We know $\mathbb Z_{210} \cong \mathbb Z_2 \times \mathbb Z_3 \times \mathbb Z_5 \times \mathbb Z_7$. Moreover,\begin{align*} 53 & \equiv 1 \pmod{2} \\ 53 & \equiv 2 \pmod{3} \\ 53 & \equiv 3 \pmod{5} \\ 53 & \equiv 4 \pmod{7}. \end{align*} Let \[ M = \left( \begin{array}{ccc} 53 & 158 & 53 \\ 23 & 93 & 53 \\ 50 & 170 & 53 \end{array} \right). \] Based on the above, find $\overline{(M \mod{2})(M \mod{3})(M \mod{5})(M \mod{7})}$.

At a conference there are $n$ mathematicians. Each of them knows exactly $k$ fellow mathematicians. Find the smallest value of $k$ such that there are at least three mathematicians that are acquainted each with the other two. [color=#BF0000]Rewording of the last line for clarification:[/color] Find the smallest value of $k$ such that there (always) exists $3$ mathematicians $X,Y,Z$ such that $X$ and $Y$ know each other, $X$ and $Z$ know each other and $Y$ and $Z$ know each other.

Elmo and Elmo's clone are playing a game. Initially, $n\geq 3$ points are given on a circle. On a player's turn, that player must draw a triangle using three unused points as vertices, without creating any crossing edges. The first player who cannot move loses. If Elmo's clone goes first and players alternate turns, who wins? (Your answer may be in terms of $n$.) [i]Proposed by Milan Haiman[/i]

Find all positive integers $k,m$ and $n$ such that $k!+3^m=3^n$

Determine whether there exists a function $f: \mathbb{Z}_{> 0} \rightarrow \mathbb{Z}_{> 0}$ such that for all positive integers $m$ and $n$, \[f(m+nf(m))=f(n)^m+2024! \cdot m.\] [i]Jaedon Whyte[/i]

For $0\leq t\leq 1$, define $f(t)=\int_0^{2\pi} |\sin x-t|dx.$ Evaluate $\int_0^1 f(t)dt.$

Let $ a,b,c$ be nonnegative real numbers. Prove that: $ \frac{1}{3}((a\minus{}b)^2\plus{}(b\minus{}c)^2\plus{}(c\minus{}a)^2) \le a^2\plus{}b^2\plus{}c^2\minus{}3 \sqrt[3]{a^2 b^2 c^2 } \le (a\minus{}b)^2\plus{}(b\minus{}c)^2\plus{}(c\minus{}a)^2.$

A census man on duty visited a house in which the lady inmates declined to reveal their individual ages, but said "We do not mind giving you the sum of the ages of any two ladies you may choose". Thereupon, the census man said, "In that case, please give me the sum of the ages of every possible pair of you". They gave the sums as: 30, 33, 41, 58, 66, 69. The census man took these figures and happily went away. How did he calculate the individual ages?

In the figure, $ PA$ is tangent to semicircle $ SAR$; $ PB$ is tangent to semicircle $ RBT$; $ SRT$ is a straight line; the arcs are indicated in the figure. Angle $ APB$ is measured by: [asy]unitsize(1.2cm); defaultpen(linewidth(.8pt)+fontsize(8pt)); dotfactor=3; pair O1=(0,0), O2=(3,0), Sp=(-2,0), R=(2,0), T=(4,0); pair A=O1+2*dir(60), B=O2+dir(85); pair Pa=rotate(90,A)*O1, Pb=rotate(-90,B)*O2; pair P=extension(A,Pa,B,Pb); pair[] dots={Sp,R,T,A,B,P}; draw(P--P+5*(A-P)); draw(P--P+5*(B-P)); clip((-2,0)--(-2,2.5)--(4,2.5)--(4,0)--cycle); draw(Arc(O1,2,0,180)--cycle); draw(Arc(O2,1,0,180)--cycle); dot(dots); label("$S$",Sp,S); label("$R$",R,S); label("$T$",T,S); label("$A$",A,NE); label("$B$",B,N); label("$P$",P,NNE); label("$a$",midpoint(Arc(O1,2,0,60)),SW); label("$b$",midpoint(Arc(O2,1,85,180)),SE); label("$c$",midpoint(Arc(O1,2,60,180)),SE); label("$d$",midpoint(Arc(O2,1,0,85)),SW);[/asy]$ \textbf{(A)}\ \frac {1}{2}(a \minus{} b) \qquad \textbf{(B)}\ \frac {1}{2}(a \plus{} b) \qquad \textbf{(C)}\ (c \minus{} a) \minus{} (d \minus{} b) \qquad \textbf{(D)}\ a \minus{} b \qquad \textbf{(E)}\ a \plus{} b$

What is the last two digits of the number $(11^2 + 15^2 + 19^2 +
...

+ 2007^2)^
2$?

Find all $ x\in \mathbb{Z} $ and $ a\in \mathbb{R} $ satisfying \[\sqrt{x^2-4}+\sqrt{x+2} = \sqrt{x-a}+a \]

In the triangle $ABC$ the length of side $AB$, and height $AH$ are known. also we know that $\angle B = 2 \angle C.$ Plot this triangle.

Is there a six-digit number where every two consecutive digits make up a certain number two-digit number that is the square of an integer? Justify your answer.

Let $ABCD$ be a rectangle and the arbitrary points $E\in (CD)$ and $F \in (AD)$. The perpendicular from point $E$ on the line $FB$ intersects the line $BC$ at point $P$ and the perpendicular from point $F$ on the line $EB$ intersects the line $AB$ at point $Q$. Prove that the points $P, D$ and $Q$ are collinear.

Each side of a convex quadrilateral is less than $20$ cm. Prove that you can specify the vertex of the quadrilateral, the distance from which to any point $Q$ inside the quadrilateral is less than $15$ cm.

In a cyclic quadrilateral $ABCD$, let $L$ and $M$ be the incenters of $ABC$ and $BCD$ respectively. Let $R$ be a point on the plane such that $LR\bot AC$ and $MR\bot BD$.Prove that triangle $LMR$ is isosceles.

A chess club consists of at least $10$ and at most $50$ members, where $G$ of them are female, and $B$ of them are male with $G>B$. In a chess tournament, each member plays with any other member exactly one time. At each game, the winner gains $1$, the loser gains $0$ and both player gains $1/2$ point when a tie occurs. At the tournament, it is observed that each member gained exactly half of his/her points from the games played against male members. How many different values can $B$ take? $ \textbf{(A)}\ 5 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 2 \qquad\textbf{(E)}\ 1 $

Lynnelle took 10 tests in her math class at Stanford. Her score on each test was an integer from 0 through 100. She noticed that, for every four consecutive tests, her average score on those four tests was at most 47.5. What is the largest possible average score she could have on all 10 tests?

Prove that there exists a right angle triangle with rational sides and area $d$ if and only if $x^2,y^2$ and $z^2$ are squares of rational numbers and are in Arithmetic Progression Here $d$ is an integer.

There are $2n$ points on a circle, $n$ are red and $n$ are blue. Janson found a red frog and a blue frog at a red point and a blue point on the circle respectively. Every minute, the red frog moves to the next red point in the clockwise direction and the blue frog moves to the next blue point in the anticlockwise direction. Prove that for any initial position of the two frogs, Janson can draw a line through the circle, such that the two frogs are always on opposite sides of the line.

There is a board with $2020$ squares in the bottom row and $2019$ in the top row, located as shown shown in the figure. [img]https://cdn.artofproblemsolving.com/attachments/f/3/516ad5485c399427638c3d1783593d79d83002.png[/img] In the bottom row the integers numbers from $ 1$ to $2020$ are placed in some order. Then in each box in the top row records the multiplication of the two numbers below it. How can they place the numbers in the bottom row so that the sum of the numbers in the top row be the smallest possible?

Harry and Terry are each told to calculate $8-(2+5)$. Harry gets the correct answer. Terry ignores the parentheses and calculates $8-2+5$. If Harry's answer is $H$ and Terry's answer is $T$, what is $H-T$? $\textbf{(A) }-10\qquad\textbf{(B) }-6\qquad\textbf{(C) }0\qquad\textbf{(D) }6\qquad \textbf{(E) }10$

Find the number of values of $x$ such that the number of square units in the area of the isosceles triangle with sides $x$, $65$, and $65$ is a positive integer.

There are $5$ people left in a game of Among Us, $4$ of whom are crewmates and the last is the impostor. None of the crewmates know who the impostor is. The person with the most votes is ejected, unless there is a tie in which case no one is ejected. Each of the $5$ remaining players randomly votes for someone other than themselves. The probability the impostor is ejected can be expressed as $\frac{m}{n}$. Find $m+n$. [i]Proposed by Sammy Charney[/i]

Coin $ A$ is flipped three times and coin $ B$ is flipped four times. What is the probability that the number of heads obtained from flipping the two fair coins is the same? $ \textbf{(A)}\ \frac {19}{128}\qquad \textbf{(B)}\ \frac {23}{128}\qquad \textbf{(C)}\ \frac {1}{4}\qquad \textbf{(D)}\ \frac {35}{128}\qquad \textbf{(E)}\ \frac {1}{2}$