Pedro must choose two irreducible fractions, each with a positive numerator and denominator such that: [list] [*]The sum of the fractions is equal to $2$. [*]The sum of the numerators of the fractions is equal to $1000$. [/list] In how many ways can Pedro do this?

Let $f$ be a real-valued function having partial derivatives and which is defined for $x^2 +y^2 \leq1$ and is such that $|f(x,y)|\leq 1.$ Show that there exists a point $(x_0, y_0 )$ in the interior of the unit circle such that $$\left( \frac{ \partial f}{\partial x}(x_0 ,y_0 ) \right)^{2}+ \left( \frac{ \partial f}{\partial y}(x_0 ,y_0 ) \right)^{2} \leq 16.$$

Suppose you are given $n$ blocks, each of which weighs an integral number of pounds, but less than $n$ pounds. Suppose also that the total weight of the $n$ blocks is less than $2n$ pounds. Prove that the blocks can be divided into two groups, one of which weighs exactly $n$ pounds.

Ari the Archer is shooting at an abnormal target. The target consists of $100$ concentric rings, each of width $1$, so that the total radius of the target is $100$. The point value of a given ring of the target is equal to its area (so getting a bull's eye would be worth $\pi$ points, but hitting on the outer ring would give $199\pi$ points). Given that Ari hits any point on the target uniformly at random, what is his expected score? [i]2017 CCA Math Bonanza Individual Round #7[/i]

2004 computers make up a network using several cables. If for a subset $S$ in the set of all computers, there isn't a cable that connects two computers in $S$, $S$ is called independant. One lets the arbitrary independant set consists at most 50 computers, and uses the least number of cables. (1) Let $c(L)$ be the number of cables which connects the computer $L$. Prove that for two computers $A,B$, $c(A)=c(B)$ if there is a cable which connects $A$ and $B$, $|c(A)-c(B)|\leq 1$ otherwise. (2) Determine the number of used cables.

Points $M$ and $N$ are given on sides $AD$ and $BC$ of rhombus $ABCD$, respectively. Line $MC$ intersects line $BD$ in point $T$, line $MN$ intersects line $BD$ in point $U$, line $CU$ intersects line $AB$ in point $Q$ and line $QT$ intersects line $CD$ in $P$. Prove that triangles $QCP$ and $MCN$ have equal area

The sales tax rate in Bergville is $6\%$. During a sale at the Bergville Coat Closet, the price of a coat is discounted $20\%$ from its \$90.00 price. Two clerks, Jack and Jill, calculate the bill independently. Jack rings up \$90.00 and adds $6\%$ sales tax, then subtracts $20\%$ from this total. Jill rings up \$90.00, subtracts $20\%$ of the price, then adds $6\%$ of the discounted price for sales tax. What is Jack's total minus Jill's total? $ \textbf{(A)}\ -\$1.06\qquad\textbf{(B)}\ -\$0.53\qquad\textbf{(C)}\ 0\qquad\textbf{(D)}\ \$0.53\qquad\textbf{(E)}\ \$1.06 $

Determine positive integers $p, q$, and $r$ such that the diagonal of a block consisting of $p\times q\times r$ unit cubes passes through exactly $1984$ of the unit cubes, while its length is minimal. (The diagonal is said to pass through a unit cube if it has more than one point in common with the unit cube.)

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$.