a) The maximal height $|AH|$ of the acute-angled triangle $ABC$ equals the median $|BM|$. Prove that the angle $ABC$ isn't greater than $60$ degrees. b) The height $|AH|$ of the acute-angled triangle $ABC$ equals the median $|BM|$ and bisectrix $|CD|$. Prove that the angle $ABC$ is equilateral.

Determine the smallest natural number $n$ for which there exist distinct nonzero naturals $a, b, c$, such that $n=a+b+c$ and $(a + b)(b + c)(c + a)$ is a perfect cube.

Which of the following is equal to $ 1\plus{}\frac{1}{1\plus{}\frac{1}{1\plus{}1}}$? $ \textbf{(A)}\ \frac{5}{4} \qquad \textbf{(B)}\ \frac{3}{2} \qquad \textbf{(C)}\ \frac{5}{3} \qquad \textbf{(D)}\ 2 \qquad \textbf{(E)}\ 3$

The midpoints of all heights of a certain tetrahedron lie on its inscribed sphere. Is this tetrahedron necessarily regular then?

[hide=Note][i]The following problem is open in the sense that no solution is currently known. Progress on the problem may be awarded points. An example of progress on the problem is a non-trivial bound on the sequence defined below.[/i][/hide] For each integer $n\ge2$, consider a regular polygon with $2n$ sides, all of length $1$. Let $C(n)$ denote the number of ways to tile this polygon using quadrilaterals whose sides all have length $1$. Determine the limit inferior and the limit superior of the sequence defined by $$\frac1{n^2}\log_2C(n).$$

Let $n$ be a positive integer and let $a_1\le a_2 \le \dots \le a_n$ be real numbers such that \[a_1+2a_2+\dots+na_n=0.\] Prove that \[a_1[x]+a_2[2x]+\dots+a_n[nx] \ge 0\] for every real number $x$. (Here $[t]$ denotes the integer satisfying $[t] \le t<[t]+1$.)

What is the probability that an integer in the set $ \{1,2,3,\ldots,100\}$ is divisible by $ 2$ and not divisible by $ 3$? $ \textbf{(A)}\ \frac{1}{6} \qquad \textbf{(B)}\ \frac{33}{100} \qquad \textbf{(C)}\ \frac{17}{50} \qquad \textbf{(D)}\ \frac{1}{2} \qquad \textbf{(E)}\ \frac{18}{25}$

Prove that for any prime $p$ in the interval $\left]n, \frac{4n}{3}\right]$, $p$ divides \[\sum^{n}_{j=0}{{n}\choose{j}}^{4}.\]

A point $M$ inside triangle $ABC$ is such that $AM=AB/2$ and $CM=BC/2$. Points $C_0$ and $A_0$ lying on $AB$ and $CB$ respectively are such that $BC_0:AC_0 = BA_0:CA_0 = 3$. Prove that the distances from $M$ to $C_0$ and $A_0$ are equal.

Let us denote $\prod = \{(x, y) | y > 0\}$. We call a [i]semicircle[/i] in $\prod$ with center on the $x-\text{axis}$ a [i]semi-line[/i]. Two intersecting [i]semi-lines [/i]determine four [i]semi-angles[/i]. A bisector of a [i]semi-angle [/i]is a [i]semi-line [/i]that bisects the [i]semi-angle[/i]. Prove that in every [i]semi-triangle [/i](determined by three [i]semi-lines[/i]) the bisectors are concurrent.

Let $\varepsilon$ be positive constant and $u$ satisfies that \[ \begin{cases} (\partial_t-\varepsilon\partial_x^2-\partial_y^2)u=0, & (t,x,y) \in \mathbb{R}_+ \times \mathbb{R} \times \mathbb{R}_+,\\ \partial_y u\vert_{y=0}=\partial_x h, &\\u\vert_{t=0}=0. & \end{cases}\] Here $h(t,x)$ is a smooth Schwartz function. Define the operator $e^{a\langle D\rangle}$ \[\mathcal{F}_x(e^{a\langle D\rangle} f)(k)=e^{a\langle k\rangle} \mathcal{F}_x(f)(k), \quad \langle k\rangle=1+\vert k\vert,\] where $\mathcal{F}_x$ stands for the Fourier transform in $x$. Show that \[\int_0^T \|e^{(1-s)\langle D\rangle} u\|_{L_{x,y}^2}^2 ds \le C \int_0^T \|e^{(1-s)\langle D\rangle} h\|_{H_x^{\frac{1}{4}}}^2 ds\] with constant $C$ independent of $\varepsilon, T$ and $h$.

Let $m$ and $n$ be positive integers integers such that $2m + 1 < n$, and let $S$ be the set of the $2^n$ subsets of $\{1,2,\ldots,n\}$. Prove that we can place the elements of $S$ on a circle, so that for any two adjacent elements $A$ and $B$, the set $A \Delta B$ has exactly $2m + 1$ elements. [b]Note[/b]: $A \Delta B = (A \cup B) - (A \cap B)$ is the set of elements that are exclusively in $A$ or exclusively in $B$.

In the figure, the outer equilateral triangle has area $16$, the inner equilateral triangle has area $1$, and the three trapezoids are congruent. What is the area of one of the trapezoids? [asy] size((70)); draw((0,0)--(7.5,13)--(15,0)--(0,0)); draw((1.88,3.25)--(9.45,3.25)); draw((11.2,0)--(7.5,6.5)); draw((9.4,9.7)--(5.6,3.25)); [/asy] $\textbf{(A)}\ 3 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 6 \qquad \textbf{(E)}\ 7$

An isosceles $\triangle ABC$ has $\angle BAC =\angle ABC =72^{o}$. The angle bisector $AL$ meets the line through $C$ parallel to $AB$ at $D$. $a)$ Prove that the circumcenter of $\triangle ADC$ lies on $BD$. $b)$ Prove that $\frac {BE} {BL}$ is irrational.

Let $ABC$ be an acute triangle with circumcenter $O$. Let $A'$ be the center of the circle passing through $C$ and tangent to $AB$ at $A$, let $B'$ be the center of the circle passing through $A$ and tangent to $BC$ at $B$, let $C'$ be the center of the circle passing through $B$ and tangent to $CA$ at $C$. a) Prove that the area of triangle $A'B'C'$ is not less than the area of triangle $ABC$. b) Let $X, Y, Z$ be the projections of $O$ onto lines $A'B', B'C', C'A'$. Given that the circumcircle of triangle $XYZ$ intersects lines $A'B', B'C', C'A'$ again at $X', Y', Z'$ ($X' \neq X, Y' \neq Y, Z' \neq Z$), prove that lines $AX', BY', CZ'$ are concurrent.

In a rectangular coordinate system we call a horizontal line parallel to the $x$ -axis triangular if it intersects the curve with equation \[y = x^4 + px^3 + qx^2 + rx + s\] in the points $A,B,C$ and $D$ (from left to right) such that the segments $AB, AC$ and $AD$ are the sides of a triangle. Prove that the lines parallel to the $x$ - axis intersecting the curve in four distinct points are all triangular or none of them is triangular.