The midpoint of triangle's side and the base of the altitude to this side are symmetric wrt the touching point of this side with the incircle. Prove that this side equals one third of triangle's perimeter. (A.Blinkov, Y.Blinkov)

Let $ABCD$ be a square of side length $5$. A circle passing through $A$ is tangent to segment $CD$ at $T$ and meets $AB$ and $AD$ again at $X\ne A$ and $Y\ne A$, respectively. Given that $XY = 6$, compute $AT$.

Construct a square from four points, one on each side.

A numerist has $n$ eurodollars and distributes them to two bank accounts $A, B$ in Germany and Switzerland, whereby the Eurodollars cannot be split into smaller monetary units due to the lack of a suitable name. In order to hide all money from the tax authorities if necessary, he would like to be able to move all of his money to account $B$. Due to the immense bureaucracy, money is only allowed to be moved between two accounts if the deposited amount in one account is double. Of course, he can carry out several such transfers in a row. Show that the number of ways to initially distribute the money appropriately is a power of two.

$ (a_n)_{n \equal{} 1}^\infty$ is defined on real numbers with $ a_n \not \equal{} 0$, $ a_na_{n \plus{} 3} \equal{} a_{n \plus{} 2}a_{n \plus{} 5}$ and $ a_1a_2 \plus{} a_3a_4 \plus{} a_5a_6 \equal{} 6$. So $ a_1a_2 \plus{} a_3a_4 \plus{} \cdots \plus{}a_{41}a_{42} \equal{} ?$ $\textbf{(A)}\ 21 \qquad\textbf{(B)}\ 42 \qquad\textbf{(C)}\ 63 \qquad\textbf{(D)}\ 882 \qquad\textbf{(E)}\ \text{None}$

Let $a$, $b$ and $c$ be positive real numbers such that $a+b+c=1$. Prove the inequality: $\frac{1}{\sqrt{(a+2b)(b+2a)}}+\frac{1}{\sqrt{(b+2c)(c+2b)}}+\frac{1}{\sqrt{(c+2a)(a+2c)}} \geq 3$

Let $n$ be a positive integer. Show that there exists an integer $m$ such that \[ 2018m^2+20182017m+2017 \] is divisible by $2^n$.

10.8 Suppose $S$ is a set of points in the plane, $|S|$ is even; no three points of $S$ are collinear. Prove that $S$ can be partitioned into two sets $S_1$ and $S_2$ so that their convex hulls have equal number of vertices.

Let $S$ be a set of size $n$, and $k$ be a positive integer. For each $1 \le i \le kn$, there is a subset $S_i \subset S$ such that $|S_i| = 2$. Furthermore, for each $e \in S$, there are exactly $2k$ values of $i$ such that $e \in S_i$. Show that it is possible to choose one element from $S_i$ for each $1 \le i \le kn$ such that every element of $S$ is chosen exactly $k$ times.

Let $ABC$ be an acute triagnle with its circumcircle $\omega$. Let point $D$ be the foot of triangle $ABC$ from point $A$. Let points $E,F$ be midpoints of sides $AB,AC$, respectively. Let points $P$ and $Q$ be the second intersections of of circle $\omega$ with circumcircle of triangles $BDE$ and $CDF$, respectively. Suppose that $A,P,B,Q$ and $C$ be on a circle in this order. Show that the lines $EF,BQ$ and $CP$ are concurrent.

Positive integers $a$, $b$, $c$ are given. It is known that $\frac{c}{b}=\frac{b}{a}$, and the number $b^2-a-c+1$ is a prime. Prove that $a$ and $c$ are double of a squares of positive integers.

Let $ABC$ be a triangle each of whose angles is greater than $30^{\circ}$. Suppose a circle centered with $P$ cuts segments $BC$ in $T,Q; CA$ in $K,L$ and $AB$ in $M,N$ such that they are on a circle in counterclockwise direction in that order.Suppose further $PQK,PLM,PNT$ are equilateral. Prove that: $a)$ The radius of the circle is $\frac{2abc}{a^2+b^2+c^2+4\sqrt{3}S}$ where $S$ is area. $b) a\cdot AP=b\cdot BP=c\cdot PC.$

Two triangles $ABC,ABD$ (with the common side $c=AB$) are given in space. Triangle $ABC$ is right with hypotenuse $AB$, $ABD$ is equilateral. Denote $\varphi$ the dihedral angle between planes $ABC,ABD$. 1) Determine the length of $CD$ in terms of $a=BC,b=CA,c$ and $\varphi$. 2) Determine all values of $\varphi$ such that the tetrahedron $ABCD$ has four sides of the same length.

A finite sequence of integers $ a_0, a_1, \ldots, a_n$ is called quadratic if for each $ i$ in the set $ \{1,2 \ldots, n\}$ we have the equality $ |a_i \minus{} a_{i\minus{}1}| \equal{} i^2.$ a.) Prove that any two integers $ b$ and $ c,$ there exists a natural number $ n$ and a quadratic sequence with $ a_0 \equal{} b$ and $ a_n \equal{} c.$ b.) Find the smallest natural number $ n$ for which there exists a quadratic sequence with $ a_0 \equal{} 0$ and $ a_n \equal{} 1996.$

$ ABC $ is a triangle with $ \angle BCA= 90^{\circ } $ and $ D,E $ on sides $ BC,CA, $ rspectively, so that $ \frac{BD}{AC}=\frac{AE}{CD}=k. $ The line $ BE $ meets $ AD $ at $ O. $ Show that $ \angle BOD =60^{\circ } $ if and only if $ k=\sqrt 3. $

Brenda and Sally run in opposite directions on a circular track, starting at diametrically opposite points. They first meet after Brenda has run $ 100$ meters. They next meet after Sally has run $ 150$ meters past their first meeting point. Each girl runs at a constant speed. What is the length of the track in meters? $ \textbf{(A)}\ 250 \qquad \textbf{(B)}\ 300 \qquad \textbf{(C)}\ 350 \qquad \textbf{(D)}\ 400\qquad \textbf{(E)}\ 500$

In a closed rectangular neighborhood there are: $S$ streets (these are straight roads of maximum length), $V$ four-arm intersections ( [img]https://cdn.artofproblemsolving.com/attachments/e/4/6a5974a30dc182b59a519a8ef4eb4f1412e05e.png[/img]), $H$ city blocks (these are rectangular areas bounded by four streets, which are no be intersected by another street) and $T$ represents the number of $T$-intersections ([img]https://cdn.artofproblemsolving.com/attachments/0/a/b390a30a0b27d83db681f70f633bdeed697163.png[/img] ). For example, in the neighborhood below, there are $15$ streets, $8$ four-arm intersections, $20$ city blocks and $22$ $T$-intersections. [img]https://cdn.artofproblemsolving.com/attachments/a/2/c1a5e463d0fb5671ac0702c91cfc2272d4e2c3.png[/img] Prove that in each district $S + V = H + 3$.

Determine all triples $(x, y, z)$ of positive integers $x > y > z > 0$, such that $x^2 = y \cdot 2^z + 1$

Let $ r$ and $ n$ be nonnegative integers such that $ r \le n$. $ (a)$ Prove that: $ \frac{n\plus{}1\minus{}2r}{n\plus{}1\minus{}r} \binom{n}{r}$ is an integer. $ (b)$ Prove that: $ \displaystyle\sum_{r\equal{}0}^{[n/2]}\frac{n\plus{}1\minus{}2r}{n\plus{}1\minus{}r} \binom{n}{r}<2^{n\minus{}2}$ for all $ n \ge 9$.

Given an equilateral triangle, cut it into four polygonal shapes so that, reassembled appropriately, these figures form a square.

Find all real triples $(a,b,c)$, for which $$a(b^2+c)=c(c+ab)$$ $$b(c^2+a)=a(a+bc)$$ $$c(a^2+b)=b(b+ca).$$

Let $ABCDEF$ be a hexagon inscribed in a circle $\Omega$ such that triangles $ACE$ and $BDF$ have the same orthocenter. Suppose that segments $BD$ and $DF$ intersect $CE$ at $X$ and $Y$, respectively. Show that there is a point common to $\Omega$, the circumcircle of $DXY$, and the line through $A$ perpendicular to $CE$. [i]Proposed by Michael Ren and Vincent Huang[/i]

Given that $(2\cos^2{7.5}-\cos{75}-1)^2$ can be expressed as $\frac{p}{q}$, what is $p+q$? [i]2022 CCA Math Bonanza Tiebreaker Round #3[/i]

A multi-digit number is written on the blackboard. Susan puts in a number of plus signs between some pairs of adjacent digits. The addition is performed and the process is repeated with the sum. Prove that regardless of what number was initially on the blackboard, Susan can always obtain a single-digit number in at most ten steps.

In quadrilateral $ABCD$, $AC = BD$ and $\measuredangle B = 60^\circ$. Denote by $M$ and $N$ the midpoints of $\overline{AB}$ and $\overline{CD}$, respectively. If $MN = 12$ and the area of quadrilateral $ABCD$ is 420, then compute $AC$. [i]Proposed by Aaron Lin[/i]