is alternating.

A point $ P$ is outside a circle and is $ 13$ inches from the center. A secant from $ P$ cuts the circle at $ Q$ and $ R$ so that the external segment of the secant $ PQ$ is $ 9$ inches and $ QR$ is $ 7$ inches. The radius of the circle is: $ \textbf{(A)}\ 3" \qquad \textbf{(B)}\ 4" \qquad \textbf{(C)}\ 5" \qquad \textbf{(D)}\ 6" \qquad \textbf{(E)}\ 7"$

A convex $N\text{-gon}$ is divided by diagonals into triangles so that no two diagonals intersect inside the polygon. The triangles are painted in black and white so that any two triangles are painted in black and white so that any two triangles with a common side are painted in different colors. For each $N$ find the maximal difference between the numbers of black and white triangles.

A right triangle and a circle are drawn such that the circle is tangent to the legs of the right triangle. The circle cuts the hypotenuse into three segments of lengths $1, 24,$ and $3,$ and the segment of length $24$ is a chord of the circle. Compute the area of the triangle.

Find all the pairs $(a,b)$ of integer numbers such that: $\triangleright$ $a-b-1|a^2+b^2$ $\triangleright$ $\frac{a^2+b^2}{2ab-1}=\frac{20}{19}$

A point $P$ lies inside a triangle $ABC$. The points $K$ and $L$ are the projections of $P$ onto $AB$ and $AC$, respectively. The point $M$ lies on the line $BC$ so that $KM = LM$, and the point $P'$ is symmetric to $P$ with respect to $M$. Prove that $\angle BAP = \angle P'AC$.

Let $ P\subseteq \mathbb{R}^m$ be a non-empty compact convex set and $ f: P\rightarrow \mathbb{R}_{ \plus{} }$ be a concave function. Prove, that for every $ \xi\in \mathbb{R}^m$ \[ \int_{P}\langle \xi,x \rangle f(x)dx\leq \left[\frac {m \plus{} 1}{m \plus{} 2}\sup_{x\in P}{\langle\xi,x\rangle} \plus{} \frac {1}{m \plus{} 2}\inf_{x\in P}{\langle\xi,x\rangle}\right] \cdot\int_{P}f(x)dx.\]

Diameter $AB$ of a circle has length a 2-digit integer (base ten). Reversing the digits gives the length of the perpendicular chord $CD$. The distance from their intersection point $H$ to the center $O$ is a positive rational number. Determine the length of $AB$.

A point $D$ is chosen in the interior of the side $BC$ of an acute triangle $ABC$, and another point $P$ in the interior of the segment $AD$, but not lying on the median through $C$. This median (through $C$) intersects the circumcircle of a triangle $CPD$ at $K(\ne C)$. Prove that the circumcircle of triangle $AKP$ always passes through a fixed point $M(\ne A)$ independent of the choices of the points $D$ and $P.$

For an $n \times n$ table filled with natural numbers, we say it is a [i]divisor table[/i] if: - the numbers in the $i$-th row are exactly all the divisors of some natural number $r_i$, - the numbers in the $j$-th column are exactly all the divisors of some natural number $c_j$, - $r_i \ne r_j$ for every $i \ne j$. A prime number $p$ is given. Determine the smallest natural number $n$, divisible by $p$, such that there exists an $n \times n$ divisor table, or prove that such $n$ does not exist. [i]Proposed by Pavle Martinović[/i]

The MIG is planning a lottery to give out prizes after the written tests, and the plan is very special. Contestants will be divided into prize groups in order to potentially receive a prize. However, based on the number of contestants, the ideal number of groups don't work. For example, when dividing into $4$ groups, there are $3$ left over. When dividing into $5$ groups, there's $2$ left over. When dividing into $6$ groups, theres $1$ left over. Finally, when dividing into $7$ groups, there are $2$ left over. With the knowledge that there are less than $300$ participants in the MIG, how many participants are there?

A marker is placed at the origin of an integer lattice. Calvin and Hobbes play the following game. Calvin starts the game and each of them takes turns alternatively. At each turn, one can choose two (not necessarily distinct) integers $a, b$, neither of which was chosen earlier by any player and move the marker by $a$ units in the horizontal direction and $b$ units in the vertical direction. Hobbes wins if the marker is back at the origin any time after the first turn. Prove or disprove that Calvin can prevent Hobbes from winning. Note: A move in the horizontal direction by a positive quantity will be towards the right, and by a negative quantity will be towards the left (and similar directions in the vertical case as well).

Let a positive integer number $n$ has $k$ different prime divisors. Prove that there exists a positive integer number $x \in \left(1, \frac{n}{k}+1 \right)$ such that $x^2-x$ divides by $n$.

One day, Rabbit was about to go for a meeting with Donkey, but Winnie the Pooh and Duck unexpectedly came to his home. Being well-bred, Rabbit offered the guests some refreshments. Pooh tied Duck’s mouth by a napkin and ate $10$ pots of honey and $22$ cups of condensed milk alone, whereby he needed two minutes for each pot of honey and $1$ minute for each cup of milk. Knowing that there was nothing sweet left in the house, Pooh released the Duck. Afflicted Rabbit observed that he wouldn’t have been late for the meeting with Donkey if Pooh had shared the refreshments with Duck. Knowing that Duck needs $5$ minutes for a pot of honey and $3$ minutes for a cup of milk, he computed the time the guests would have needed to devastate his supplies. What is that time?

Let $A$ and $B$ be points on circle $\Gamma$ such that $AB=\sqrt{10}.$ Point $C$ is outside $\Gamma$ such that $\triangle ABC$ is equilateral. Let $D$ be a point on $\Gamma$ and suppose the line through $C$ and $D$ intersects $AB$ and $\Gamma$ again at points $E$ and $F \neq D.$ It is given that points $C, D, E, F$ are collinear in that order and that $CD=DE=EF.$ What is the area of $\Gamma?$ [i]Proposed by Kyle Lee[/i]

Find all prime numbers $ p,q,r$, such that $ \frac{p}{q}\minus{}\frac{4}{r\plus{}1}\equal{}1$

If $x$ is a real number such that $(x - 3)(x - 1)(x + 1)(x + 3) + 16 = 116^2$, what is the largest possible value of $x$?

Solve the system $$ \left\{\begin{matrix} ax=b\\bx=a \end{matrix}\right. $$ independently of the fixed elements $ a,b $ of a group of odd order. [i]Marian Andronache[/i]

The graph $G$ with 2014 vertices doesn’t contain any 3-cliques. If the set of the degrees of the vertices of $G$ is $\{1,2,...,k\}$, find the greatest possible value of $k$.

Prove that every integer $ k$ greater than 1 has a multiple that is less than $ k^4$ and can be written in the decimal system with at most four different digits.

A [i]quadratic[/i] number is a real root of the equations $ax^2 + bx + c = 0$ where $|a|,|b|,|c|\in\{1,2,\ldots,10\}$. Find the smallest positive integer $n$ for which at least one of the intervals$$\left(n-\dfrac{1}{3}, n\right)\quad \text{and}\quad\left(n, n+\dfrac{1}{3}\right)$$does not contain any quadratic number.