Let $s_1,s_2,s_3,s_4,...$ be a sequence (infinite list) of $1$s and $0$s. For example $1,0,1,0,1,0,...$, that is, $s_n=1$ if $n$ is odd and $s_n=0$ if $n$ is even, is such a sequence. Prove that it is possible to delete infinitely many terms in $s_1,s_2,s_3,s_4,...$ so that the resulting sequence is the original sequence. For the given example, one can delete $s_3,s_4,s_7,s_8,s_{11},s_{12},...$

There are 2000 cities in the country. Every city is connected by non-stop two-way airlines with some other cities, and for each city, the number of airlines originating from it is a factor of two. (i.e. $1$, $2$, $4$, $8$, $...$). For each city $A$, the statistician calculated the number routes with no more than one transfer connecting $A$ with other cities, and then summed up the results for all $2000$ cities. He got $100,000$. Prove that the statistician was wrong.

Prove that the sum of the plane angles at each of the vertices of a given tetrahedron is $ 180^{\circ} $ if and only if all its faces are congruent.

Let be two $ 2\times 2 $ real matrices $ A,B, $ such that $ AB=\begin{pmatrix} 1&1\\1&2 \end{pmatrix} . $ Calculate $ \left((BA)^{-1} +BA\right)^{2019 } . $ [i]Dan Nedeianu[/i]

$a, b$ are two fixed lines through $O$. Variable lines $x, y$ are parallel. $x$ intersects a at $A$ and $b$ at $C$, $y$ intersects $a$ at $B$ and $b$at $D$. The lines $AD$ and $BC$ meet at $M$. The line through $M$ parallel to $x$ meets $a$ at $L$ and $b$ at $N$. What can you say about $L, M, N$? Find the locus $M$.

In triangle $ABC$ there are points $D\in(AC)$ and $F\in(AB)$ such that $AD=AB$ and line $BC$ splits the segment $[CF]$ in half. Prove that $BF=CD$.

Let $f: \ [0,\ 1] \rightarrow \mathbb{R}$ be an increasing function satisfying the following conditions: a) $f(0)=0$; b) $f\left(\frac{x}{3}\right)=\frac{f(x)}{2}$; c) $f(1-x)=1-f(x)$. Determine $f\left(\frac{18}{1991}\right)$.

Let $n \in \mathbb{N}^+,$ $x_1,x_2,...,x_{n+1},p,q\in \mathbb{R}^+ $ , $p<q$ and $x^p_{n+1}>\sum_{i=1}^{n}x^p_{i}.$ Prove that $(1)x^q_{n+1}>\sum_{i=1}^{n}x^q_{i};$ $(2)\left(x^p_{n+1}-\sum_{i=1}^{n}x^p_{i}\right)^{\frac{1}{p}}<\left(x^q_{n+1}-\sum_{i=1}^{n}x^q_{i}\right)^{\frac{1}{q}}.$

For any positive integer $x$, we set $$ g(x) = \text{ largest odd divisor of } x, $$ $$ f(x) = \begin{cases} \frac{x}{2} + \frac{x}{g(x)} & \text{ if } x \text{ is even;} \\ 2^{\frac{x+1}{2}} & \text{ if } x \text{ is odd.} \end{cases} $$ Consider the sequence $(x_n)_{n \in \mathbb{N}}$ defined by $x_1 = 1$, $x_{n + 1} = f(x_n)$. Show that the integer $2018$ appears in this sequence, determine the least integer $n$ such that $x_n = 2018$, and determine whether $n$ is unique or not.

Find all integers $a$ and $b$ satisfying the equality $3^a - 5^b = 2$. (I. Gorodnin)

There is an arithmetic progression of $7$ terms in which all the terms are different prime numbers. Determine the smallest possible value of the last term of such a progression. Clarification: In an arithmetic progression of difference $d$ each term is equal to the previous one plus $d$.

Say a positive integer $n$ is [i]radioactive[/i] if one of its prime factors is strictly greater than $\sqrt{n}$. For example, $2012 = 2^2 \cdot 503$, $2013 = 3 \cdot 11 \cdot 61$ and $2014 = 2 \cdot 19 \cdot 53$ are all radioactive, but $2015 = 5 \cdot 13 \cdot 31$ is not. How many radioactive numbers have all prime factors less than $30$? [i]Proposed by Evan Chen[/i]

[u]Bouncy Balls[/u] In the following problems, you will consider the trajectories of balls moving and bouncing off of the boundaries of various containers. The balls are small enough that you can treat them as points. Let us suppose that a ball starts at a point $X$, strikes a boundary (indicated by the line segment $AB$) at $Y$ , and then continues, moving along the ray $Y Z$. Balls always bounce in such a way that $\angle XY A = \angle BY Z$. This is indicated in the above diagram. [img]https://cdn.artofproblemsolving.com/attachments/4/6/42ad28823d839f804d618a1331db43a9ebdca1.png[/img] Balls bounce off of boundaries in the same way light reflects off of mirrors - if the ball hits the boundary at point P, the trajectory after $P$ is the reflection of the trajectory before $P$ through the perpendicular to the boundary at P. A ball inside a rectangular container of width $7$ and height $12$ is launched from the lower-left vertex of the container. It first strikes the right side of the container after traveling a distance of $\sqrt{53}$ (and strikes no other sides between its launch and its impact with the right side). [b]p4.[/b] Find the height at which the ball first contacts the right side. [b]p5.[/b] How many times does the ball bounce before it returns to a vertex? (The final contact with a vertex does not count as a bounce.) Now a ball is launched from a vertex of an equilateral triangle with side length $5$. It strikes the opposite side after traveling a distance of $\sqrt{19}$. [b]p6.[/b] Find the distance from the ball's point of rst contact with a wall to the nearest vertex. [b]p7.[/b] How many times does the ball bounce before it returns to a vertex? (The final contact with a vertex does not count as a bounce.) In this final problem, a ball is again launched from the vertex of an equilateral triangle with side length $5$. [b]p8.[/b] In how many ways can the ball be launched so that it will return again to a vertex for the first time after $2009$ bounces?

In $ \triangle ABC$ with right angle at $ C$, altitude $ CH$ and median $ CM$ trisect the right angle. If the area of $ \triangle CHM$ is $ K$, then the area of $ \triangle ABC$ is $ \textbf{(A)}\ 6K \qquad \textbf{(B)}\ 4\sqrt3\ K \qquad \textbf{(C)}\ 3\sqrt3\ K \qquad \textbf{(D)}\ 3K \qquad \textbf{(E)}\ 4K$

In a quadrilateral $ABCD$, we have $\angle ACB = \angle ADB = 90$ and $CD < BC$. Denote $E$ as the intersection of $AC$ and $BD$, and let the perpendicular bisector of $BD$ hit $BC$ at $F$. The circle with center $F$ which passes through $B$ hits $AB$ at $P (\neq B)$ and $AC$ at $Q$. Let $M$ be the midpoint of $EP$. Prove that the circumcircle of $EPQ$ is tangent to $AB$ if and only if $B, M, Q$ are colinear.

Let $O$ be the circumcircle of acute triangle $ABC$, $AD$-altitude of $ABC$ ($ D \in BC$), $ AD \cap CO =E$, $M$-midpoint of $AE$, $F$-feet of perpendicular from $C$ to $AO$. Proved that point of intersection $OM$ and $BC$ lies on circumcircle of triangle $BOF$

I've come across a challenging graph theory problem. Roughly translated, it goes something like this: There are n lines drawn on a plane; no two lines are parallel to each other, and no three lines meet at a single point. Those lines would partition the plane down into many 'area's. Suppose we select one point from each area. Also, should two areas share a common side, we connect the two points belonging to the respective areas with a line. A graph consisted of points and lines will have been made. Find all possible 'n' that will make a hamiltonian circuit exist for the given graph

Prove that for any natural number $k{}$ there is a natural number $n{}$ such that $\mathrm{lcm}(1,2,\ldots,n)=\mathrm{lcm}(1,2,\ldots,n+k).$ [i]From the folklore[/i]

Can an arbitrary polygon be cut into isosceles trapezoids?

The three digit number $n=CCA$ (in base $10$), where $C\neq A$, is divisible by $14$. How many possible values for $n$ are there? [i]2016 CCA Math Bonanza Individual #4[/i]

Given a right triangle $ABC$, the point $M$ is the midpoint of the hypotenuse $AB$. A circle is circumscribed around the triangle $BCM$, which intersects the segment $AC$ at a point $Q$ other than $C$. It turned out that the segment $QA$ is twice as large as the side $BC$. Find the acute angles of triangle $ABC$. (Mykola Moroz)

The points $A, B, C, D$ lie in this order on a circle $o$. The point $S$ lies inside $o$ and has properties $\angle SAD=\angle SCB$ and $\angle SDA= \angle SBC$. Line which in which angle bisector of $\angle ASB$ in included cut the circle in points $P$ and $Q$. Prove $PS =QS$.

Let S be a sphere cirucmscribed on a regular tetrahedron with an edge length greater than 1. The sphere $ S $ is represented as the sum of four sets. Prove that one of these sets includes points $ P $, $ Q $ such that the length of the segment $ PQ $ exceeds 1.

The football tournament, held in one round, involved $16$ teams, each two of which scored a different number of points. ($3$ points were given for a victory, $1$ point for a draw, $0$ points for a defeat.) It turned out that the Chisel team lost to all the teams that ultimately scored fewer points. What is the best result that the Chisel team could achieve (insert location)?

It is known that from segments of lengths $a$, $b$ and $c$, a triangle can be formed. Could it happen that from segments of lengths $$\sqrt{a^2 + \frac{2}{3} bc},\quad \sqrt{b^2 + \frac{2}{3} ca}\quad \text{and} \quad \sqrt{c^2 + \frac{2}{3} ab},$$ a right-angled triangle can be formed?