Prove that no matter what digits are placed in the four empty boxes, the eight-digit number \[ \textbf{9999}\Box\Box\Box\Box \] is not a perfect square. (A $\textit{perfect square}$ is a whole number times itself. For example, $25$ is a perfect square because $25 = 5 \times 5$.)

Let $s$ be any arc of the unit circle lying entirely in the first quadrant. Let $A$ be the area of the region lying below $s$ and above the $x$-axis and let $B$ be the area of the region lying to the right of the $y$-axis and to the left of $s$. Prove that $A+B$ depends only on the arc length, and not on the position, of $s$.

[b]p1.[/b] The three little pigs are learning about fractions. They particularly like the number x = $1/5$, because when they add the denominator to the numerator, add the denominator to the denominator, and form a new fraction, they obtain $6/10$, which equals $3x$ (so each little pig can have his own $x$). The $101$ Dalmatians hear about this and want their own fraction. Your job is to help them. (a) Find a fraction $y$ such that when the denominator is added to the numerator and also added to the denominator, the result is $101y$. (b) Prove that the fraction $y$ (put into lowest terms) in part (a) is the only fraction in lowest terms with this property. [b]p2.[/b] A small kingdom consists of five square miles. The king, who is not very good at math, wants to divide the kingdom among his $9$ sons. He tells each son to mark out a region of $1$ square mile. Prove that there are two sons whose regions overlap by at least $1/9$ square mile. [b]p3.[/b] Let $\pi (n)$ be the number of primes less than or equal to n. Sometimes $n$ is a multiple of $\pi (n)$. It is known that $\pi (4) = 2$ (because of the two primes $2, 3$) and $\pi (64540) = 6454$. Show that there exists an integer $n$, with $4 < n < 64540$, such that $\pi (n) = n/8$. [b]p4.[/b] Two circles of radii $R$ and $r$ are externally tangent at a point $A$. Their common external tangent is tangent to the circles at $B$ and $C$. Calculate the lengths of the sides of triangle $ABC$ in terms of $R$ and $r$. [img]https://cdn.artofproblemsolving.com/attachments/e/a/e5b79cb7c41e712602ec40edc037234468b991.png[/img] [b]p5.[/b] There are $2005$ people at a meeting. At the end of the meeting, each person who has shaken hands with at most $10$ people is given a red T-shirt with the message “I am unfriendly.” Then each person who has shaken hands only with people who received red T-shirts is given a blue T-shirt with the message “All of my friends are unfriendly.” (Some lucky people might get both red and blue T-shirts, for example, those who shook no one’s hand.) Prove that the number of people who received blue T-shirts is less than or equal to the number of people who received red T-shirts. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $M$ be the midpoint of side $BC$ of $\triangle ABC$. Using the [i]smallest possible[/i] $n$, described a method for cutting $\triangle AMB$ into $n$ triangles which can be reassembled to form a triangle congruent to $\triangle AMC$.

How many divisors of $12!$ are perfect squares? [i]2015 CCA Math Bonanza Lightning Round #4.1[/i]

For each lattice point on the Cartesian Coordinate Plane, one puts a positive integer at the lattice such that [b]for any given rectangle with sides parallel to coordinate axes, the sum of the number inside the rectangle is not a prime. [/b] Find the minimal possible value of the maximum of all numbers.

There are $168$ primes below $1000$. Then sum of all primes below $1000$ is, $\textbf{(A)}~11555$ $\textbf{(B)}~76127$ $\textbf{(C)}~57298$ $\textbf{(D)}~81722$

Pete wrote down $21$ pairwise distinct positive integers, each not greater than $1,000,000$. For every pair $(a, b)$ of numbers written down by Pete, Nick wrote the number $$F(a;b)=a+b -\gcd(a;b)$$ on his piece of paper. Prove that one of Nick’s numbers differs from all of Pete’s numbers.

Given positive integers $a,b$, let the integers $q,r$, with $0 \leq r < ab$, be such that $a^2 + b^2 = abq + r$. Prove that $q + r \leq ab + 1$ and find all equality cases.

Each entry in a $2000\times 2000$ array is $0$, $1$, or $-1$. Show that it's possible for all $4000$ row sums and column sums to be distinct.

Triangle $ABC$ has an obtuse angle at $\angle A$. Points $D$ and $E$ are placed on $\overline{BC}$ in the order $B$, $D$, $E$, $C$ such that $\angle BAD=\angle BCA$ and $\angle CAE=\angle CBA$. If $AB=10$, $AC=11$, and $DE=4$, determine $BC$.

let $n>2$ be a fixed integer.positive reals $a_i\le 1$(for all $1\le i\le n$).for all $k=1,2,...,n$,let $A_k=\frac{\sum_{i=1}^{k}a_i}{k}$ prove that $|\sum_{k=1}^{n}a_k-\sum_{k=1}^{n}A_k|<\frac{n-1}{2}$.

Function $f$ is defined on the positive integers by $f(1)=1$, $f(2)=2$ and $$f(n+2)=f(n+2-f(n+1))+f(n+1-f(n))\enspace\text{for }n\ge1.$$ (a) Prove that $f(n+1)-f(n)\in\{0,1\}$ for each $n\ge1$. (b) Show that if $f(n)$ is odd then $f(n+1)=f(n)+1$. (c) For each positive integer $k$ find all $n$ for which $f(n)=2^{k-1}+1$.

Consider an acute triangle $ABC$ and it's circumcircle $\omega$. With center $A$, we construct a circle $\gamma$ that intersects arc $AB$ of circle $\omega$ , that doesn't contain $C$, at point $D$ and arc $AC$ , that doesn't contain $B$, at point $E$. Suppose that the intersection point $K$ of lines $BE$ and $CD$ lies on circle $\gamma$. Prove that line $AK$ is perpendicular on line $BC$.

At a certain grocery store, cookies may be bought in boxes of $10$ or $21.$ What is the minimum positive number of cookies that must be bought so that the cookies may be split evenly among $13$ people? [i]Author: Ray Li[/i]

If $a$ and $b$ are positive real numbers such that $a \cdot 2^b=8$ and $a^b=2,$ compute $a^{\log_2 a} 2^{b^2}.$

Let $f$ be a non-constant function from the set of positive integers into the set of positive integer, such that $a-b$ divides $f(a)-f(b)$ for all distinct positive integers $a$, $b$. Prove that there exist infinitely many primes $p$ such that $p$ divides $f(c)$ for some positive integer $c$. [i]Proposed by Juhan Aru, Estonia[/i]

For all reals $a, b, c,d $ prove the following inequality: $$\frac{a + b + c + d}{(1 + a^2)(1 + b^2)(1 + c^2)(1 + d^2)}< 1$$

Given real $a$. Find the least possible area of the rectangle with the sides parallel to the coordinate axes and containing the figure determined by the system of inequalities $$y \le -x^2 \,\,\, and \,\,\, y \ge x^2 - 2x + a$$

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 $S$.

Show that there are no integers $a,b,c$ for which $a^2+b^2-8c=6$.

Let $k\ge 2$ be an integer and let $a_1 ,a_2 ,\cdots ,a_n,b_1 ,b_2 ,\cdots ,b_n$ be non-negative real numbers. Prove that\[\left(\frac{n}{n-1}\right)^{n-1}\left(\frac{1}{n} \sum_{i\equal{}1}^{n} a_i^2\right)+\left(\frac{1}{n} \sum_{i\equal{}1}^{n} b_i\right)^2\ge\prod_{i=1}^{n}(a_i^{2}+b_i^{2})^{\frac{1}{n}}.\]

Let $a_1,a_2,a_3,\ldots$ be a sequence of integers, with the property that every consecutive group of $a_i$'s averages to a perfect square. More precisely, for every positive integers $n$ and $k$, the quantity \[\frac{a_n+a_{n+1}+\cdots+a_{n+k-1}}{k}\] is always the square of an integer. Prove that the sequence must be constant (all $a_i$ are equal to the same perfect square). [i]Evan O'Dorney and Victor Wang[/i]

Find all function $f:\mathbb R^+ \rightarrow \mathbb R^+$ such that: \[f\left(\frac{f(x)}{x}+y\right)=1+f(y), \quad \forall x,y \in \mathbb R^+.\]

Consider a grid with $n{}$ lines and $m{}$ columns $(n,m\in\mathbb{N},m,n\ge2)$ made of $n\cdot m \; 1\times1$ squares called ${cells}$. A ${snake}$ is a sequence of cells with the following properties: the first cell is on the first line of the grid and the last cell is on the last line of the grid, starting with the second cell each has a common side with the previous cell and is not above the previous cell. Define the ${length}$ of a snake as the number of cells it's made of. Find the arithmetic mean of the lengths of all the snakes from the grid.