Let a point $M$ not lying on coordinates axes be given. Points $Q$ and $P$ move along $Y$ - and $X$-axis respectively so that angle $P M Q$ is always right. Find the locus of points symmetric to $M$ wrt $P Q$.

Supoose $A$ is a set of integers which contains all integers that can be written as $2^a-2^b$, $a,b\in \mathbb{Z}_{\ge 1}$ and also has the property that $a+b\in A$ whenever $a,b\in A$. Prove that if $A$ contains at least an odd number, then $A=\mathbb{Z}$. [i] (Andrei Bâra)[/i]

Find the sum of the two smallest odd primes $p$ such that for some integers $a$ and $b$, $p$ does not divide $b$, $b$ is even, and $p^2=a^3+b^2$. [i]2021 CCA Math Bonanza Individual Round #13[/i]

In triangle $ABC$, $D$ is a point on $AB$ and $E$ is a point on $AC$ such that $BE$ and $CD$ are bisectors of $\angle B$ and $\angle C$ respectively. Let $Q,M$ and $N$ be the feet of perpendiculars from the midpoint $P$ of $DE$ onto $BC,AB$ and $AC$, respectively. Prove that $PQ=PM+PN$.

Consider two overlapping regular tetrahedrons of side length $2$ in space. They are centered at the same point, and the second one is oriented so that the lines from its center to its vertices are perpendicular to the faces of the first tetrahedron. What is the volume encompassed by the combined solid?

Circle $\omega_1$ with radius $6$ centered at point $A$ is internally tangent at point $B$ to circle $\omega_2$ with radius $15$. Points $C$ and $D$ lie on $\omega_2$ such that $\overline{BC}$ is a diameter of $\omega_2$ and $\overline{BC} \perp \overline{AD}$. The rectangle $EFGH$ is inscribed in $\omega_1$ such that $\overline{EF} \perp \overline{BC}$, $C$ is closer to $\overline{GH}$ than to $\overline{EF}$, and $D$ is closer to $\overline{FG}$ than to $\overline{EH}$, as shown. Triangles $\triangle DGF$ and $\triangle CHG$ have equal areas. The area of rectangle $EFGH$ is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. [asy] size(5cm); defaultpen(fontsize(10pt)); pair A = (9, 0), B = (15, 0), C = (-15, 0), D = (9, 12), E = (9+12/sqrt(5), -6/sqrt(5)), F = (9+12/sqrt(5), 6/sqrt(5)), G = (9-12/sqrt(5), 6/sqrt(5)), H = (9-12/sqrt(5), -6/sqrt(5)); filldraw(G--H--C--cycle, lightgray); filldraw(D--G--F--cycle, lightgray); draw(B--C); draw(A--D); draw(E--F--G--H--cycle); draw(circle(origin, 15)); draw(circle(A, 6)); dot(A); dot(B); dot(C); dot(D); dot(E); dot(F); dot(G); dot(H); label("$A$", A, (.8, -.8)); label("$B$", B, (.8, 0)); label("$C$", C, (-.8, 0)); label("$D$", D, (.4, .8)); label("$E$", E, (.8, -.8)); label("$F$", F, (.8, .8)); label("$G$", G, (-.8, .8)); label("$H$", H, (-.8, -.8)); label("$\omega_1$", (9, -5)); label("$\omega_2$", (-1, -13.5)); [/asy]

In a room, there are $n$ lights numbered with positive integers $1, 2, \ldots, n$. At the beginning of the game subsets $S_1, S_2,\ldots,S_k$ of $\{1,\ldots, n\}$ can be chosen. For every integer $1\le i\le k$, there is a button that turns on the lights corresponding to the elements of $S_i$ and also a button that turns off all the lights corresponding to the elements of $S_i$. For any positive integer $n$, determine the smallest $k$ for which it is possible to choose the sets $S_1, S_2, \ldots, S_n$ in such a way that allows any combination of the $n$ lights to be turned on, starting from the state where all the lights are off. [i]Proposed by Kristóf Zólomy, Budapest[/i]

Let $ABCDE$ be a convex pentagon with $CD= DE$ and $\angle EDC \ne 2 \cdot \angle ADB$. Suppose that a point $P$ is located in the interior of the pentagon such that $AP =AE$ and $BP= BC$. Prove that $P$ lies on the diagonal $CE$ if and only if area $(BCD)$ + area $(ADE)$ = area $(ABD)$ + area $(ABP)$. (Hungary)

Let be three positive real numbers $ x,y,z. $ Prove that there is a group of real numbers that contain the elements $ x+y/z $ and $ x+z/y $ and in which these two elements are inverses to each other. [i]D.M. Bătinețu[/i]

Let $f(x) = (x^4 + 2x^3 + 4x^2 + 2x + 1)^5$. Compute the prime $p$ satisfying $f(p) = 418{,}195{,}493$. [i]Proposed by Eugene Chen[/i]

Eight consecutive numbers are chosen from the Fibonacci sequence $1, 2, 3, 5, 8, 13, 21,...$. Prove that the sequence does not contain the sum of chosen numbers.

Petya drew a pentagon $ABCDE$ on the plane. After that, Vasya marked all the points $S$ in a given half-space relative to the plane of the pentagon so that in the pyramid $SABCD$ exactly two side faces are perpendicular to the plane of the base $ABCD$, and the height is $1$. How many points could have Vasya?

Let be three discs $ D_1,D_2,D_3. $ For each $ i,j\in\{1,2,3\} , $ denote $ a_{ij} $ as being the area of $ D_i\cap D_j. $ If $ x_1,x_2,x_3\in\mathbb{R} $ such that $ x_1x_2x_3\neq 0, $ then $$ a_{11} x_1^2+a_{22} x_2^2+a_{33} x_3^2+2a_{12} x_1x_2+2a_{23 }x_2x_3+2a_{31} x_3x_1>0. $$ [i]Vasile Pop[/i]

Let $AL$ be the bisector of triangle $ABC$. Circle $\omega_1$ is circumscribed around triangle $ABL$. Tangent to $\omega_1$ at point $B$ intersects the extension of $AL$ at point $K$. The circle $\omega_2$ circumscribed around the triangle $CKL$ intersects $\omega_1$ a second time at point $Q$, with $Q$ lying on the side $AC$. Find the value of the angle $ABC$. (Vladislav Radomsky)

Let $ABC$ be an equilateral triangle with side length $1$. This triangle is rotated by some angle about its center to form triangle $DEF.$ The intersection of $ABC$ and $DEF$ is an equilateral hexagon with an area that is $\frac{4} {5}$ the area of $ABC.$ The side length of this hexagon can be expressed in the form $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. What is $m+n$? [i]Author: Ray Li[/i]

For a given positive integer $ k$ denote the square of the sum of its digits by $ f_1(k)$ and let $ f_{n\plus{}1}(k) \equal{} f_1(f_n(k)).$ Determine the value of $ f_{1991}(2^{1990}).$

For the real number $a$ find the number of solutions $(x, y, z)$ of a system of the equations: $\left\{\begin{array}{lll} x+y^2+z^2=a \\ x^2+y+z^2=a \\ x^2+y^2+z=a\end{array}\right.$

How many three-digit perfect squares are there such that if each digit is increased by one, the resulting number is also a perfect square? (A): $1$, (B): $2$, (C): $4$, (D): $8$, (E) None of the above.

Let $S$ be the set $\{-1, 1\}^n$, that is, $n$-tuples such that each coordinate is either $-1$ or $1$. For \[s = (s_1, s_2, \ldots, s_n), t = (t_1, t_2, \ldots, t_n) \in \{-1, 1\}^n,\] define $s \odot t = (s_1t_1, s_2t_2, \ldots, s_nt_n)$. Let $c$ be a positive constant, let $f : S \to \{-1, 1\}$ be a function such that there are at least $(1-c) \cdot 2^{2n}$ pairs $(s, t)$ with $s, t \in S$ such that $f(s \odot t) = f(s)f(t)$. Show that there exists a function $f'$ such that $f'(s \odot t) = f'(s)f'(t)$ for all $s, t \in S$ and $f(s) = f'(s)$ for at least $(1-10c) \cdot 2^n$ values of $s \in S$.

A regular hexagon has positive integer side length. A laser is emitted from one of the hexagon’s corners, and is reflected off the edges of the hexagon until it hits another corner. Let $a$ be the distance that the laser travels. What is the smallest possible value of $a^2$ such that $a > 2019$? You need not simplify/compute exponents.

How many nonnegative integers with at most $40$ digits consisting of entirely zeroes and ones are divisible by $11$?

Triangle $ ABC$ has incentre $ I$ and the incircle touches $ BC, CA$ at $ D, E$ respectively. Let $ BI$ meet $ DE$ at $ G$. Show that $ AG$ is perpendicular to $ BG$.

Consider a board having 2 rows and $n$ columns. Thus there are $2n$ cells in the board. Each cell is to be filled in by $0$ or $1$ . [list=a] [*] In how many ways can this be done such that each row sum and each column sum is even? [*] In how many ways can this be done such that each row sum and each column sum is odd? [/list]

For what values of $a$ does the system of equations \[x^2 = y^2,(x-a)^2 +y^2 = 1\]have exactly 2 solutions?

The integer $202020$ is a multiple of $91$. For each positive integer $n$, show how $n$ additional $2$'s may be inserted into the digits of $202020$ such that the resulting $(n+6)$-digit number is also a multiple of $91$. For example, a possible way to do this when $n=5$ is [u]2[/u]2020[u]2[/u]20[u]222[/u] (the inserted $2$'s are underlined).