An isosceles trapezoid $ABCD$, inscribed in $\omega$, satisfies $AB=CD, AD<BC, AD<CD$. A circle with center $D$ and passing $A$ hits $BD, CD, \omega$ at $E, F, P(\not= A)$, respectively. Let $AP \cap EF = Q$, and $\omega$ meet $CQ$ and the circumcircle of $\triangle BEQ$ at $R(\not= C), S(\not= B)$, respectively. Prove that $\angle BER= \angle FSC$.

Joaquim, José and João participate of the worship of triangle $ABC$. It is well known that $ABC$ is a random triangle, nothing special. According to the dogmas of the worship, when they form a triangle which is similar to $ABC$, they will get immortal. Nevertheless, there is a condition: each person must represent a vertice of the triangle. In this case, Joaquim will represent vertice $A$, José vertice $B$ and João will represent vertice $C$. Thus, they must form a triangle which is similar to $ABC$, in this order. Suppose all three points are in the Euclidean Plane. Once they are very excited to become immortal, they act in the following way: in each instant $t$, Joaquim, for example, will move with constant velocity $v$ to the point in the same semi-plan determined by the line which connects the other two points, and which would create a triangle similar to $ABC$ in the desired order. The other participants act in the same way. If the velocity of all of them is same, and if they initially have a finite, but sufficiently large life, determine if they can get immortal. [i]Observation: Initially, Joaquim, José and João do not represent three collinear points in the plane[/i]

Prove that there are infinitely many integers $k\geqslant 2024$ for which there exists a set $\{a_1,\ldots,a_k\}$ with the following properties:[list] [*]$a_1{}$ is a positive integer and $a_{i+1}=a_i+1$ for all $1\leqslant i<k,$ and [*]$2(a_1\cdots a_{k-2}-1)^2$ is divisible by $2(a_1+\cdots+a_k)+a_1-a_1^2.$ [/list][i](Proposed by Prajit Adhikari, Nepal)[/i]

Inside a convex quadrilateral \( ABCD \), a point \( P \) is chosen such that \[ \angle PAD = \angle PAB = \angle PBC = \angle PCB = \angle PDA = 30^\circ. \] Prove that \( \angle CDP = 30^\circ \). [i]Proposed by Vadym Solomka[/i]

In the classroom of at least four students the following holds: no matter which four of them take seats around a round table, there is always someone who either knows both of his neighbours, or does not know either of his neighbours. Prove that it is possible to divide the students into two groups such that in one of them, all students know one another, and in the other, none of the students know each other. (Note: if student A knows student B, then student B knows student A as well.)

Andy the Ant lives on a coordinate plane and is currently at $(-20, 20)$ facing east (that is, in the positive $x$-direction). Andy moves $1$ unit and then turns $90^{\circ}$ degrees left. From there, Andy moves $2$ units (north) and then turns $90^{\circ}$ degrees left. He then moves $3$ units (west) and again turns $90^{\circ}$ degrees left. Andy continues his progress, increasing his distance each time by $1$ unit and always turning left. What is the location of the point at which Andy makes the $2020$th left turn? $\textbf{(A)}\ (-1030, -994)\qquad\textbf{(B)}\ (-1030, -990)\qquad\textbf{(C)}\ (-1026, -994)\qquad\textbf{(D)}\ (-1026, -990)\qquad\textbf{(E)}\ (-1022, -994)$

Let $ABCD$ be a quadrilateral inscribed in circle $\omega$ with $\overline{AC} \perp \overline{BD}$. Let $E$ and $F$ be the reflections of $D$ over lines $BA$ and $BC$, respectively, and let $P$ be the intersection of lines $BD$ and $EF$. Suppose that the circumcircle of $\triangle EPD$ meets $\omega$ at $D$ and $Q$, and the circumcircle of $\triangle FPD$ meets $\omega$ at $D$ and $R$. Show that $EQ = FR$.

Line segment $\overline{AB}$ is a diameter of a circle with $AB=24$. Point $C$, not equal to $A$ or $B$, lies on the circle. As point $C$ moves around the circle, the centroid (center of mass) of $\triangle{ABC}$ traces out a closed curve missing two points. To the nearest positive integer, what is the area of the region bounded by this curve? $\textbf{(A)} \text{ 25} \qquad \textbf{(B)} \text{ 38} \qquad \textbf{(C)} \text{ 50} \qquad \textbf{(D)} \text{ 63} \qquad \textbf{(E)} \text{ 75}$

Prove that there is a constant $C$ such that, if $p(x)$ is a polynomial of degree $1999$, then \[|p(0)|\leq C\int_{-1}^1|p(x)|\,dx.\]

Classify up to homeomorphism the topological spaces of the support of functions that are real quadratic polynoms of three variables and and irreducible over the set of real numbers.

There are $4$ pairs of men and women, and all $8$ people are arranged in a row so that in each pair the woman is somewhere to the left of the man. How many such arrangements are there?

For any positive integer $n$, let $f(n)$ be the number of possible choices of signs $+\ \text{or}\ - $ in the algebraic expression $\pm 1\pm 2\ldots \pm n$, such that the obtained sum is zero. Show that $f(n)$ satisfies the following conditions: a) $f(n)=0$ for $n=1\pmod{4}$ or $n=2\pmod{4}$. b) $2^{\frac{n}{2}-1}\le f(n)\le 2^n-2^{\lfloor\frac{n}{2}\rfloor+1}$, for $n=0\pmod{4}$ or $n=3\pmod{4}$. [i]Ioan Tomsecu[/i]

Given a triangle $ABC$ with symmedians $AD,BE,CF$ concurrent at [i]Lemoine[/i] point $L(D,E,F$ are on the sides $BC,CA,AB,$ respectively$).$ Prove that: $LA+LB+LC\ge 2(LD+LE+LF).$

In the isosceles triangle $ ABC$ the angle $ BAC$ is a right angle. Point $ D$ lies on the side $ BC$ and satisfies $ BD \equal{} 2 \cdot CD$. Point $ E$ is the foot of the perpendicular of the point $ B$ on the line $ AD$. Find the angle $ CED$.

In a triangle $ABC$ points $A_1,B_1,C_1$ are the midpoints of $BC,CA,AB$ respectively,and $A_2,B_2,C_2$ are the midpoints of the altitudes from $A,B,C$ respectively. Show that the lines $A_1A_2,B_1B_2,C_1,C_2$ are concurrent.

Assume that the set of all positive integers is decomposed into $r$ disjoint subsets $A_{1}, A_{2}, \cdots, A_{r}$ $A_{1} \cup A_{2} \cup \cdots \cup A_{r}= \mathbb{N}$. Prove that one of them, say $A_{i}$, has the following property: There exist a positive integer $m$ such that for any $k$ one can find numbers $a_{1}, \cdots, a_{k}$ in $A_{i}$ with $0 < a_{j+1}-a_{j} \le m \; (1\le j \le k-1)$.