Let $ ABCD $ be a tetahedron and $ M,N $ the middlepoints of $ AB, $ respectively, $ CD. $ Show that any plane that contains $ M $ and $ N $ cuts the tetrahedron in two polihedra that have same volume.

One member of an infinite arithmetic sequence in the set of natural numbers is a perfect square. Show that there are infinitely many members of this sequence having this property.

a) Show that $\forall n \in \mathbb{N}_0, \exists A \in \mathbb{R}^{n\times n}: A^3=A+I$. b) Show that $\det(A)>0, \forall A$ fulfilling the above condition.

Let $ P$ be a plane. Prove that there exists no function $ f: P\rightarrow P$ such that for every convex quadrilateral $ ABCD$, the points $ f(A),f(B),f(C),f(D)$ are the vertices of a concave quadrilateral. [i]Dinu Şerbănescu[/i]

If $x$ and $y$ are real numbers, prove that $\frac{4x^2+1}{y^2+2}$ is not integer

$1989$ equal circles are arbitrarily placed on the table without overlap. What is the least number of colors are needed such that all the circles can be painted with any two tangential circles colored differently.

In a forest there are $5$ trees $A, B, C, D, E$ that are in that order on a straight line. At the midpoint of $AB$ there is a daisy, at the midpoint of $BC$ there is a rose bush, at the midpoint of $CD$ there is a jasmine, and at the midpoint of $DE$ there is a carnation. The distance between $A$ and $E$ is $28$ m; the distance between the daisy and the carnation is $20$ m. Calculate the distance between the rose bush and the jasmine.

Prove that there exist positive constants $ c$ and $ n_0$ with the following property. If $ A$ is a finite set of integers, $ |A| \equal{} n > n_0$, then \[ |A \minus{} A| \minus{} |A \plus{} A| \leq n^2 \minus{} c n^{8/5}.\]

Consider the function $f(x, y, z) = (x-y +z,y -z +x, z-x+y)$ and denote by $f^{(n)}(x, y,z)$ the function $f$ applied $n$ times to the tuple $(x,y,z)$. Let $r_1$, $r_2$, $r_3$ be the three roots of the equation $x^3- 4x^2 + 12 = 0$ and let $g(x) = x^3 + a_2x^2 + a_1x + a_0$ be the cubic polynomial with the tuple $f^{(3)}(r_1, r_2, r_3)$ as roots. Find the value of $a_1$.

A subalgebra $\mathfrak{h}$ of a Lie algebra $\mathfrak g$ is said to have the $\gamma$ property with respect to a scalar product ${\langle \cdot,\cdot \rangle}$ given on ${\mathfrak g}$ if ${X \in \mathfrak{h}}$ implies ${\langle [X,Y],X\rangle =0}$ for all ${Y \in \mathfrak g}$. Prove that the maximum dimension of ${\gamma}$-property subalgebras of a given ${2}$ step nilpotent Lie algebra with respect to a scalar product is independent of the selection of the scalar product. [i]Proposed by Péter Nagy Tibor[/i]

Points $X$ and $Y$ are on a parabola of the form $y=\frac{x^2}{a^2}$ and $A$ is the point $(x, y) = (0, a)$. Assume $XY$ passes through $A$ and hits the line $y=-a$ at a point $B$. Let $\omega$ be the circle passing through $(0, -a)$, $A$, and $B$. A point $P$ is chosen on $\omega$ such that $PA = 8$. Given that $X$ is between $A$ and $B$, $AX=2$, and $XB=10$, find $PX \cdot PY$. [i]Proposed by Kevin Zhao[/i]

Let $a$ and $b$ satisfy the conditions $\begin{cases} a^3 - 6a^2 + 15a = 9 \\ b^3 - 3b^2 + 6b = -1 \end{cases}$ . The value of $(a - b)^{2014}$ is: (A): $1$, (B): $2$, (C): $3$, (D): $4$, (E) None of the above.

Find all positive integers $p$ and $q$ such that all the roots of the polynomial $(x^2 - px+q)(x^2 -qx+ p)$ are positive integers.

Find all such functions $f:\mathbb{R} \to \mathbb{R}$ that for any real $x,y$ the following equation is true. $$f(f(x)+y)+1=f(x^2+y)+2f(x)+2y$$

Let $n$ be a positive integer and let $A$, $B$ be two complex nonsingular $n \times n$ matrices such that \[A^2B-2ABA+BA^2=0.\] Prove that the matrix $AB^{-1}A^{-1}B-I_n$ is nilpotent.

Find all pairs $(n, q)$ such that $n$ is a positive integer, $q$ is a not integer rational and $$n^q-q$$ is an integer.

Fix a circle $\Gamma$, a line $\ell$ to tangent $\Gamma$, and another circle $\Omega$ disjoint from $\ell$ such that $\Gamma$ and $\Omega$ lie on opposite sides of $\ell$. The tangents to $\Gamma$ from a variable point $X$ on $\Omega$ meet $\ell$ at $Y$ and $Z$. Prove that, as $X$ varies over $\Omega$, the circumcircle of $XYZ$ is tangent to two fixed circles.

For how many integers $n$ between $1$ and $50$, inclusive, is \[ \frac{(n^2-1)!}{(n!)^n} \]an integer? (Recall that $0! = 1$.) $\textbf{(A) } 31 \qquad \textbf{(B) } 32 \qquad \textbf{(C) } 33 \qquad \textbf{(D) } 34 \qquad \textbf{(E) } 35$

Find all the integers $x$ in $[20, 50]$ such that $6x+5\equiv 19 \mod 10$, that is, $10$ divides $(6x+15)+19$.

The following game is played with a group of $n$ people and $n+1$ hats are numbered from $1$ to $n+1.$ The people are blindfolded and each of them puts one of the $n+1$ hats on his head (the remaining hat is hidden). Now, a line is formed with the $n$ people, and their eyes are uncovered: each of them can see the numbers on the hats of the people standing in front of him. Now, starting from the last person (who can see all the other players) the players take turns to guess the number of the hat on their head, but no two players can guess the same number (each player hears all the guesses from the other players). What is the highest number of guaranteed correct guesses, if the $n$ people can discuss a common strategy? [i]Proposed by Viktor Kiss, Budapest[/i]

The circumferences $C_1$ and $C_2$ cut each other at different points $A$ and $K$. The common tangent to $C_1$ and $C_2$ nearer to $K$ touches $C_1$ at $B$ and $C_2$ at $C$. Let $P$ be the foot of the perpendicular from $B$ to $AC$, and let $Q$ be the foot of the perpendicular from $C$ to $AB$. If $E$ and $F$ are the symmetric points of $K$ with respect to the lines $PQ$ and $BC$, respectively, prove that $A, E$ and $F$ are collinear.

Consider the set $I=\{ 1,2, \cdots, 2020 \}$. Let $W= \{w(a,b)=(a+b)+ab | a,b \in I \} \cap I$, $Y=\{y(a,b)=(a+b) \cdot ab | a,b \in I \} \cap I$ be its two subsets. Further, let $X= W \cap Y$. [b](1)[/b] Find the sum of maximal and minimal elements in $X$. [b](2)[/b] An element $n=y(a,b) (a \le b)$ in $Y$ is called [i]excellent[/i], if its representation is not unique (for instance, $20=y(1,5)=y(2,3)$). Find the number of [i]excellent[/i] elements in $Y$. [hide=Note][b](2)[/b] is only for Grade 11.[/hide]

A sequence starts at some rational number $x_1>1$, and is subsequently defined using the recurrence relation \[x_{n+1}=\frac{x_n\cdot n}{\lfloor x_n\cdot n\rfloor }\] Show that $k>0$ exists with $x_k=1$.

Let $ ABC$ be an equilateral triangle. Let $ D,E, F,M,N,$ and $ P$ be the mid-points of $ BC, CA, AB, FD, FB,$ and $ DC$ respectively. [b](a)[/b] Show that the line segments $ AM,EN,$ and $ FP$ are concurrent. [b](b)[/b] Let $ O$ be the point of intersection of $ AM,EN,$ and $ FP.$ Find $ OM : OF : ON : OE : OP : OA.$

Prove that for all primes $p$ true is equality $$\sum_{k=1}^{p-1}\left\lfloor\frac{k^3}{p}\right\rfloor=\frac{(p-2)(p-1)(p+1)}{4}$$