We call a coloring of an $8\times 8$ board in three colors good if in any corner of five cells contains cells of all three colors. (A five-square corner is a shape made from a $3 \times 3$ square by cutting square $ 2\times 2$.) Prove that the number of good colorings is not less than than $68$.

Determine all triples $(a, b, c)$ of positive integers such that $$a! + b! = 2^{c!}.$$ [i](Walther Janous)[/i]

Fill in each cell of the grid with one of the numbers 1, 2, or 3. After all numbers are filled in, if a row, column, or any diagonal has a number of cells equal to a multiple of 3, then it must have the same amount of 1’s, 2’s, and 3’s. (There are 10 such diagonals, and they are all marked in the grid by a gray dashed line.) Some numbers have been given to you. [asy] defaultpen(linewidth(0.45)); real[][] arr = { {0, 2, 1, 0, 0, 0, 0, 0, 0}, {3, 0, 0, 2, 0, 0, 0, 0, 0}, {0, 0, 0, 2, 0, 0, 3, 2, 0}, {0, 2, 1, 0, 0, 1, 0, 0, 3}, {3, 0, 0, 0, 0, 3, 0, 0, 3}, {2, 0, 0, 0, 0, 0, 2, 3, 0}, {3, 2, 3, 2, 0, 2, 0, 0, 3}, {0, 0, 0, 0, 0, 3, 0, 0, 1}, {0, 0, 0, 0, 0, 0, 1, 3, 0}}; for (int i=0; i<9; ++i){ for (int j=0; j<9; ++j){ draw((i,j)--(i+1,j)--(i+1, j+1)--(i,j+1)--cycle); if(arr[8-j][i] != 0){ label((string) arr[8-j][i], (i+0.5, j+0.5)); } } } draw((3,0)--(0,3), linetype(new real[] {4,4})+grey); draw((6,0)--(0,6), linetype(new real[] {4,4})+grey); draw((9,0)--(0,9), linetype(new real[] {4,4})+grey); draw((3,9)--(9,3), linetype(new real[] {4,4})+grey); draw((6,9)--(9,6), linetype(new real[] {4,4})+grey); draw((6,0)--(9,3), linetype(new real[] {4,4})+grey); draw((3,0)--(9,6), linetype(new real[] {4,4})+grey); draw((0,0)--(9,9), linetype(new real[] {4,4})+grey); draw((0,3)--(6,9), linetype(new real[] {4,4})+grey); draw((0,6)--(3,9), linetype(new real[] {4,4})+grey); [/asy] You do not need to prove that your answer is the only one possible; you merely need to find an answer that satisfies the constraints above. (Note: In any other USAMTS problem, you need to provide a full proof. Only in this problem is an answer without justification acceptable.)

One triangular face $F$ of a tetrahedron $\mathcal{T}$ has side lengths $\sqrt{5}$, $\sqrt{65}$, and $2\sqrt{17}$. The other three faces of $\mathcal{T}$ are right triangles whose hypotenuses coincide with the sides of $F$. There exists a sphere inside $\mathcal{T}$ tangent to all four of its faces. Compute the radius of this sphere.

Through a point located on a side of a triangle of area $1$, two straight lines are drawn parallel to the two remaining sides. They divided the triangle into three parts. Let $s$ be the largest of the areas of these parts. Find the smallest possible value of $s$.

If $x\in\left[-\frac{5\pi}{12},-\frac{\pi}{3}\right]$, then the maximum value of $y=\tan\left(x+\frac{2\pi}{3}\right)-\tan\left(x+\frac{\pi}{6}\right)+\cos\left(x+\frac{\pi}{6}\right)$ is $\text{(A)}\frac{12}{5}\sqrt2\qquad\text{(B)}\frac{11}{6}\sqrt2\qquad\text{(C)}\frac{11}{6}\sqrt3\qquad\text{(D)}\frac{12}{5}\sqrt3$

If $x,y,z$ are positive numbers, prove that $$\frac x{\sqrt{y+z}}+\frac y{\sqrt{z+x}}+\frac z{\sqrt{x+y}}\ge\sqrt{\frac32(x+y+z)}.$$

A trio of lousy salespeople charge increasing prices on tomatoes as you buy more. The first charges you $x^1_1$ dollars for the $x_1$[i]th [/i]tomato you buy from him, the second charges $x^2_2$ dollars for the $x_2$[i]th[/i] tomato, and the third charges $x^3_3$ dollars for the $x_3$[i]th [/i]tomato. If you want to buy $100$ tomatoes for as cheap as possible, how many should you buy from the first salesperson?

Laura added two three-digit positive integers. All six digits in these numbers are different. Laura's sum is a three-digit number $S$. What is the smallest possible value for the sum of the digits of $S$? $\textbf{(A)}\ 1\qquad\textbf{(B)}\ 4\qquad\textbf{(C)}\ 5\qquad\textbf{(D)}\ 15\qquad\textbf{(E)}\ 21$

If there are several heaps of stones on the table, it is said that there are $\textit{many}$ stones on the table, if we can find $50$ piles and number them with the numbers from $1$ to $50$ so that the first pile contains at least one stone, the second - at least two stones,..., the $50$-th has at least $50$ stones. Let the table be initially contain $100$ piles of $100$ stones each. Find the largest $n \leq 10 000$ such that after removing any $n$ stones, there will still be $\textit{many}$ stones left on the table.

[b]p1.[/b] If $P$ is a (convex) polygon, a triangulation of $P$ is a set of line segments joining pairs of corners of $P$ in such a way that $P$ is divided into non-overlapping triangles, each of which has its corners at corners of $P$. For example, the following are different triangulations of a square. (a) Prove that if $P$ is an $n$-gon with $n > 3$, then every triangulation of $P$ produces at least two triangles $T_1$, $T_2$ such that two of the sides of $T_i$, $i = 1$ or $2$ are also sides of $P$. (b) Find the number of different possible triangulations of a regular hexagon. [img]https://cdn.artofproblemsolving.com/attachments/9/d/0f760b0869fafc882f293846c05d182109fb78.png[/img] [b]p2.[/b] There are $n$ students, $n \ge 2$, and $n + 1$ cubical cakes of volume $1$. They have the use of a knife. In order to divide the cakes equitably they make cuts with the knife. Each cut divides a cake (or a piece of a cake) into two pieces. (a) Show that it is possible to provide each student with a volume $(n + 1)/n$ of a cake while making no more than $n - 1$ cuts. (b) Show that for each integer $k$ with $2 \le k \le n$ it is possible to make $n - 1$ cuts in such a way that exactly $k$ of the $n$ students receive an entire (uncut) cake in their portion. [b]p3. [/b]The vertical lines at $x = 0$, $x = \frac12$ , $x = 1$, $x = \frac32$ ,$...$ and the horizontal lines at $y = 0$, $y = \frac12$ , $y = 1$, $y = \frac32$ ,$ ...$ subdivide the first quadrant of the plane into $\frac12 \times \frac12$ square regions. Color these regions in a checkerboard fashion starting with a black region near the origin and alternating black and white both horizontally and vertically. (a) Let $T$ be a rectangle in the first quadrant with sides parallel to the axes. If the width of $T$ is an integer, prove that $T$ has equal areas of black and white. Note that a similar argument works to show that if the height of $T$ is an integer, then $T$ has equal areas of black and white. (b) Let $R$ be a rectangle with vertices at $(0, 0)$, $(a, 0)$, $(a, b)$, and $(0, b)$ with $a$ and $b$ positive. If $R$ has equal areas of black and white, prove that either $a$ is an integer or that $b$ is an integer. (c) Suppose a rectangle $R$ is tiled by a finite number of rectangular tiles. That is, the rectangular tiles completely cover $R$ but intersect only along their edges. If each of the tiles has at least one integer side, prove that $R$ has at least one integer side. [b]p4.[/b] Call a number [i]simple [/i] if it can be expressed as a product of single-digit numbers (in base ten). (a) Find two simple numbers whose sum is $2014$ or prove that no such numbers exist. (b) Find a simple number whose last two digits are $37$ or prove that no such number exists. [b]p5.[/b] Consider triangles for which the angles $\alpha$, $\beta$, and $\gamma$ form an arithmetic progression. Let $a, b, c$ denote the lengths of the sides opposite $\alpha$, $\beta$, $\gamma$ , respectively. Show that for all such triangles, $$\frac{a}{c}\sin 2\gamma +\frac{c}{a} \sin 2\alpha$$ has the same value, and determine an algebraic expression for this value. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find all the pairs of real numbers $(x,y)$ that are solutions of the system: $(x^{2}+y^{2})^{2}-xy(x+y)^{2}=19 $ $| x - y | = 1$

You have a collection of small wooden blocks that are rectangular solids measuring $3$×$4$×$6$. Each of the six faces of each block is to be painted a solid color, and you have three colors of paint to use. Find the number of distinguishable ways you could paint the blocks. (Two blocks are distinguishable if you cannot rotate one block so that it looks identical to the other block.)

This year's gift idea from BabyMath consists of a series of nine colored plastic containers of decreasing size, alternating in shape like a cube and a sphere. All containers can open and close with a convenient hinge, and each container can hold just about anything next in line. The largest and smallest container are both cubes. Determine the relationship between the edge lengths of these cubes.

We have two piles with $2000$ and $2017$ coins respectively. Ann and Bob take alternate turns making the following moves: The player whose turn is to move picks a pile with at least two coins, removes from that pile $t$ coins for some $2\le t \le 4$, and adds to the other pile $1$ coin. The players can choose a different $t$ at each turn, and the player who cannot make a move loses. If Ann plays first determine which player has a winning strategy.

Call a natural number [i]simple[/i] if it is not divisible by any square of a prime number (in other words it is square-free). Prove that there are infinitely many positive integers $n$ such that both $n$ and $n+1$ are [i]simple[/i].

Prove that: there exists a positive constant $K$, and an integer series $\{a_n\}$, satisfying: $(1)$ $0<a_1<a_2<\cdots <a_n<\cdots $; $(2)$ For any positive integer $n$, $a_n<1.01^n K$; $(3)$ For any finite number of distinct terms in $\{a_n\}$, their sum is not a perfect square.

Let $a, b > 0$ be reals such that \[ a^3=a+1\\ b^6=b+3a \] Show that $a>b$

If $ \log_{10} (x^2\minus{}3x\plus{}6)\equal{}1$, the value of $ x$ is: $ \textbf{(A)}\ 10\text{ or }2 \qquad\textbf{(B)}\ 4\text{ or }\minus{}2 \qquad\textbf{(C)}\ 3\text{ or }\minus{}1 \qquad\textbf{(D)}\ 4\text{ or }\minus{}1\\ \textbf{(E)}\ \text{none of these}$

Let $ABC$ be a triangle and $D$ a point on $BC$ such that $AB =\sqrt2$, $AC =\sqrt3$, $\angle BAD = 30^o$, and $\angle CAD = 45^o$. Find $AD$.

Let $a$ and $b$ be two real numbers. Find these numbers given that the graphs of $f:\mathbb{R} \to \mathbb{R} , f(x)=2x^4-a^2x^2+b-1$ and $g:\mathbb{R} \to \mathbb{R} ,g(x)=2ax^3-1$ have exactly two points of intersection.

Does the bellow depicted figure fit into a square $5\times5$.

Suppose there are $n\geq{5}$ different points arbitrarily arranged on a circle, the labels are $1, 2,\dots $, and $n$, and the permutation is $S$. For a permutation , a “descending chain” refers to several consecutive points on the circle , and its labels is a clockwise descending sequence (the length of sequence is at least $2$), and the descending chain cannot be extended to longer .The point with the largest label in the chain is called the "starting point of descent", and the other points in the chain are called the “non-starting point of descent” . For example: there are two descending chains $5, 2$and $4, 1$ in $5, 2, 4, 1, 3$ arranged in a clockwise direction, and $5$ and $4$ are their starting points of descent respectively, and $2, 1$ is the non-starting point of descent . Consider the following operations: in the first round, find all descending chains in the permutation $S$, delete all non-starting points of descent , and then repeat the first round of operations for the arrangement of the remaining points, until no more descending chains can be found. Let $G(S)$ be the number of all descending chains that permutation $S$ has appeared in the operations, $A(S)$ be the average value of $G(S)$of all possible n-point permutations $S$. (1) Find $A(5)$. (2)For $n\ge{6}$ , prove that $\frac{83}{120}n-\frac{1}{2} \le A(S) \le \frac{101}{120}n-\frac{1}{2}.$

Let $a$ and $b$ be real numbers greater than $1$ for which there exists a positive real number $c$, different from $1$, such that \[2(\log_ac+\log_bc)=9\log_{ab}c.\] Find the largest possible value of $\log_ab$. $\textbf{(A) }\sqrt2\qquad\textbf{(B) }\sqrt3\qquad\textbf{(C) }2\qquad\textbf{(D) }\sqrt6\qquad\textbf{(E) }3$

The right triangles $ABC$ and $AB_1C_1$ are similar and have opposite orientation. The right angles are at $C$ and $C_1$, and we also have $ \angle CAB = \angle C_1AB_1$. Let $M$ be the point of intersection of the lines $BC_1$ and $B_1C$. Prove that if the lines $AM$ and $CC_1$ exist, they are perpendicular.