Point $A$ is located in this circle of radius $1$. An arbitrary chord is drawn through it, and then a circle of radius $2$ is drawn through the ends of this chord. Prove that all such circles touch some fixed circle, not depending from the initial choice of the chord.

Let $\Gamma$ be the circumcircle of acute triangle $ABC$. Points $D$ and $E$ are on segments $AB$ and $AC$ respectively such that $AD = AE$. The perpendicular bisectors of $BD$ and $CE$ intersect minor arcs $AB$ and $AC$ of $\Gamma$ at points $F$ and $G$ respectively. Prove that lines $DE$ and $FG$ are either parallel or they are the same line. [i]Proposed by Silouanos Brazitikos, Evangelos Psychas and Michael Sarantis, Greece[/i]

On the Cartesian plane the curve $(C)$ has equation $x^2=y^3$. A line $d$ varies on the plane such that $d$ always cut $(C)$ at three distinct points with $x$-coordinates $x_1,\, x_2,\, x_3$. a. Prove that the following quantity is a constant: $$\sqrt[3]{\frac{x_1x_2}{x_3^2}}+\sqrt[3]{\frac{x_2x_3}{x_1^2}}+\sqrt[3]{\frac{x_3x_1}{x_2^2}}.$$ b. Prove the following inequality: $$\sqrt[3]{\frac{x_1^2}{x_2x_3}}+\sqrt[3]{\frac{x_2^2}{x_3x_1}}+\sqrt[3]{\frac{x_3^2}{x_3x_1}}<-\frac{15}{4}.$$

$.4+.02+.006=$ $\text{(A)}\ .012 \qquad \text{(B)}\ .066 \qquad \text{(C)}\ .12 \qquad \text{(D)}\ .24 \qquad \text{(E)} .426$

Let $ABC$ be a triangle and let $P$ be a point in its interior. Suppose $ \angle B A P = 10 ^ { \circ } , \angle A B P = 20 ^ { \circ } , \angle P C A = 30 ^ { \circ } $ and $ \angle P A C = 40 ^ { \circ } $. Find $ \angle P B C $.

Let $f:] 0. \infty [ \to R$ be a strictly increasing function, such that $$f (x) f\left(f (x) +\frac{1}{x} \right)= 1.$$ Find $f (1)$.

Let $E$ be a family of subsets of $\{1,2,\ldots,n\}$ with the property that for each $A\subset \{1,2,\ldots,n\}$ there exist $B\in F$ such that $\frac{n-d}2\leq |A \bigtriangleup B| \leq \frac{n+d}2$. (where $A \bigtriangleup B = (A\setminus B) \cup (B\setminus A)$ is the symmetric difference). Denote by $f(n,d)$ the minimum cardinality of such a family. a) Prove that if $n$ is even then $f(n,0)\leq n$. b) Prove that if $n-d$ is even then $f(n,d)\leq \lceil \frac n{d+1}\rceil$. c) Prove that if $n$ is even then $f(n,0) = n$

In the scalene triangle $ABC$,$\angle ACB=60$ and $\Omega$ is its cirumcirle.On the bisectors of the angles $BAC$ and $CBA$ points $A^\prime$,$B^\prime$ are chosen respectively such that $AB^\prime \parallel BC$ and $BA^\prime \parallel AC$.$A^\prime B^\prime$ intersects with $\Omega$ at $D,E$.Prove that triangle $CDE$ is isosceles.(A. Kuznetsov)

Prove that for any polynomial $f(x)$ (with real coefficients) there exist polynomials $g(x)$ and $h(x)$ (with real coefficients) such that $f(x) = g(h(x)) - h(g(x))$.

[i]Each problem in this section will depend on the previous one! The values $A, B, C$, and $D$ refer to the answers to problems $1, 2, 3$, and $4$, respectively.[/i] [b]TR1.[/b] The number $2020$ has three different prime factors. What is their sum? [b]TR2.[/b] Let $A$ be the answer to the previous problem. Suppose$ ABC$ is a triangle with $AB = 81$, $BC = A$, and $\angle ABC = 90^o$. Let $D$ be the midpoint of $BC$. The perimeter of $\vartriangle CAD$ can be written as $x + y\sqrt{z}$, where $x, y$, and $z$ are positive integers and $z$ is not divisible by the square of any prime. What is $x + y$? [b]TR3.[/b] Let $B$ the answer to the previous problem. What is the unique real value of $k$ such that the parabola $y = Bx^2 + k$ and the line $y = kx + B$ are tangent? [b]TR4.[/b] Let $C$ be the answer to the previous problem. How many ordered triples of positive integers $(a, b, c)$ are there such that $gcd(a, b) = gcd(b, c) = 1$ and $abc = C$? [b]TR5.[/b] Let $D$ be the answer to the previous problem. Let $ABCD$ be a square with side length $D$ and circumcircle $\omega$. Denote points $C'$ and $D'$ as the reflections over line $AB$ of $C$ and $D$ respectively. Let $P$ and $Q$ be the points on $\omega$, with$ A$ and $P$ on opposite sides of line $BC$ and $B$ and $Q$ on opposite sides of line $AD$, such that lines $C'P$ and $D'Q$ are both tangent to $\omega$. If the lines $AP$ and $BQ$ intersect at $T$, what is the area of $\vartriangle CDT$? PS. You had better use hide for answers.

Given $n^2$ numbers $x_{i,j}$ ($i,j=1,2,...,n$) satisfying the system of $n^3$ equations $$x_{i,j}+x_{j,k}+x_{k,i}=0 \,\,\, (i,j,k = 1,...,n)$$Prove that there exist such numbers $a_1,a_2,...,a_n$, that $x_{i,j}=a_i-a_j$ for all $i,j=1,...n$.

Point $A$ and segment $BC$ are given. Determine the locus of points in space which are vertices of right angles with one side passing through $A$, and the other side intersecting segment $BC$.

Let $a_1,b_1,c_1$ be natural numbers. We define \[a_2=\gcd(b_1,c_1),\,\,\,\,\,\,\,\,b_2=\gcd(c_1,a_1),\,\,\,\,\,\,\,\,c_2=\gcd(a_1,b_1),\] and \[a_3=\operatorname{lcm}(b_2,c_2),\,\,\,\,\,\,\,\,b_3=\operatorname{lcm}(c_2,a_2),\,\,\,\,\,\,\,\,c_3=\operatorname{lcm}(a_2,b_2).\] Show that $\gcd(b_3,c_3)=a_2$.

On the sides of a convex, non-regular $m$-gon are built externally regular heptagons. It is known that their centers are vertices of a regular $m$-gon. What’s the least possible value of $m$?

In how many ways can Eve fill each of the six squares of a $2 \times 3$ grid with either a $0$ or a $1$, such that Anne can then divide the grid into three congruent rectangles: one containing two $0$s, one containing two $1$s, and one containing a $0$ and a $1$? [i]Proposed by Michael Tang[/i]

Each of the $100$ students in a certain summer camp can either sing, dance, or act. Some students have more than one talent, but no student has all three talents. There are $42$ students who cannot sing, $65$ students who cannot dance, and $29$ students who cannot act. How many students have two of these talents? $\textbf{(A)}\ 16\qquad\textbf{(B)}\ 25\qquad\textbf{(C)}\ 36\qquad\textbf{(D)}\ 49\qquad\textbf{(E)}\ 64$