Consider points $C_1, C_2$ on the side $AB$ of a triangle $ABC$, points $A_1, A_2$ on the side $BC$ and points $B_1 , B_2$ on the side $CA$ such that these points divide the corresponding sides to three equal parts. It is known that all the points $A_1, A_2, B_1, B_2 , C_1$ and $C_2$ are concyclic. Prove that triangle $ABC$ is equilateral.

Let $D$ be a point in the interior of $ABC$. Let $BD$ and $AC$ intersect at $E$ while $CD$ and $AB$ intersect at $F$. Let $EF$ intersect $BC$ at $G$. Let $H$ be an arbitrary point on $AD$. Let $HF$ and $BD$ intersect at $I$. Let $HE$ and $CD$ intersect at $J$ . prove that $G$,$I$ and $J$ are collinear

A [i]magic square[/i] is a table [img]https://cdn.artofproblemsolving.com/attachments/7/9/3b1e2b2f5d2d4c486f57c4ad68b66f7d7e56dd.png[/img] in which all the natural numbers from $1$ to $16$ appear and such that: $\bullet$ all rows have the same sum $s$. $\bullet$ all columns have the same sum $s$. $\bullet$ both diagonals have the same sum $s$ . It is known that $a_{22} = 1$ and $a_{24} = 2$. Calculate $a_{44}$.

Three unit circles $\omega_1$, $\omega_2$, and $\omega_3$ in the plane have the property that each circle passes through the centers of the other two. A square $S$ surrounds three circles in such a way that each of its four sides is tangent to at least one of $\omega_1$, $\omega_2$, and $\omega_3$. Find the side length of the square $S$.

Given $n \ (n \geq 3)$ points in space such that every three of them form a triangle with one angle greater than or equal to $120^\circ$, prove that these points can be denoted by $A_1,A_2, \ldots,A_n$ in such a way that for each $i, j, k, 1 \leq i < j < k \leq n$, angle $A_iA_jA_k$ is greater than or equal to $120^\circ . $

The sequence of numbers $a_1, a_2, a_3, ...$ is defined recursively by $a_1 = 1, a_{n + 1} = \lfloor \sqrt{a_1+a_2+...+a_n} \rfloor $ for $n \ge 1$. Find all numbers that appear more than twice at this sequence.

In a room there are nine men. Among every three of them there are two mutually acquainted. Prove that some four of them are mutually acquainted.

Integers $1 \le n \le 200$ are written on a blackboard just one by one. We surrounded just $100$ integers with circle. We call a square of the sum of surrounded integers minus the sum of not surrounded integers $score$ of this situation. Calculate the average score in all ways.

Let $P$ be a point inside a triangle $ABC$ such that $\angle PAB = \angle PCA$ and $\angle PAC = \angle PBA$. If $O \ne P$ is the circumcenter of $\triangle ABC$, prove that $\angle APO$ is right.

Yannick has a bicycle lock with a $4$-digit passcode whose digits are between $0$ and $9$ inclusive. (Leading zeroes are allowed.) The dials on the lock is currently set at $0000$. To unlock the lock, every second he picks a contiguous set of dials, and increases or decreases all of them by one, until the dials are set to the passcode. For example, after the first second the dials could be set to $1100$, $0010$, or $9999$, but not $0909$ or $0190$. (The digits on each dial are cyclic, so increasing $9$ gives $0$, and decreasing $0$ gives $9$.) Let the complexity of a passcode be the minimum number of seconds he needs to unlock the lock. What is the maximum possible complexity of a passcode, and how many passcodes have this maximum complexity? Express the two answers as an ordered pair.

Consider the Pascal's triangle in the figure where the binomial coefficients are arranged in the usual manner. Select any binomial coefficient from anywhere except the right edge of the triangle and labet it $C$. To the right of $C$, in the horizontal line, there are $t$ numbers, we denote them as $a_1,a_2,\cdots,a_t$, where $a_t = 1$ is the last number of the series. Consider the line parallel to the left edge of the triangle containing $C$, there will only be $t$ numbers diagonally above $C$ in that line. We successively name them as $b_1,b_2,\cdots,b_t$, where $b_t = 1$. Show that \[b_ta_1-b_{t-1}a_2+b_{t-2}a_3-\cdots+(-1)^{t-1}b_1a_t = 1\]. For example, Suppose you choose $\binom41 = 4$ (see figure), then $t = 3$, $a_1 = 6, a_2 = 4, a_3 = 1$ and $b_1 = 3, b_2 = 2, b_3 = 1$. \[\begin{array}{ccccccccccc} & & & & & 1 & & & & & \\ & & & & 1 & & \underset{b_3}{1} & & & & \\ & & & 1 & & \underset{b_2}{2} & & 1 & & & \\ & & 1 & & \underset{b_1}{3} & & 3 & & 1 & & \\ & 1 & & \boxed{4} & & \underset{a_1}{6} & & \underset{a_2}{4} & & \underset{a_3}{1} & \\ \ldots & & \ldots & & \ldots & & \ldots & & \ldots & & \ldots \\ \end{array}\]

Does there exist an integer containing only digits $2$ and $0$ which is a $k$-th power of a positive integer ($k \ge2$)?

While finding the sine of a certain angle, an absent-minded professor failed to notice that his calculator was not in the correct angular mode. He was lucky to get the right answer. The two least positive real values of $x$ for which the sine of $x$ degrees is the same as the sine of $x$ radians are $\frac{m\pi}{n-\pi}$ and $\frac{p\pi}{q+\pi},$ where $m,$ $n,$ $p$ and $q$ are positive integers. Find $m+n+p+q.$

Let $\pi$ be a given permutation of the set $\{1, 2, \dots, n\}$. Determine the smallest possible value of \[ \sum_{i=1}^n |\pi(i) - \sigma(i)|, \] where $\sigma$ is a permutation chosen from the set of all $n$-cycles. Express the result in terms of the number and lengths of the cycles in the disjoint cycle decomposition of $\pi$, including the fixed points.

The altitudes $AA_1$ and $BB_1$ of an acute-angled triangle ABC meet at point $O$. Let $A_1A_2$ and $B_1B_2$ be the altitudes of triangles $OBA_1$ and $OAB_1$ respectively. Prove that $A_2B_2$ is parallel to $AB$. (L.Steingarts)

Let $\triangle ABC$ be a triangle with $AB=6, BC=8, AC=10$, and let $D$ be a point such that if $I_A, I_B, I_C, I_D$ are the incenters of the triangles $BCD,$ $ ACD,$ $ ABD,$ $ ABC$, respectively, the lines $AI_A, BI_B, CI_C, DI_D$ are concurrent. If the volume of tetrahedron $ABCD$ is $\frac{15\sqrt{39}}{2}$, then the sum of the distances from $D$ to $A,B,C$ can be expressed in the form $\frac{a}{b}$ for some positive relatively prime integers $a,b$. Find $a+b$. [i]Proposed by [b]FedeX333X[/b][/i]

Let $ABC$ a triangle with $AB=BC$ and incircle $\omega$. Let $M$ the mindpoint of $BC$; $P, Q$ points in the sides $AB, AC$ such that $PQ\parallel BC$, $PQ$ is tangent to $\omega$ and $\angle CQM=\angle PQM$. Find the perimeter of triangle $ABC$ knowing that $AQ=1$.

Do there exist numbers $a,b \in R$ and surjective function $f: R \to R$ such that $f(f(x)) = bx f(x) +a$ for all real $x$? I.Voronovich

Fill in each empty white circle with a number from $1$ to $16$ so that each number is used exactly once. One number has been given to you. If a square has a given number inside and its four vertices contain the numbers $a, b, c, d$ in clockwise order, then the number inside the square must be equal to $(a + c)(b + d)$. There is a unique solution, but you do not need to prove that your answer is the only one possible. You merely need to find an answer that satisfies the constraints above. (Note: In any other USAMTS problem, you need to provide a full proof. Only in this problem is an answer without justification acceptable.) [asy] unitsize(1cm); draw((0,0)--(3,0)--(3,3)--(0,3)--(0,0)); draw((0,1)--(3,1)); draw((0,2)--(3,2)); draw((1,0)--(1,3)); draw((2,0)--(2,3)); string[][] givens = {{"","638","650"},{"50","","338"},{"77","130",""}}; string[][] numbers = {{"","","",""},{"","","",""},{"","","",""},{"","","","5"}}; for(int i=0; i < 4; ++i) { for(int j=0; j < 4; ++j) { filldraw(circle((i,j),0.3), white); label(numbers[3-j][i], (i,j)); } } for(int i=0; i < 3; ++i){ for(int j=0; j < 3; ++j){ label(givens[2-j][i], (i + 0.5, j + 0.5)); } } [/asy]

How many ordered triples of integers $ (a, b, c)$ satisfy\[|a \plus{} b| \plus{} c \equal{} 19\quad\text{and}\quad ab \plus{} |c| \equal{} 97?\] $\textbf{(A)}\ 0\qquad \textbf{(B)}\ 4\qquad \textbf{(C)}\ 6\qquad \textbf{(D)}\ 10\qquad \textbf{(E)}\ 12$

Compute the expected sum of elements in a subset of $\{1, 2, 3, . . . , 2020\}$ (including the empty set) chosen uniformly at random.

Let $ABCDE$ be a regular pentagon, and $F$ some point on the arc $AB$ of the circumcircle of $ABCDE$. Show that \[ \frac{FD}{FE + FC} = \frac{FB + FA}{FD} = \frac{-1 + \sqrt{5}}{2}, \] and that $FD + FB + FA = FE + FC$.

Let $S$ be the set of points inside and on the boarder of a regular haxagon with side length 1. Find the least constant $r$, such that there exists one way to colour all the points in $S$ with three colous so that the distance between any two points with same colour is less than $r$.

Quadratic trinomials with positive leading coefficients are arranged in the squares of a $6 \times 6$ table. Their $108$ coefficients are all integers from $-60$ to $47$ (each number is used once). Prove that at least in one column the sum of all trinomials has a real root. [i]K. Kokhas & F. Petrov[/i]

Prove that there is a constant $c>0$ with the following property: If $a, b, n$ are positive integers such that $\gcd(a+i, b+j)>1$ for all $i, j\in\{0, 1, \ldots n\}$, then\[\min\{a, b\}>c^n\cdot n^{\frac{n}{2}}.\]