If a certain number is doubled and the result is increased by $11$, the final number is $23$. What is the original number?

Let $q(n)$ denote the sum of the digits of a natural number $n$. Determine $q(q(q(2000^{2000})))$.

Find all integer solutions of the equation $(x^2 - y^2)^2 = 1 + 16y$. [i]M. Sonkin[/i]

All positive integers are coloured either red or green, such that the following conditions are satisfied: - There are equally many red as green integers. - The sum of three (not necessarily distinct) red integers is red. - The sum of three (not necessarily distinct) green integers is green. Find all colourings that satisfy these conditions.

Determine all integers $a$ such that $a^k + 1$ is divisible by $12321$ for some $k$

Let $n\ge2$ and let $p(x)=x^n+a_{n-1}x^{n-1} \cdots a_1x+a_0$ be a polynomial with real coefficients. Prove that if for some positive integer $k(<n)$ the polynomial $(x-1)^{k+1}$ divides $p(x)$ then $$\sum_{i=0}^{n-1}|a_i| \ge 1 +\frac{2k^2}{n}$$

Find all positive integers $m, n$ satisfying $n!+2^{n-1}=2^m$.

Alice and Bob play the following game. It begins with a set of $1000$ $1\times 2$ rectangles. A [i]move[/i] consists of choosing two rectangles (a rectangle may consist of one or several $1\times 2$ rectangles combined together) that share a common side length and combining those two rectangles into one rectangle along those sides sharing that common length. The first player who cannot make a move loses. Alice moves first. Describe a winning strategy for Bob.

Let a tetrahedron $ ABCD$ and $ A',B',C',D'$ be the circumcenters of triangles $ BCD,CDA,DAB,ABC$ respectively. Denote planes $ (P_A),(P_B),(P_C),(P_D)$ be the planes which pass through $ A,B,C,D$ and perpendicular to $ C'D',D'A',A'B',B'C'$ respectively. Prove that these planes have a common point called $ I.$ If $ P$ is the center of the circumsphere of the tetrahedron, must this tetrahedron be regular?

Let $OABCDEF$ be a hexagonal pyramid with base $ABCDEF$ circumscribed around a sphere $\omega$. The plane passing through the touching points of $\omega$ with faces $OFA$, $OAB$ and $ABCDEF$ meets $OA$ at point $A_1$, points $B_1$, $C_1$, $D_1$, $E_1$ and $F_1$ are defined similarly. Let $\ell$, $m$ and $n$ be the lines $A_1D_1$, $B_1E_1$ and $C_1F_1$ respectively. It is known that $\ell$ and $m$ are coplanar, also $m$ and $n$ are coplanar. Prove that $\ell$ and $n$ are coplanar.

For a finite set $C$ of integer numbers, we define $S(C)$ as the sum of the elements of $C$. Find two non-empty sets $A$ and $B$ whose intersection is empty, whose union is the set $\{1,2,\ldots, 2021\}$ and such that the product $S(A)S(B)$ is a perfect square.

Given a prime number $p$ congruent to $1$ modulo $5$ such that $2p+1$ is also prime, show that there exists a matrix of $0$s and $1$s containing exactly $4p$ (respectively, $4p+2$) $1$s no sub-matrix of which contains exactly $2p$ (respectively, $2p+1$) $1$s.

A library has one exit and one entrance and a blackboard at each. Only one person enters or leaves at a time. As he does so he records the number of people found/remaining in the library on the blackboard. Prove that at the end of the day exactly the same numbers will be found on the two blackboards (possibly in a different order).

Let $\Gamma$ denote the circumcircle and $O$ the circumcentre of the acute-angled triangle $ABC$, and let $M$ be the midpoint of the segment $BC$. Let $T$ be the second intersection point of $\Gamma$ and the line $AM$, and $D$ the second intersection point of $\Gamma$ and the altitude from $A$. Let further $X$ be the intersection point of the lines $DT$ and $BC$. Let $P$ be the circumcentre of the triangle $XDM$. Prove that the circumcircle of the triangle $OPD$ passes through the midpoint of $XD$.

Let $ABCD$ be a rectangle with sides $AB=a$ and $BC=b$. Let $O$ be the intersection point of it's diagonals. Extent side $BA$ towards $A$ at a segment $AE=AO$, and diagonal $DB$ towards $B$ at a segment $BZ=BO$. If the triangle $EZC$ is an equilateral, then prove that: i) $b=a\sqrt3$ ii) $AZ=EO$ iii) $EO \perp ZD$

Let $a$ and $b$ be 2019-digit numbers. Exactly 12 digits of $a$ are non-zero: the five leftmost and seven rightmost, and exactly 14 digits of $b$ are non-zero: the five leftmost and nine rightmost. Prove that the largest common divisor of $a$ and $b$ has no more than 14 digits. [i]Proposed by L. Samoilov[/i]

Let $\triangle ABC$ be a scalene triangle, $O$ be the midpoint of $BC$, and $M$ and $N$ be the intersections of the circle with diameter $BC$ and $AB$ and $BC$, respectively. The bisectors of $\angle BAC$ and $\angle MON$ intersect at $R$. Prove that the circumcircles of $\triangle BMR$ and $\triangle CNR$ intersect on $BC$.

Let $m$ and $n$ be positive integers. Prove that the number $2n-1$ is divisible by $(2^m-1)^2$ if and only if $n$ is divisible by $m(2^m-1)$.

Find the number of ways that $7$ different guests can be seated at a round table with exactly 10 seats, without removing any empty seats. Here two seatings are considered to be the same if they can be obtained from each other by a rotation.

What is the maximal number of crosses than can fit in a $10\times 11$ board without overlapping? Is this problem well-known? [asy] size(4.58cm); real labelscalefactor = 0.5; /* changes label-to-point distance */ pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */ pen dotstyle = black; /* point style */ real xmin = -3.18, xmax = 1.4, ymin = -0.22, ymax = 3.38; /* image dimensions */ /* draw figures */ draw((-3.,2.)--(1.,2.)); draw((-2.,3.)--(-2.,0.)); draw((-2.,0.)--(-1.,0.)); draw((-1.,0.)--(-1.,3.)); draw((-1.,3.)--(-2.,3.)); draw((-3.,1.)--(1.,1.)); draw((1.,1.)--(1.,2.)); draw((-3.,2.)--(-3.,1.)); draw((0.,2.)--(0.,1.)); draw((-1.,2.)--(-1.,1.)); draw((-2.,2.)--(-2.,1.)); /* dots and labels */ clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); /* end of picture */ [/asy]

For all $x,y,z>0$ satisfying $\frac{x}{yz}+\frac{y}{zx}+\frac{z}{xy}\le x+y+z$, prove that $$\frac{1}{x^2+y+z}+\frac{1}{y^2+z+x}+\frac{1}{z^2+x+y} \le 1$$

Triangle $ABC$ lies in a regular decagon as shown in the figure. What is the ratio of the area of the triangle to the area of the entire decagon? Write the answer as a fraction of integers. [img]https://1.bp.blogspot.com/-Ld_-4u-VQ5o/Xzb-KxPX0wI/AAAAAAAAMWg/-qPtaI_04CQ3vvVc1wDTj3SoonocpAzBQCLcBGAsYHQ/s0/2007%2BMohr%2Bp1.png[/img]

Find the sum of the $23$ smallest positive integers that are $4$ more than a multiple of $23$ and whose last two digits are $23.$

Find the maximum number $E$ such that the following holds: there is an edge-colored graph with 60 vertices and $E$ edges, with each edge colored either red or blue, such that in that coloring, there is no monochromatic cycles of length 3 and no monochromatic cycles of length 5.

Let C be the number of ways to arrange the letters of the word CATALYSIS, T be the number of ways to arrange the letters of the word TRANSPORT, S be the number of ways to arrange the letters of the word STRUCTURE, and M be the number of ways to arrange the letters of the word MOTION. What is $\frac{C - T + S}{M}$ ?