In the octagon below all sides have the length $1$ and all angles are equal. Determine the distance between the corners $A$ and $B$. [img]https://1.bp.blogspot.com/-i6TAFDvcQ8w/XzXCRhnV_kI/AAAAAAAAMVw/rKrQMfPYYJIaCwl8hhdVHdqO4fIn8O7cwCLcBGAsYHQ/s0/2011%2BMogh%2Bp2.png[/img]

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) Can $1000$ queens be placed on a $2017\times2017$ chessboard such that every square is attacked by some queen? A square is attacked by a queen if it lies on the same row, column, or diagonal as the queen. (b) A $2017\times2017$ grid of squares originally contains a $0$ in each square. At any step, Kelvin the Frog choose two adjacent squares (two squares are adjacent if they share a side) and increments the numbers in both of them by $1$. Can Kelvin make every square contain a different power of $2$? (c) A [i]tournament[/i] consists of single games between every pair of players, where each game has a winner and a loser with no ties. A set of people is [i]dominated[/i] if there exists a player who beats all of them. Does there exist a tournament in which every set of $2017$ people is dominated? (d) Every cell of a $19\times19$ grid is colored either red, yellow, green, or blue. Does there necessarily exist a rectangle whose sides are parallel to the grid, all of whose vertices are the same color? (e) Does there exist a $c\in\mathbb{R}^+$ such that $\max(|A\cdot A|, |A+A|)\ge c|A|\log^2|A|$ for all finite sets $A\subset \mathbb{Z}$? (f) Can the set $\{1, 2, \dots, 1093\}$ be partitioned into $7$ subsets such that each subset is sum-free (i.e. no subset contains $a,b,c$ with $a+b=c$)? [color = red]The USAYNO disclaimer is only included in problem 33. I have included it here for convenience.[/color]

Does a parallelogram exist such that all pairwise meets of bisectors of its angles are situated outside it?

Suppose $f$ is a nonconstant polynomial with integer coefficients with the following property: [list] [*]$f(0)$ and $f(1)$ are both odd. [*]Define a sequence of integers with $a_k = f(1)f(2) \cdots f(k)+1$ [/list] Prove that there are infinitely many prime numbers dividing at least one element of the sequence. [i]Proposed by Sayandeep Shee[/i]

Let $A$ be a solid $a\times b\times c$ rectangular brick, where $a,b,c>0$. Let $B$ be the set of all points which are a distance of at most one from some point of $A$. Express the volume of $B$ as a polynomial in $a,b,$ and $c$.

Find the value of \[\sum_{a = 1}^{\infty} \sum_{b = 1}^{\infty} \sum_{c = 1}^{\infty} \frac{ab(3a + c)}{4^{a+b+c} (a+b)(b+c)(c+a)}.\]

Find all $f: \mathbb{N}\rightarrow \mathbb{N}$, for which $f(f(n)+m)=n+f(m+2014)$ for $\forall$ $m,n\in \mathbb{N}$.

Given the following real numbers $a. b, c $ greater than one that $a + b + c = 6$. Prove the inequality $$\frac{a}{b^2-1}+\frac{b}{c^2-1}+\frac{c}{a^2-1}\ge 2$$

The continuous function $f:[0,1]\rightarrow\mathbb{R}$ has the property: \[\lim_{x\rightarrow\infty}\ n\left(f\left(x+\frac{1}{n}\right)-f(x)\right)=0 \] for every $x\in [0,1)$. Show that: a) For every $\epsilon >0$ and $\lambda\in (0,1)$, we have: \[ \sup\ \{x\in[0,\lambda )\mid |f(x)-f(0)|\le \epsilon x \}=\lambda \] b) $f$ is a constant function.

A group of $6$ friends sit in the back row of an otherwise empty movie theater. Each row in the theater contains $8$ seats. Euler and Gauss are best friends, so they must sit next to each other, with no empty seat between them. However, Lagrange called them names at lunch, so he cannot sit in an adjacent seat to either Euler or Gauss. In how many different ways can the $6$ friends be seated in the back row? $\text{(A) }2520\qquad\text{(B) }3600\qquad\text{(C) }4080\qquad\text{(D) }5040\qquad\text{(E) }7200$

It is known that, for all positive integers $k,$ \[1^{2}+2^{2}+3^{2}+\cdots+k^{2}=\frac{k(k+1)(2k+1)}{6}. \]Find the smallest positive integer $k$ such that $1^{2}+2^{2}+3^{2}+\cdots+k^{2}$ is a multiple of $200.$

Let $\mathcal{F}$ be the set of the differentiable functions $f: \mathbb{R} \to \mathbb{R}$ satisfying $f(x) \ge f(x+ \sin x)$ for any $x \in \mathbb{R}.$ a) Prove that there exist nonconstant functions in $\mathcal{F}.$ b) Prove that if $f \in \mathcal{F},$ then the set of solutions of the equation $f'(x)=0$ is infinite.

Given trapezoid $ ABCD$ with parallel sides $ AB$ and $ CD$, assume that there exist points $ E$ on line $ BC$ outside segment $ BC$, and $ F$ inside segment $ AD$ such that $ \angle DAE \equal{} \angle CBF$. Denote by $ I$ the point of intersection of $ CD$ and $ EF$, and by $ J$ the point of intersection of $ AB$ and $ EF$. Let $ K$ be the midpoint of segment $ EF$, assume it does not lie on line $ AB$. Prove that $ I$ belongs to the circumcircle of $ ABK$ if and only if $ K$ belongs to the circumcircle of $ CDJ$. [i]Proposed by Charles Leytem, Luxembourg[/i]

The leader of an IMO team chooses positive integers $n$ and $k$ with $n > k$, and announces them to the deputy leader and a contestant. The leader then secretly tells the deputy leader an $n$-digit binary string, and the deputy leader writes down all $n$-digit binary strings which differ from the leader’s in exactly $k$ positions. (For example, if $n = 3$ and $k = 1$, and if the leader chooses $101$, the deputy leader would write down $001, 111$ and $100$.) The contestant is allowed to look at the strings written by the deputy leader and guess the leader’s string. What is the minimum number of guesses (in terms of $n$ and $k$) needed to guarantee the correct answer?

Consider rectangle $ABCD$.$ E$ is the mid-point of $AD$ and $F$ is the mid-point of $ED$. $CE$ cuts $AB$ in $G$ and $BF$ cuts $CD$ in $H$ point. We can write ratio of areas of $BCG$ and $BCH$ triangles as $\frac{m}{n}$. Find the value of $10m + 10n + mn$.

On a square board with $m$ rows and $n$ columns, where $m\le n$, some squares are colored black in such a way that no two rows are alike. Find tha biggest integer $k$ such that, for every possible coloring to start with, one can always color $k$ columns entirely red in such a way that still no two rows are alike.

Consider the sequence $(a_n)_{n\geqslant 1}$ defined by $a_1=1/2$ and $2n\cdot a_{n+1}=(n+1)a_n.$[list=a] [*]Determine the general formula for $a_n.$ [*]Let $b_n=a_1+a_2+\cdots+a_n.$ Prove that $\{b_n\}-\{b_{n+1}\}\neq \{b_{n+1}\}-\{b_{n+2}\}.$ [/list]

Show that the set of positive integers that cannot be represented as a sum of distinct perfect squares is finite.

For any positive integer $n,$ let $f(n)$ be the number of ordered triples $(a,b,c)$ of positive integers such that [list] [*] $\max(a,b,c)$ divides $n$ and [*] $\gcd(a,b,c)=1.$ [/list] Compute $f(1)+f(2)+\cdots+f(100).$

While waiting for their food at a restaurant in Harvard Square, Ana and Banana draw $3$ squares $\square_1, \square_2, \square_3$ on one of their napkins. Starting with Ana, they take turns filling in the squares with integers from the set $\{1,2,3,4,5\}$ such that no integer is used more than once. Ana's goal is to minimize the minimum value that the polynomial $a_1x^2 + a_2x + a_3$ attains over all real $x$, where $a_1, a_2, a_3$ are the integers written in $\square_1, \square_2, \square_3$ respectively. Banana aims to maximize $M$. Assuming both play optimally, compute the final value of $100a_1+10a_2+a_3$.

Find the remainder when $$\prod_{i = 1}^{1903} (2^i + 5)$$ is divided by $1000$.

On a Cartesian coordinate plane, points $(1, 2)$ and $(7, 4)$ are opposite vertices of a square. What is the area of the square?

Find the number of ordered triples of nonnegative integers $(a, b, c)$ that satisfy \[(ab + 1)(bc + 1)(ca + 1) = 84.\]

Let $n_1, n_2$ be positive integers. Consider in a plane $E$ two disjoint sets of points $M_1$ and $M_2$ consisting of $2n_1$ and $2n_2$ points, respectively, and such that no three points of the union $M_1 \cup M_2$ are collinear. Prove that there exists a straightline $g$ with the following property: Each of the two half-planes determined by $g$ on $E$ ($g$ not being included in either) contains exactly half of the points of $M_1$ and exactly half of the points of $M_2.$

Let $x_1 , \dots , x_{2024}$ be non negative real numbers with $\displaystyle{\sum_{i=1}^{2024}}x_i = 1$. Find, with proof, the minimum and maximum possible values of the following expression \[\sum_{i=1}^{1012} x_i + \sum_{i=1013}^{2024} x_i^2 .\]