Samuel Tsui and Jason Yang each chose a different integer between $1$ and $60$, inclusive. They don’t know each others’ numbers, but they both know that the other person’s number is between $1$ and $60$ and distinct from their own. They have the following conversation: Samuel Tsui: Do our numbers have any common factors greater than $1$? Jason Yang: Definitely not. However their least common multiple must be less than$ 2023$. Samuel Tsui: Ok, thismeans that the sumof the factors of our two numbers are equal. What is the sumof Samuel Tsui’s and Jason Yang’s numbers? [i]Proposed by Samuel Tsui[/i]

The cells of a $8 \times 8$ table are initially white. Alice and Bob play a game. First Alice paints $n$ of the fields in red. Then Bob chooses $4$ rows and $4$ columns from the table and paints all fields in them in black. Alice wins if there is at least one red field left. Find the least value of $n$ such that Alice can win the game no matter how Bob plays.

From an equilateral triangle with side 2014 we cut off another equilateral triangle with side 214, such that both triangles have one common vertex and two of the side of the smaller triangles lie on two of the side of the bigger triangle. Is it possible to cover this figure with figures in the picture without overlapping (rotation is allowed) if all figures are made of equilateral triangles with side 1? Explain the answer! [asy] import olympiad; unitsize(20); pair A,B,C,D,E,F,G,H; A=(0,0); B=(1,0); C=rotate(60)*B; D=rotate(60)*C; E=rotate(60)*D; F=rotate(60)*E; G=rotate(60)*F; draw(A--B); draw(A--C); draw(A--D); draw(A--E); draw(A--F); draw(A--G); draw(B--C--D--E--F--G--B); A=(2,0); B=A+(1,0); C=A+rotate(60)*(B-A); D=A+rotate(60)*(C-A); E=A+rotate(120)*(D-A); F=A+rotate(60)*(E-A); G=2*F-E; H=2*C-D; draw(A--D--C--A--B--C--H--B--G--F--E--A--F--B); A=(4,0); B=A+(1,0); C=A+rotate(-60)*(B-A); D=B+rotate(60)*(C-B); E=B+rotate(60)*(D-B); F=B+rotate(60)*(E-B); G=E+rotate(60)*(D-E); H=E+rotate(60)*(G-E); draw(A--B--C--A); draw(C--D--B); draw(D--E--B); draw(B--F--E); draw(E--G--D); draw(E--H--G); A=(8.5,0.5); B=A+(1,0); C=A+rotate(60)*(B-A); D=A+rotate(60)*(C-A); E=A+rotate(60)*(D-A); F=A+rotate(60)*(E-A); G=A+rotate(60)*(F-A); H=G+rotate(60)*(F-G); draw(A--B); draw(A--C); draw(A--D); draw(A--E); draw(A--F); draw(A--G); draw(B--C); draw(D--E--F--G--B); draw(G--H--F);[/asy]

Let $c$ be a positive real number. Find all functions $f:\mathbb{R}^+\to\mathbb{R}^+$ that satisfy $$x^2f(xf(y))f(x)f(y)=c$$ for all positive reals $x$ and $y$.

One hundred students at Century High School participated in the AHSME last year, and their mean score was $100$. The number of non-seniors taking the AHSME was $50\%$ more than the number of seniors, and the mean score of the seniors was $50\%$ higher than that of the non-seniors. What was the mean score of the seniors? $ \textbf{(A)}\ 100\qquad\textbf{(B)}\ 112.5\qquad\textbf{(C)}\ 120\qquad\textbf{(D)}\ 125\qquad\textbf{(E)}\ 150 $

For each nonnegative integer $k$ find all nonnegative integers $x,y,z$ such that $x^2 +y^2 +z^2 = 8^k$

Given the integer $n > 1$ and the real number $a > 0$ determine the maximum of $\sum_{i=1}^{n-1} x_i x_{i+1}$ taken over all nonnegative numbers $x_i$ with sum $a.$

Given triangle $ABC$ with $AB<AC$ and its circumcircle $\Omega$. Let $I$ be the incenter of $ABC$, and the feet from $I$ to $BC$ is $D$. The circle with center $A$ and radius $AI$ intersects $\Omega$ at $E$ and $F$. $P$ is a point on $EF$ such that $DP$ is parallel to $AI$. Prove that $AP$ and $MI$ intersects on $\Omega$ where $M$ is the midpoint of arc $BAC$. [hide = Remark] In the test, the incenter called $O$ and the circumcircle called $Luna$ $Cabrera$ You have to prove $AP \cap MO \in Luna$ $Cabrera$ [/hide] [i]Proposed by BlessingOfHeaven[/i]

In the space, three lines $a,b,c$ that any two in them are skew lines. Then the number of lines that intersect all of $a,b,c$ is $\text{(A)}0\qquad\text{(B)}1\qquad\text{(C)}\text{more than one, but finitely many}\qquad\text{(D)} \text{infinitely many}$

Write down one of the following integers: $1, 2, 4, 8, 16.$ If your team is the only one that submits this integer, you will receive that number of points; otherwise, you receive zero. [b][color=#f00]There's no real way to solve this but during the competition, each of the 5 available scores were submitted at least twice by the 16 teams competing. [/color][/b]

For $m$ and $n$ positive integers that are prime to each other, determine the possible values ​​of $$\gcd (5^m + 7^m, 5^n + 7^n)$$

Petya and Vasya play the following game. Petya conceives a polynomial $P(x)$ having integer coefficients. On each move, Vasya pays him a ruble, and calls an integer $a$ of his choice, which has not yet been called by him. Petya has to reply with the number of distinct integer solutions of the equation $P(x)=a$. The game continues until Petya is forced to repeat an answer. What minimal amount of rubles must Vasya pay in order to win? [i](Anant Mudgal)[/i] (Translated from [url=http://sasja.shap.homedns.org/Turniry/TG/index.html]here.[/url])

There are $6$ different bus lines in a city, each stopping at exactly $5$ stations and running in both directions. Nevertheless, for every two different stations there is always a bus line connecting these two stations. Determine the maximum number of stations in this city. [i](Karl Czakler)[/i]

Find the smallest positive integer $n$ such that \[0< \sqrt[4]{n}-\lfloor \sqrt[4]{n}\rfloor < 0.00001.\]

A triangle can be cut into three similar triangles. Prove that it can be cut into any number of triangles similar to each other.

A set $X$ consisting of $n$ positive integers is called $\textit{good}$ if the following condition holds: For any two different subsets of $X$, say $A$ and $B$, the number $s(A) - s(B)$ is not divisible by $2^n$. (Here, for a set $A$, $s(A)$ denotes the sum of the elements of $A$) Given $n$, find the number of good sets of size $n$, all of whose elements is strictly less than $2^n$.

Find all positive real numbers $r<1$ such that there exists a set $\mathcal{S}$ with the given properties: i) For any real number $t$, exactly one of $t, t+r$ and $t+1$ belongs to $\mathcal{S}$; ii) For any real number $t$, exactly one of $t, t-r$ and $t-1$ belongs to $\mathcal{S}$.

Evaluate $ \int_0^{\frac{\pi}{3}} \frac{1}{\cos ^ 7 x}\ dx$.

Prove that there do not exist polynomials $ P$ and $ Q$ such that \[ \pi(x)\equal{}\frac{P(x)}{Q(x)}\] for all $ x\in\mathbb{N}$.

Does there exist an infinite set of triples of integers $x, y, z$ (not necessarily positive) such that $$x^2 + y^2 + z^2 = x^3 + y^3+z^3?$$ (NB Vassiliev)

A choir director must select a group of singers from among his $6$ tenors and $8$ basses. The only requirements are that the difference between the number of tenors and basses must be a multiple of $4$, and the group must have at least one singer. Let $N$ be the number of groups that can be selected. What is the remainder when $N$ is divided by $100$? $\textbf{(A)}\ 47 \qquad\textbf{(B)}\ 48 \qquad\textbf{(C)}\ 83 \qquad\textbf{(D)}\ 95 \qquad\textbf{(E)}\ 96$

The infinite sequence $a_0, a_1, a_2, a_3,... $ is defined by $a_0 = 2$ and $$a_n =\frac{2a_{n-1} + 1}{a_{n-1} + 2}$$ , $n = 1, 2, 3, ...$ Prove that $1 < a_n < 1 + \frac{1}{3^n}$ for all $n = 1, 2, 3, . .$

A rectangular table $ 9$ rows $ \times$ $ 2008$ columns is fulfilled with numbers $ 1$, $ 2$, ...,$ 2008$ in a such way that each number appears exactly $ 9$ times in table and difference between any two numbers from same column is not greater than $ 3$. What is maximum value of minimum sum in column (with minimal sum)?

The real numbers $a_0,a_1,a_2,\ldots$ satisfy $1=a_0\le a_1\le a_2\le\ldots. b_1,b_2,b_3,\ldots$ are defined by $b_n=\sum_{k=1}^n{1-{a_{k-1}\over a_k}\over\sqrt a_k}$. [b]a.)[/b] Prove that $0\le b_n<2$. [b]b.)[/b] Given $c$ satisfying $0\le c<2$, prove that we can find $a_n$ so that $b_n>c$ for all sufficiently large $n$.

A convex polygon on a plane contains at least $m^2+1$ points with integer coordinates. Prove that it contains $m+1$ points with integer coordinates that lie on the same line.