In an acute-angled triangle $ ABC $ consider $ A_1,B_1,C_1 $ to be the symmetric points of the orthocenter of $ ABC $ to the sides $ BC,AC,AB, $ respectively. Show that if the centroids of the triangles $ ABC,A_1B_1C_1 $ are the same, then $ ABC $ is equilateral. [i]Carmen Botea[/i]

In the night, stars in the sky are seen in different time intervals. Suppose for every $k$ stars ($k>1$), at least $2$ of them can be seen in one moment. Prove that we can photograph $k-1$ pictures from the sky such that each of the mentioned stars is seen in at least one of the pictures. (The number of stars is finite. Define the moments that the $n^{th}$ star is seen as $[a_n,b_n]$ that $a_n<b_n$.)

Let $ A_1,A_2,...,A_n$ be the vertices of a closed convex $ n$-gon $ K$ numbered consecutively. Show that at least $ n\minus{}3$ vertices $ A_i$ have the property that the reflection of $ A_i$ with respect to the midpoint of $ A_{i\minus{}1}A_{i\plus{}1}$ is contained in $ K$. (Indices are meant $ \textrm{mod} \;n\ .$)

In the convex pentagon $ ABCDE,$ the sides $ BC, CD, DE$ are equal. Moreover each diagonal of the pentagon is parallel to a side ($ AC$ is parallel to $ DE$, $ BD$ is parallel to $ AE$ etc.). Prove that $ ABCDE$ is a regular pentagon.

There are $N$ points in the plane such that the [b]total number[/b] of pairwise distances of these $N$ points is at most $n$. Prove that $N \le (n + 1)^2$.

a) Let $a$ and $b$ be two non-negative real numbers. Show that $a+b \ge \sqrt{\frac{a^2+b^2}{2}}+ \sqrt{ab}$ b) Let $a$ and $b$ be two real numbers in $[0, 3]$. Show that $\sqrt{\frac{a^2+b^2}{2}}+ \sqrt{ab} \ge \frac{(a+b)^2}{2}$

Two sequences $b_i$, $c_i$, $0 \le i \le 100$ contain positive integers, except $c_0=0$ and $b_{100}=0$. Some towns in Graphland are connected with roads, and each road connects exactly two towns and is precisely $1$ km long. Towns, which are connected by a road or a sequence of roads, are called [i]neighbours[/i]. The length of the shortest path between two towns $X$ and $Y$ is denoted as [i]distance[/i]. It is known that the greatest [i]distance[/i] between two towns in Graphland is $100$ km. Also the following property holds for every pair $X$ and $Y$ of towns (not necessarily distinct): if the [i]distance[/i] between $X$ and $Y$ is exactly $k$ km, then $Y$ has exactly $b_k$ [i]neighbours[/i] that are at the [i]distance[/i] $k+1$ from $X$, and exactly $c_k$ [i]neighbours[/i] that are at the [i]distance[/i] $k-1$ from $X$. Prove that $$\frac{b_0b_1 \cdot \cdot \cdot b_{99}}{c_1c_2 \cdot \cdot \cdot c_{100}}$$ is a positive integer.

A solid tetrahedron is sliced off a solid wooden unit cube by a plane passing through two nonadjacent vertices on one face and one vertex on the opposite face not adjacent to either of the first two vertices. The tetrahedron is discarded and the remaining portion of the cube is placed on a table with the cut surface face down. What is the height of this object? $ \textbf{(A)}\ \dfrac{\sqrt{3}}{3}\qquad\textbf{(B)}\ \dfrac{2\sqrt{2}}{3}\qquad\textbf{(C)}\ 1\qquad\textbf{(D)}\ \dfrac{2\sqrt{3}}{3}\qquad\textbf{(E)}\ \sqrt{2} $

Find the function $f(x)$ which satisfies the following integral equation. \[f(x)=\int_0^x t(\sin t-\cos t)dt+\int_0^{\frac{\pi}{2}} e^t f(t)dt\]

Parabolas $P_1, P_2$ share a focus at $(20,22)$ and their directrices are the $x$ and $y$ axes respectively. They intersect at two points $X,Y.$ Find $XY^2.$ [i]Proposed by Evan Chang[/i]

Let $\omega_1, \omega_2, \omega_3, \ldots, \omega_{2020!}$ be the distinct roots of $x^{2020!} - 1$. Suppose that $n$ is the largest integer such that $2^n$ divides the value $$\sum_{k=1}^{2020!} \frac{2^{2019!}-1}{\omega_{k}^{2020}+2}.$$ Then $n$ can be written as $a! + b$, where $a$ and $b$ are positive integers, and $a$ is as large as possible. Find the remainder when $a+b$ is divided by $1000$. [i]Proposed by vsamc[/i]

Some squares of the infinite sheet of cross-lined paper are red. Each $2\times 3$ rectangle (of $6$ squares) contains exactly two red squares. How many red squares can be in the $9\times 11$ rectangle of $99$ squares?

Let $T$ be the answer to question $16$. Compute the number of distinct real roots of the polynomial $x^4 + 6x^3 +\frac{T}{2}x^2 + 6x + 1$.

Each point in the plane is labeled with a real number. Show that there exist two distinct points $P$ and $Q$ whose labels differ by less than the distance from $P$ to $Q$. [i]Holden Mui[/i]

Let $(a_n)_{n\in\mathbb N}$ be an increasing sequence of positive integers. A term $a_k$ in the sequence is said to be good if it a sum of some other terms (not necessarily distinct). Prove that all terms of the sequence, apart from finitely many of them, are good.

Let $ a\in\mathbb{R}. $ Prove the following proposition: $$ \left( x,y\in\mathbb{R}\implies x^4+y^4+axy+2\ge 0 \right)\iff |a|\le 4. $$