We have four white equilateral triangles of $3$ cm on each side and join them by their sides to obtain a triangular base pyramid. At each edge of the pyramid we mark two red dots that divide it into three equal parts. Number the red dots, so that when you scroll them in the order they were numbered, result a path with the smallest possible perimeter. How much does that path measure?

Find all pairs of prime numbers $(p, q)$ for which $7pq^2 + p = q^3 + 43p^3 + 1$

An acute triangle $\bigtriangleup ABC$ is given which $BC>CA>AB$. $I$ is the interior and the incircle of $\bigtriangleup ABC$ meets $BC, CA, AB$ at $D,E,F$. $AD$ and $BE$ meet at $P$. Let $l_{1}$ be a tangent from D to the circumcircle of $\bigtriangleup DIP$, and define $l_{2}$ and $l_{3}$ on $E$ and $F$, respectively. Prove $l_{1},l_{2},l_{3}$ meet at one point.

In triangle $ABC$, $I$ is the centre of the incircle. There is a circle tangent to $AI$ at $I$ which passes through $B$. This circle intersects $AB$ once more in $P$ and intersects $BC$ once more in $Q$. The line $QI$ intersects $AC$ in $R$. Prove that $|AR|\cdot |BQ|=|P I|^2$

Solve the equation in $\mathbb{R}$ $$\left\{\left\{\frac{x^2-x}{2021}\right \}-\left\{\frac{x^2+x}{2022}\right \} \right \}=0.$$

As shown in the figure, it is known that $BC = AC$ in $ABC$, $M$ is the midpoint of $AB$, points $D$ and $E$ lie on $AB$ satisfying $\angle DCE = \angle MCB$, the circumscribed circle of $\vartriangle BDC$ and the circumscribed circle of $\vartriangle AEC$ intersect at point $F$ (different from point $C$), point $H$ lies on $AB$ such that the straight line $CM$ bisects the line segment $HF$. Let the circumcenters of $\vartriangle HFE$ and $\vartriangle BFM$ be $O_1$ and $O_2$ respectively. Prove that $O_1O_2\perp CF$. [img]https://cdn.artofproblemsolving.com/attachments/e/4/e8fc62735b8cfbd382e490617f26d335c46823.png[/img]

A trapezoid $ABCD$ ($AB || CF$,$AB > CD$) is circumscribed.The incircle of the triangle $ABC$ touches the lines $AB$ and $AC$ at the points $M$ and $N$,respectively.Prove that the incenter of the trapezoid $ABCD$ lies on the line $MN$.

The sum of $10$ consecutive integers is $75$. What is the smallest of these $10$ integers?

Let $ABC$ be an equilateral triangle. $D$ and $E$ are midpoints of $[AB]$ and $[AC]$. The ray $[DE$ cuts the circumcircle of $\triangle ABC$ at $F$. What is $\frac {|DE|}{|DF|}$? $ \textbf{(A)}\ \frac 12 \qquad\textbf{(B)}\ \frac {\sqrt 3}3 \qquad\textbf{(C)}\ \frac 23(\sqrt 3 - 1) \qquad\textbf{(D)}\ \frac 23 \qquad\textbf{(E)}\ \frac {\sqrt 5 - 1}2 $

Given a triangle $ABC$ and a point $M$ such that the lines $MA,MB,MC$ intersect the lines $BC,CA,AB$ in this order in points $D,E$ and $F,$ respectively. Prove that there are numbers $\epsilon_1, \epsilon_2, \epsilon_3 \in \{-1, 1\}$ such that: \[\epsilon_1 \cdot \frac{MD}{AD} + \epsilon_2 \cdot \frac{ME}{BE} + \epsilon_3 \cdot \frac{MF}{CF} = 1.\]

The incircle of tangential quadrilateral $ABCD$ intersects diagonal $BD$ at $P$ and $Q$ $(BP<BQ).$ Let $UV$ be the diameter of the incircle perpendicular to $AC$ $(BU<BV).$ Show that the lines $AC,PV,$ and $QU$ pass through one point. [i]Based on problem 2 of IOM 2018, Moscow[/i]

Let $\omega$ be a circle centered at $O$ with radius $R=2018$. For any $0 < r < 1009$, let $\gamma$ be a circle of radius $r$ centered at a point $I$ satisfying $OI =\sqrt{R(R-2r)}$. Choose any $A,B,C\in \omega$ with $AC, AB$ tangent to $\gamma$ at $E,F$, respectively. Suppose a circle of radius $r_A$ is tangent to $AB,AC$, and internally tangent to $\omega$ at a point $D$ with $r_A=5r$. Let line $EF$ meet $\omega$ at $P_1,Q_1$. Suppose $P_2,P_3,Q_2,Q_3$ lie on $\omega$ such that $P_1P_2,P_1P_3,Q_1Q_2,Q_1Q_3$ are tangent to $\gamma$. Let $P_2P_3,Q_2Q_3$ meet at $K$, and suppose $KI$ meets $AD$ at a point $X$. Then as $r$ varies from $0$ to $1009$, the maximum possible value of $OX$ can be expressed in the form $\frac{a\sqrt{b}}{c}$, where $a,b,c$ are positive integers such that $b$ is not divisible by the square of any prime and $\gcd (a,c)=1$. Compute $10a+b+c$. [i]Proposed by Vincent Huang

A sequence $x_0, x_1, x_2, . . .$ is given by $x_0 = 8$ and $x_{n+1} =\frac{1 + x_n}{1- x_n}$ for $n = 0, 1, 2, . . . .$ Determine the number $x_{2013}$.

Let $f: [0,1]\to\mathbb{R}$ be a continuous function such that \[ \int_{0}^{1}f(x)dx=0. \] Prove that there is $c\in (0,1)$ such that \[ \int_{0}^{c}xf(x)dx=0. \] [i]Cezar Lupu, Tudorel Lupu[/i]

From a regular octahedron with edge $1$, cut off a pyramid about each vertex. The base of each pyramid is a square with edge $\frac 13$. Can copies of the polyhedron so obtained, whose faces are either regular hexagons or squares, be used to tile space?

Suppose that $N$ is a three digit number divisible by $7$ such that upon removing its middle digit, the remaining two digit number is also divisible by $7$. What is the minimum possible value of $N$? [i]2019 CCA Math Bonanza Lightning Round #3.1[/i]

Consider $h+1$ chess boards. Number the squares of each board from 1 to 64 in such a way that when the perimeters of any two boards of the collection are brought into coincidence in any possible manner, no two squares in the same position have the same number. What is the maximum value of $h?$

Let $n$ and $m$ be positive integers. The daycare nanny uses $n \times m$ square floor mats to construct an $n \times m$ rectangular area, with a baby on each of the mats. Each baby initially faces toward one side of the rectangle. When the nanny claps, all babies crawl one mat forward in the direction it is facing at, and then turn 90 degrees clockwise. If a baby crawls outside of the rectangle, it cries. If two babies simultaneously crawl onto the same mat, they bump into each other and cry. Suppose that it is possible for the nanny to arrange the initial direction of each baby so that, no matter how many times she claps, no baby would cry. Find all possible values of $n$ and $m$. [i]Proposed by Chu-Lan Kao[/i]

The real numbers $a,b,c,x,y,$ and $z$ are such that $a>b>c>0$ and $x>y>z>0$. Prove that \[\frac {a^2x^2}{(by+cz)(bz+cy)}+\frac{b^2y^2}{(cz+ax)(cx+az)}+\frac{c^2z^2}{(ax+by)(ay+bx)}\ge \frac{3}{4}\]

A set of teams held a round-robin tournament in which every team played every other team exactly once. Every team won $10$ games and lost $10$ games; there were no ties. How many sets of three teams $\{A, B, C\}$ were there in which $A$ beat $B$, $B$ beat $C$, and $C$ beat $A?$ $\textbf{(A)}\ 385 \qquad \textbf{(B)}\ 665 \qquad \textbf{(C)}\ 945 \qquad \textbf{(D)}\ 1140 \qquad \textbf{(E)}\ 1330$

Consider the trinomial $f(x) = x^2 + 2bx + c$ with integer coefficients $b$ and $c$. Prove that if $f(n) \ge 0$ for all integers $n$, then $f(x) \ge 0$ even for all rational numbers $x$.

On the side $BC$ of the rectangle $ABCD$, a point $P{}$ is marked so that $\angle APD = 90^\circ$. On the straight line $AD$, points $Q{}$ and $R{}$ are selected outside the segment $AD$ such that $AQ = BP$ and $CP = DR$. The circle $\omega$ passes through the points $Q, D$ and the circumcenter of the triangle $PDQ$. The circle $\gamma$ passes through the points $A, R$ and the circumcenter of the triangle $APR$. Prove that the radius of one of the circles touching the line $AD$ and the circles $\omega$ and $\gamma$ is $2AB$.

$\textbf{(Caos)}$ A cao [sic] has 6 legs, 3 on each side. A walking pattern for the cao is defined as an ordered sequence of raising and lowering each of the legs exactly once (altogether 12 actions), starting and ending with all legs on the ground. The pattern is safe if at any point, he has at least 3 legs on the ground and not all three legs are on the same side. Estimate $N$, the number of safe patterns. An estimate of $E > 0$ earns $\left\lfloor 20\min(N/E, E/N)^4 \right\rfloor$ points.

Prove for positive numbers: $$(a^2+b+\frac{3}{4})(b^2+a+\frac{3}{4}) \geq (2a+\frac{1}{2})(2b+\frac{1}{2})$$

By dividing $2023$ by a natural number $m$, the remainder is $23$. How many numbers $m$ are there with this property?