Let $\theta=\frac{2\pi}{2015}$, and suppose the product \[\prod_{k=0}^{1439}\left(\cos(2^k\theta)-\frac{1}{2}\right)\] can be expressed in the form $\frac{b}{2^a}$, where $a$ is a non-negative integer and $b$ is an odd integer (not necessarily positive). Find $a+b$. [i]2017 CCA Math Bonanza Tiebreaker Round #3[/i]

A number is said to be [i]balanced [/i] if one of its digits is average of the others. How many [i]balanced [/i]$3$-digit numbers are there?

The diagonals of a trapezoid are perpendicular, and its altitude is equal to the medial line. Prove that this trapezoid is isosceles

Welcome to the [b]USAYNO[/b], where each question has a yes/no answer. Choose any subset of the following six problems to answer. If you answer $n$ problems and get them [b]all[/b] correct, you will receive $\max(0, (n-1)(n-2))$ points. If any of them are wrong (or you leave them all blank), you will receive $0$ points. Your answer should be a six-character string containing 'Y' (for yes), 'N' (for no), or 'B' (for blank). For instance if you think 1, 2, and 6 are 'yes' and 3 and 4 are 'no', you should answer YYNNBY (and receive $12$ points if all five answers are correct, 0 points if any are wrong). (a) $a,b,c,d,A,B,C,$ and $D$ are positive real numbers such that $\frac{a}{b} > \frac{A}{B}$ and $\frac{c}{d} > \frac{C}{D}$. Is it necessarily true that $\frac{a+c}{b+d} > \frac{A+C}{B+D}$? (b) Do there exist irrational numbers $\alpha$ and $\beta$ such that the sequence $\lfloor\alpha\rfloor+\lfloor\beta\rfloor, \lfloor2\alpha\rfloor+\lfloor2\beta\rfloor, \lfloor3\alpha\rfloor+\lfloor3\beta\rfloor, \dots$ is arithmetic? (c) For any set of primes $\mathbb{P}$, let $S_\mathbb{P}$ denote the set of integers whose prime divisors all lie in $\mathbb{P}$. For instance $S_{\{2,3\}}=\{2^a3^b \; | \; a,b\ge 0\}=\{1,2,3,4,6,8,9,12,\dots\}$. Does there exist a finite set of primes $\mathbb{P}$ and integer polynomials $P$ and $Q$ such that $\gcd(P(x), Q(y))\in S_\mathbb{P}$ for all $x,y$? (d) A function $f$ is called [b]P-recursive[/b] if there exists a positive integer $m$ and real polynomials $p_0(n), p_1(n), \dots, p_m(n)$[color = red], not all zero,[/color] satisfying \[p_m(n)f(n+m)=p_{m-1}(n)f(n+m-1)+\dots+p_0(n)f(n)\] for all $n$. Does there exist a P-recursive function $f$ satisfying $\lim_{n\to\infty} \frac{f(n)}{n^{\sqrt{2}}}=1$? (e) Does there exist a [b]nonpolynomial[/b] function $f: \mathbb{Z}\to\mathbb{Z}$ such that $a-b$ divides $f(a)-f(b)$ for all integers $a\neq b$? (f) Do there exist periodic functions $f, g:\mathbb{R}\to\mathbb{R}$ such that $f(x)+g(x)=x$ for all $x$? [color = red]A clarification was issued for problem 33(d) during the test. I have included it above.[/color]

Let $A,B\in\mathcal{M}_3(\mathbb{R})$ de matrices such that $A^2+B^2=O_3.$ Prove that $\det(aA+bB)=0$ for any real numbers $a$ and $b.$

For each real number $x$, let $f(x)$ be the minimum of the numbers $4x+1$, $x+2$, and $-2x+4$. Then the maximum value of $f(x)$ is $\text{(A)} \ \frac 13 \qquad \text{(B)} \ \frac 12 \qquad \text{(C)} \ \frac 23 \qquad \text{(D)} \ \frac 52 \qquad \text{(E)} \ \frac 83$

A lame rook lies on a $9\times 9$ chessboard. It can move one cell horizontally or vertically. The rook made $n{}$ moves, visited each cell at most once, and did not make two moves consecutively in the same direction. What is the largest possible value of $n{}$? [i]From the folklore[/i]

Let $ABCD$ be a cyclic quadrilateral which is not a trapezoid and whose diagonals meet at $E$. The midpoints of $AB$ and $CD$ are $F$ and $G$ respectively, and $\ell$ is the line through $G$ parallel to $AB$. The feet of the perpendiculars from E onto the lines $\ell$ and $CD$ are $H$ and $K$, respectively. Prove that the lines $EF$ and $HK$ are perpendicular.

Evaluate $\sum_{n=1}^{\infty}\frac{1}{(2n - 1)(3n - 1)}$.

Is it possible to complete the following square knowning that each row and column make an aritmetic progression?

Let $a,b,c\in \mathbb{R}_{\ge 0}$ satisfy $a+b+c=1$. Prove the inequality : \[ \frac{3-b}{a+1}+\frac{a+1}{b+1}+\frac{b+1}{c+1}\ge 4 \]

Solve the equation $(x + 1)^3(x - 2)^3 + (x -1)^3(x + 2)^3 = 8(x^2 -2)^3.$

The numbers from $1$ to $2025$ are arranged in some order in the cells of the $1 \times 2025$ strip. Let's call a [i]flip[/i] an operation that takes two arbitrary cells of a strip and swaps the numbers written in them, but only if the larger of these numbers is located to the left of the smaller one. A [i]flop[/i] is a set of several flips that do not contain common cells that are executed simultaneously. (For example, a simultaneous flip between the 2nd and 8th cells and a flip between the 5th and 101st cells.) Prove that there exists a sequence of $66$ flops such that for any initial arrangement, applying this sequence of flops to it will result in the numbers being ordered from left to right in ascending order.

In the right triangle shown the sum of the distances $ BM$ and $ MA$ is equal to the sum of the distances $ BC$ and $ CA$. If $ MB \equal{} x$, $ CB \equal{} h$, and $ CA \equal{} d$, then $ x$ equals: [asy]size(200); defaultpen(linewidth(.8pt)+fontsize(10pt)); dotfactor=4; draw((0,0)--(8,0)--(0,5)--cycle); label("C",(0,0),SW); label("A",(8,0),SE); label("M",(0,5),N); dot((0,3.5)); label("B",(0,3.5),W); label("$x$",(0,4.25),W); label("$h$",(0,1),W); label("$d$",(4,0),S);[/asy]$ \textbf{(A)}\ \frac {hd}{2h \plus{} d} \qquad \textbf{(B)}\ d \minus{} h \qquad \textbf{(C)}\ \frac {1}{2}d \qquad \textbf{(D)}\ h \plus{} d \minus{} \sqrt {2d} \qquad \textbf{(E)}\ \sqrt {h^2 \plus{} d^2} \minus{} h$

In the movie ”Prison break $4$”. Michael Scofield has to break into The Company. There, he encountered a kind of code to protect Scylla from being taken away. This code require picking out every number in a $2015\times 2015$ grid satisfying: i) The number right above of this number is $\equiv 1 \mod 2$ ii) The number right on the right of this number is $\equiv 2 \mod 3$ iii) The number right below of this number is $\equiv 3 \mod 4$ iv) The number right on the right of this number is $\equiv 4 \mod 5$ . How many number does Schofield have to choose? Also, in a $n\times n$ grid, the numbers from $ 1$ to $n^2$ are arranged in the following way : On the first row, the numbers are written in an ascending order $1, 2, 3, 4, ..., n$, each cell has one number. On the second row, the number are written in descending order $2n, 2n -1, 2n- 2, ..., n + 1$. On the third row, it is ascending order again $2n + 1, 2n + 2, ..., 3n$. The numbers are written like that until $n$th row. For example, this is how a $3$ $\times$ $3$ board looks like:[img]https://cdn.artofproblemsolving.com/attachments/8/7/0a5c8aba6543fd94fd24ae4b9a30ef8a32d3bd.png[/img]

What is the last digit of $2021^{2021}$? [i]Proposed by Yifan Kang[/i]

$a_1,…,a_n$ are real numbers such that $a_1+…+a_n=0$ and $|a_1|+…+|a_n|=1$. Prove that : $$|a_1+2a_2+…+na_n| \leq \frac{n-1}{2}$$

Prove that there exists a positive integer $n < 10^6$ such that $5^n$ has six consecutive zeros in its decimal representation. [i]Proposed by Evan Chen[/i]

For a natural number $p$, one can move between two points with integer coordinates if the distance between them equals $p$. Find all prime numbers $p$ for which it is possible to reach the point $(2002,38)$ starting from the origin $(0,0)$.

Given a prime number $p{}$ and integers $x{}$ and $y$, find the remainder of the sum $x^0y^{p-1}+x^1y^{p-2}+\ldots+x^{p-2}y^1+x^{p-1}y^0$ upon division by $p{}$.

Given positive reals $a,b,c;$ show that we have \[\left(a+\frac 1b\right)\left(b+\frac 1c\right)\left(c+\frac 1a\right)\geq 8.\]

Which of the following is true if $S = \dfrac 1{1^2} + \dfrac 1{2^2} + \dfrac 1{3^2} + \cdots + \dfrac 1{2001^2} + \dfrac 1{2002^2}$? $ \textbf{a)}\ 1\leq S < \dfrac 43 \qquad\textbf{b)}\ \dfrac 43 \leq S < 2 \qquad\textbf{c)}\ 2 \leq S < \dfrac 73$ $\textbf{d)}\ \dfrac 73 \leq S < \dfrac 52 \qquad\textbf{e)}\ \dfrac 52 \leq S < 3 $

[b]p1.[/b] Solve the equation $$\frac{\sqrt{x^2 - 2x + 1}}{x^2 - 1}+\frac{x^2 - 1}{\sqrt{x^2 - 2x + 1}}=\frac52.$$ [b]p2.[/b] Solve the inequality: $\frac{1 - 2\sqrt{1-x^2}}{x} \le 1$. [b]p3.[/b] Let $ABCD$ be a convex quadrilateral such that the length of the segment connecting midpoints of the two opposite sides $AB$ and $CD$ equals $\frac{|AD| + |BC|}{2}$. Prove that $AD$ is parallel to $BC$. [b]p4.[/b] Solve the equation: $\frac{1}{\cos x}+\frac{1}{\sin x}= 2\sqrt2$. [b]p5.[/b] Long, long ago, far, far away there existed the Old Republic Galaxy with a large number of stars. It was known that for any four stars in the galaxy there existed a point in space such that the distance from that point to any of these four stars was less than or equal to $R$. Master Yoda asked Luke Skywalker the following question: Must there exist a point $P$ in the galaxy such that all stars in the galaxy are within a distance $R$ of the point $P$? Give a justified argument that will help Like answer Master Yoda’s question. [b]p6.[/b] The Old Republic contained an odd number of inhabited planets. Some pairs of planets were connected to each other by space flights of the Trade Federation, and some pairs of planets were not connected. Every inhabited planet had at least one connections to some other inhabited planet. Luke knew that if two planets had a common connection (they are connected to the same planet), then they have a different number of total connections. Master Yoda asked Luke if there must exist a planet that has exactly two connections. Give a justified argument that will help Luke answer Master Yoda’s question. PS. You should use hide for answers.

Find all functions $f:Q\rightarrow Q$ such that \[ f(x+y)+f(y+z)+f(z+t)+f(t+x)+f(x+z)+f(y+t)\ge 6f(x-3y+5z+7t) \] for all $x,y,z,t\in Q.$

In a regular 100-gon, 41 vertices are colored black and the remaining 59 vertices are colored white. Prove that there exist 24 convex quadrilaterals $Q_{1}, \ldots, Q_{24}$ whose corners are vertices of the 100-gon, so that [list] [*] the quadrilaterals $Q_{1}, \ldots, Q_{24}$ are pairwise disjoint, and [*] every quadrilateral $Q_{i}$ has three corners of one color and one corner of the other color. [/list]