How many triangles appear in the diagram below: [asy] import graph; size(6cm); real lsf=0.5; pen dps=linewidth(0.6)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=0,xmax=8,ymin=0,ymax=8; draw((0,8)--(0,0)); draw((0,0)--(8,0)); draw((8,0)--(8,8)); draw((8,8)--(0,8)); draw((0,8)--(1,7)); draw((1,7)--(2,8)); draw((2,8)--(3,7)); draw((3,7)--(4,8)); draw((4,8)--(5,7)); draw((5,7)--(6,8)); draw((6,8)--(7,7)); draw((7,7)--(8,8)); draw((8,6)--(7,7)); draw((0,6)--(1,7)); draw((1,7)--(2,6)); draw((2,6)--(3,7)); draw((3,7)--(4,6)); draw((4,6)--(5,7)); draw((5,7)--(6,6)); draw((6,6)--(7,7)); draw((1,5)--(0,6)); draw((1,5)--(2,6)); draw((2,6)--(3,5)); draw((3,5)--(4,6)); draw((4,6)--(5,5)); draw((5,5)--(6,6)); draw((6,6)--(7,5)); draw((7,5)--(8,6)); draw((7,5)--(8,4)); draw((0,4)--(1,5)); draw((1,5)--(2,4)); draw((2,4)--(3,5)); draw((3,5)--(4,4)); draw((4,4)--(5,5)); draw((5,5)--(6,4)); draw((6,4)--(7,5)); draw((1,3)--(0,4)); draw((1,3)--(2,4)); draw((3,3)--(4,4)); draw((3,3)--(2,4)); draw((5,3)--(4,4)); draw((5,3)--(6,4)); draw((6,4)--(7,3)); draw((7,3)--(8,4)); draw((8,2)--(7,3)); draw((0,2)--(1,3)); draw((1,3)--(2,2)); draw((2,2)--(3,3)); draw((3,3)--(4,2)); draw((5,3)--(4,2)); draw((5,3)--(6,2)); draw((7,3)--(6,2)); draw((7,1)--(6,2)); draw((7,1)--(8,2)); draw((7,1)--(8,0)); draw((6,0)--(7,1)); draw((4,0)--(5,1)); draw((5,1)--(6,0)); draw((2,0)--(3,1)); draw((3,1)--(4,0)); draw((0,0)--(1,1)); draw((1,1)--(2,0)); draw((1,1)--(0,2)); draw((1,1)--(2,2)); draw((2,2)--(3,1)); draw((3,1)--(4,2)); draw((4,2)--(5,1)); draw((5,1)--(6,2)); dot((8,0),ds); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); [/asy]

Let $x$, $y$, $z$ be positive reals satisfying $\left(x+y+z\right)^{3}=32xyz$ Find the minimum and the maximum of $P=\frac{x^{4}+y^{4}+z^{4}}{\left(x+y+z\right)^{4}}$

Let $a,b,c$ be real numbers satisfying $a<b<c,a+b+c=6,ab+bc+ac=9$. Prove that $0<a<1<b<3<c<4$ [hide="Solution"] Let $abc=k$, then $a,b,c\ (a<b<c)$ are the roots of cubic equation $x^3-6x^2+9x-k=0\Longleftrightarrow x(x-3)^2=k$ that is to say, $a,b,c\ (a<b<c)$ are the $x$-coordinates of the interception of points between $y=x(x-3)^2$ and $y=k$. $y=x(x-3)^2$ have local maximuml value of $4$ at $x=1$ and local minimum value of $0$ at $x=3$. Since the $x$-coordinate of the interception point between $y=x(x-3)^2$ and $y=4$ which is the tangent line at local maximum point $(1,4)$ is a point $(4,4)$,Moving the line $y=k$ so that the two graphs $y=x(x-3)^2$ and $y=k$ have the distinct three interception points,we can find that the range of $a,b,c$ are $0<a<1,1<b<3,3<c<4 $,we are done.[/hide]

Let $(x_1,x_2,\dots,x_{100})$ be a permutation of $(1,2,...,100)$. Define $$S = \{m \mid m\text{ is the median of }\{x_i, x_{i+1}, x_{i+2}\}\text{ for some }i\}.$$ Determine the minimum possible value of the sum of all elements of $S$.

A box contains $5$ chips, numbered $1$, $2$, $3$, $4$, and $5$. Chips are drawn randomly one at a time without replacement until the sum of the values drawn exceeds $4$. What is the probability that $3$ draws are required? $\textbf{(A)} \frac{1}{15} \qquad \textbf{(B)} \frac{1}{10} \qquad \textbf{(C)} \frac{1}{6} \qquad \textbf{(D)} \frac{1}{5} \qquad \textbf{(E)} \frac{1}{4}$

let ${a_{n}}$ be a sequence of integers,$a_{1}$ is odd,and for any positive integer $n$,we have $n(a_{n+1}-a_{n}+3)=a_{n+1}+a_{n}+3$,in addition,we have $2010$ divides $a_{2009}$ find the smallest $n\ge\ 2$,so that $2010$ divides $a_{n}$

For $ s,t \in \mathbb{N}$, consider the set $ M\equal{}\{ (x,y) \in \mathbb{N} ^2 | 1 \le x \le s, 1 \le y \le t \}$. Find the number of rhombi with the vertices in $ M$ and the diagonals parallel to the coordinate axes.

Given any positive integer $n$, prove that: [b](a)[/b] Every $n$ points in the closed unit square $[0,1]\times [0,1]$ can be joined by a path of length less than $2\sqrt{n}+4$; and [b](b)[/b] There exist $n$ points in the closed unit square $[0,1]\times [0,1]$ that cannot be joined by a path of length less than $\sqrt{n}-1$.

Is there exist a sequence $a_0,a_1,a_2,\cdots $ consisting of non-zero integers that satisfies the following condition? [b]Condition[/b]: For all integers $n$ ($\ge 2020$), equation $$a_n x^n+a_{n-1}x^{n-1}+\cdots +a_0=0$$ has a real root with its absolute value larger than $2.001$.

Determine all integers $n\ge 2$ such that for any two infinite sequences of positive integers $a_1<a_2< \cdots $ and $b_1, b_2, \cdots$, such that $a_i\mid a_j$ for all $i<j$, there always exists a real number $c$ such that $$\lfloor{ca_i}\rfloor \equiv b_i \pmod {n}$$ for all $i\ge 1$. [i]Proposed by Wong Jer Ren & Ivan Chan Kai Chin[/i]

Let $L,E,T,M,$ and $O$ be digits that satisfy $LEET+LMT=TOOL.$ Given that $O$ has the value of $0,$ digits may be repeated, and $L\neq0,$ what is the value of the $4$-digit integer $ELMO?$

Adam, Benin, Chiang, Deshawn, Esther, and Fiona have internet accounts. Some, but not all, of them are internet friends with each other, and none of them has an internet friend outside this group. Each of them has the same number of internet friends. In how many different ways can this happen? $ \textbf{(A)}\ 60 \qquad\textbf{(B)}\ 170 \qquad\textbf{(C)}\ 290 \qquad\textbf{(D)}\ 320 \qquad\textbf{(E)}\ 660 $

The sequence $1, 2, \dots, 2023, 2024$ is written on a whiteboard. Every second, Megavan chooses two integers $a$ and $b$, and four consecutive numbers on the whiteboard. Then counting from the left, he adds $a$ to the 1st and 3rd of those numbers, and adds $b$ to the 2nd and 4th of those numbers. Can he achieve the sequence $2024, 2023, \dots, 2, 1$ in a finite number of moves? [i](Proposed by Avan Lim Zenn Ee)[/i]

Let $\{ a_1, a_2, \ldots, a_n\} = \{1, 2, \ldots, n\}$. Prove that \[\frac 12 +\frac 23 +\cdots+\frac{n-1}{n} \leq \frac{a_1}{a_2} + \frac{a_2}{a_3} +\cdots+\frac{a_{n-1}}{a_n}.\]

Real numbers $x,y,z, t$ satisfy $x + y + z +t = 0$ and $x^2+ y^2+ z^2+t^2 = 1$. Prove that $- 1 \le xy + yz + zt + tx \le 0$.

Determine the number of functions $f$ from the integers to $\{1,2,\cdots,15\}$ which satisfy $$f(x)=f(x+15)$$ and $$f(x+f(y))=f(x-f(y))$$ for all $x,y$. [i]Proposed by Vijay Srinivasan[/i]

In rectangle $ ABCD$, $ AB\equal{}100$. Let $ E$ be the midpoint of $ \overline{AD}$. Given that line $ AC$ and line $ BE$ are perpendicular, find the greatest integer less than $ AD$.

Find the sum of the real roots of the polynomial \[ \prod_{k=1}^{100} \left( x^2-11x+k \right) = \left( x^2-11x+1 \right)\left( x^2-11x+2 \right)\dots\left(x^2-11x+100\right). \][i]Proposed by Evan Chen[/i]

A point $O$ lies inside of the square $ABCD$. Prove that the difference between the sum of angles $OAB, OBC, OCD , ODA$ and $180^{\circ}$ does not exceed $45^{\circ}$.

Consolidated Megacorp is planning to send a salesperson to Elbonia who needs to visit every town there. It is possible to travel between any two towns of Elbonia directly either by barge or by mule cart (the same type of travel is available in either direction, and these are the only types of travel available). Show that it is possible to choose a starting town so that the salesperson can complete a round trip visiting each town exactly once and returning to her starting point, while changing the type of transportation used at most one time (this is desirable, since it’s hard to arrange for the merchandise to be transferred from barge to cart or vice versa).

Calculate the following integarls. [1] $\int x\cos ^ 2 x dx$ [2] $\int \frac{x-1}{(3x-1)^2}dx$ [3] $\int \frac{x^3}{(2-x^2)^4}dx$ [4] $\int \left({\frac{1}{4\sqrt{x}}+\frac{1}{2x}}\right)dx$ [5] $\int (\ln x)^2 dx$

Determine all functions $f : \mathbb{R} \to \mathbb{R}$ such that the following equation holds for every real $x,y$: \[ f(f(x) + y) = \lfloor x + f(f(y)) \rfloor. \] [b]Note:[/b] $\lfloor x \rfloor$ denotes the greatest integer not greater than $x$.

A fixed number of people are dividing an inheritance among themselves. An heir will be called poor if he gets less than $\$99$ and rich if he gets more than $\$10 000$ (some heirs may be neither rich nor poor). The total inheritance and the number of heirs are such that the total income of the rich heirs will be no less than that of the poor ones no matter how the inheritance is divided. Prove that the total income of the rich heirs is no less than $100$ times that of the poor ones. (F Nazarov)

If $m,~n,~p,$ and $q$ are real numbers and $f(x)=mx+n$ and $g(x)=px+q$, then the equation $f(g(x))=g(f(x))$ has a solution $\textbf{(A) }\text{for all choices of }m,~n,~p, \text{ and } q\qquad$ $\textbf{(B) }\text{if and only if }m=p\text{ and }n=q\qquad$ $\textbf{(C) }\text{if and only if }mq-np=0\qquad$ $\textbf{(D) }\text{if and only if }n(1-p)-q(1-m)=0\qquad$ $\textbf{(E) }\text{if and only if }(1-n)(1-p)-(1-q)(1-m)=0$

A positive integer is called [i]squarefree[/i] if its only perfect square factor is $1$. Call a set of positive integers [i]squarefreeful[/i] if each product of two of its elements is squarefree, and [i]squarefreefullest[/i] if no positive integer less than the maximum element of the set can be added while preserving the set's squarefreefulness. What is the minimum number of elements in a squarefreefullest set containing $31$?