For $r> 0$ denote by $B_r$ the set of points at distance at most $r$ length units from the origin. If $P_r$ is the set of the points in $B_r$ whit integer coordinates, show that the equation $$xy^3z + 2x^3z^3-3x^5y = 0$$ has an odd number of solutions $(x, y, z)$ in $P_r$.

What is the sum of all two-digit odd numbers whose digits are all greater than $6$?

[u]Round 1[/u] [b]p1.[/b] What is $2 + 22 + 1 + 3 - 31 - 3$? [b]p2.[/b] Let $ABCD$ be a rhombus. Given $AB = 5$, $AC = 8$, and $BD = 6$, what is the perimeter of the rhombus? [b]p3.[/b] There are $2$ hats on a table. The first hat has $3$ red marbles and 1 blue marble. The second hat has $2$ red marbles and $4$ blue marbles. Jordan picks one of the hats randomly, and then randomly chooses a marble from that hat. What is the probability that she chooses a blue marble? [u]Round 2[/u] [b]p4.[/b] There are twelve students seated around a circular table. Each of them has a slip of paper that they may choose to pass to either their clockwise or counterclockwise neighbor. After each person has transferred their slip of paper once, the teacher observes that no two students exchanged papers. In how many ways could the students have transferred their slips of paper? [b]p5.[/b] Chad wants to test David's mathematical ability by having him perform a series of arithmetic operations at lightning-speed. He starts with the number of cubic centimeters of silicon in his 3D printer, which is $109$. He has David perform all of the following operations in series each second: $\bullet$ Double the number $\bullet$ Subtract $4$ from the number $\bullet$ Divide the number by $4$ $\bullet$ Subtract $5$ from the number $\bullet$ Double the number $\bullet$ Subtract $4$ from the number Chad instructs David to shout out after three seconds the result of three rounds of calculations. However, David computes too slowly and fails to give an answer in three seconds. What number should David have said to Chad? [b]p6.[/b] Points $D, E$, and $F$ lie on sides $BC$, $CA$, and $AB$ of triangle $ABC$, respectively, such that the following length conditions are true: $CD = AE = BF = 2$ and $BD = CE = AF = 4$. What is the area of triangle $ABC$? [u]Round 3[/u] [b]p7.[/b] In the $2, 3, 5, 7$ game, players count the positive integers, starting with $1$ and increasing, which do not contain the digits $2, 3, 5$, and $7$, and also are not divisible by the numbers $2, 3, 5$, and $7$. What is the fifth number counted? [b]p8.[/b] If A is a real number for which $19 \cdot A = \frac{2014!}{1! \cdot 2! \cdot 2013!}$ , what is $A$? Note: The expression $k!$ denotes the product $k \cdot (k - 1) \cdot ...\cdot 2 \cdot 1$. [b]p9.[/b] What is the smallest number that can be written as both $x^3 + y^2$ and $z^3 + w^2$ for positive integers $x, y, z,$ and $w$ with $x \ne z$? [u]Round 4[/u] [i]Each of the three problems in this round depends on the answer to one of the other problems. There is only one set of correct answers to these problems; however, each problem will be scored independently, regardless of whether the answers to the other problems are correct. In addition, it is given that the answer to each of the following problems is a positive integer less than or equal to the problem number. [/i] [b]p10.[/b] Let $B$ be the answer to problem $11$ and let $C$ be the answer to problem $12$. What is the sum of a side length of a square with perimeter $B$ and a side length of a square with area $C$? [b]p11.[/b] Let $A$ be the answer to problem $10$ and let $C$ be the answer to problem $12$. What is $(C - 1)(A + 1) - (C + 1)(A - 1)$? [b]p12.[/b] Let $A$ be the answer to problem $10$ and let $B$ be the answer to problem $11$. Let $x$ denote the positive difference between $A$ and $B$. What is the sum of the digits of the positive integer $9x$? PS. You should use hide for answers. Rounds 5-8 have been posted [url=https://artofproblemsolving.com/community/c3h2915810p26040675]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Consider the sequence $1, 2, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 1, ...$ Find $n$ such that the first $n$ terms sum up to $2010.$

Let $ABCD$ be a trapezoid with bases $AB$ and $CD$. Points $P$ and $Q$ lie on diagonals $AC$ and $BD$, respectively and $\angle APD = \angle BQC$. Prove that $\angle AQD = \angle BPC$.

Prove that the equation $4^x +6^x =9^x$ has no rational solutions.

A rational number $ x$ is called [i]good[/i] if it satisfies: $ x\equal{}\frac{p}{q}>1$ with $ p$, $ q$ being positive integers, $ \gcd (p,q)\equal{}1$ and there exists constant numbers $ \alpha$, $ N$ such that for any integer $ n\geq N$, \[ |\{x^n\}\minus{}\alpha|\leq\dfrac{1}{2(p\plus{}q)}\] Find all the good numbers.

If $a,b,c$ are triangle sides then prove that $(\sum_{cyc}\sqrt{\frac{a}{-a+b+c}} \geq 3$

Let $ABCDE$ be a convex pentagon such that $AB=BC=CD$, $\angle{EAB}=\angle{BCD}$, and $\angle{EDC}=\angle{CBA}$. Prove that the perpendicular line from $E$ to $BC$ and the line segments $AC$ and $BD$ are concurrent.

Each point on the boundary of a square has to be colored in one color. Consider all right triangles with the vertices on the boundary of the square. Determine the least number of colors for which there is a coloring such that no such triangle has all its vertices of the same color.

Suppose $a_1,a_2,\ldots,a_n>0 $, where $n>1$ and $\sum_{i=1}^{n}a_i=1$. For each $i=1,2,\ldots,n $, let $b_i=\frac{a^2_i}{\sum\limits_{j=1}^{n}a^2_j}$. Prove that \[\sum_{i=1}^{n}\frac{a_i}{1-a_i}\le \sum_{i=1}^{n}\frac{b_i}{1-b_i} .\] When does equality occur ?

Each of the quadratic polynomials $P(x), Q(x)$ and $P(x)+Q(x)$ with real coefficients has a repeated root. Is it guaranteed that those roots coincide? [i]Boris Frenkin[/i]

(a) Show that if $f:[-1,1]\to \mathbb{R}$ is a convex and $C^2$ function such that $f(1),f(-1)\geq 0$, then: \[\min_{x\in[-1,1]} \{f(x)\} \geq - \int_{-1}^1 f''\] (b) Let $B\subset \mathbb{R}^2$ the closed ball with center $0$ and radius $1$. Show that if $f: B \to \mathbb{R}$ is a convex and $C^2$ function and $f\geq 0$ in $\partial B$, then: \[f(0)\geq -\frac{1}{\sqrt{\pi}} \left( \int_{B} (f_{xx}f_{yy}-f_{xy}^2) \right)^{1/2}\]

A [i]lattice point[/i] in the plane is a point with integer coordinates. Let $T$ be a triangle in the plane whose vertice are lattice points, but with no other lattice points on its sides. Furthermore, suppose $T$ contains exactly four lattice points in its interior. Prove that these four points lie on a straight line.

An angle bisector $AD$ was drawn in triangle $ABC$. It turned out that the center of the inscribed circle of triangle $ABC$ coincides with the center of the inscribed circle of triangle $ABD$. Find the angles of the original triangle.

Let $n$ be a positive integer greater than $1$ and let us call an arbitrary set of cells in a $n\times n$ square $\textit{good}$ if they are the intersection cells of several rows and several columns, such that none of those cells lie on the main diagonal. What is the minimum number of pairwise disjoint $\textit{good}$ sets required to cover the entire table without the main diagonal?

There are $n$ boys and $n$ girls sitting around a circular table, where $n>3$. In every move, we are allowed to swap the places of $2$ adjacent children. The [b]entropy[/b] of a configuration is the minimal number of moves such that at the end of them each child has at least one neighbor of the same gender. Find the maximal possible entropy over the set of all configurations. [i]Authored by Viktor Simjanoski[/i]

Let $ n$ be a given positive integer. Find the smallest positive integer $ u_n$ such that for any positive integer $ d$, in any $ u_n$ consecutive odd positive integers, the number of them that can be divided by $ d$ is not smaller than the number of odd integers among $ 1, 3, 5, \ldots, 2n \minus{} 1$ that can be divided by $ d$.

There are $40$ points on the two parallel lines. We divide it to pairs, such that line segments, that connects point in pair, do not intersect each other ( endpoint from one segment cannot lies on another segment). Prove, that number of ways to do it is less than $3^{39}$

Let $P(x)$ be a polynomial of degree $5$ and suppose that a and b are real numbers different from zero. Suppose the remainder when $P(x)$ is divided by $x^3 + ax + b$ equals the remainder when $P(x)$ is divided by $x^3 + ax^2 + b$. Then determine $a + b$.

Let $ f$ be a polynomial of degree 2 with integer coefficients. Suppose that $ f(k)$ is divisible by 5 for every integer $ k$. Prove that all coefficients of $ f$ are divisible by 5.

Prove that the intersection of a plane and a regular tetrahedron can be an obtuse-angled triangle and that the obtuse angle in any such triangle is always smaller than $ 120^{\circ}.$

The incircle of the triangle $ABC$, with the center $I$ , touches the sides $AB, AC$ and $BC$ in the points $K, L$ and $M$ respectively. Points $P$ and $Q$ are taken on the sides $AC$ and $BC$ respectively, such that $|AP| = |CL|$ and $|BQ| = |CM|$. Prove that the difference of areas of the figures $APIQB$ and $CPIQ$ is equal to the area of the quadrangle $CLIM$

a) Is it true that for every bijection $f:\mathbb N\to\mathbb N$ the series $$\sum_{n=1}^\infty\frac1{nf(n)}$$is convergent? b) Prove that there exists a bijection $f:\mathbb N\to\mathbb N$ such that the series $$\sum_{n=1}^\infty\frac1{n+f(n)}$$is convergent. ($\mathbb N$ is the set of all positive integers.)

Find all nonnegative solutions $(x,y,z)$ to the system $$\begin{cases} \sqrt{x+y}+\sqrt{z} = 7 \\ \sqrt{x+z}+\sqrt{y} = 7 \\ \sqrt{y+z}+\sqrt{x} = 5 \end{cases}$$