Found problems: 4
2017 Harvard-MIT Mathematics Tournament, 34
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]
2017 Harvard-MIT Mathematics Tournament, 33
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]
2017 Harvard-MIT Mathematics Tournament, 35
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) Does there exist a finite set of points, not all collinear, such that a line between any two points in the set passes through a third point in the set?
(b) Let $ABC$ be a triangle and $P$ be a point. The [i]isogonal conjugate[/i] of $P$ is the intersection of the reflection of line $AP$ over the $A$-angle bisector, the reflection of line $BP$ over the $B$-angle bisector, and the reflection of line $CP$ over the $C$-angle bisector. Clearly the incenter is its own isogonal conjugate. Does there exist another point that is its own isogonal conjugate?
(c) Let $F$ be a convex figure in a plane, and let $P$ be the largest pentagon that can be inscribed in $F$. Is it necessarily true that the area of $P$ is at least $\frac{3}{4}$ the area of $F$?
(d) Is it possible to cut an equilateral triangle into $2017$ pieces, and rearrange the pieces into a square?
(e) Let $ABC$ be an acute triangle and $P$ be a point in its interior. Let $D,E,F$ lie on $BC, CA, AB$ respectively so that $PD$ bisects $\angle{BPC}$, $PE$ bisects $\angle{CPA}$, and $PF$ bisects $\angle{APB}$. Is it necessarily true that $AP+BP+CP\ge 2(PD+PE+PF)$?
(f) Let $P_{2018}$ be the surface area of the $2018$-dimensional unit sphere, and let $P_{2017}$ be the surface area of the $2017$-dimensional unit sphere. Is $P_{2018}>P_{2017}$?
[color = red]The USAYNO disclaimer is only included in problem 33. I have included it here for convenience.[/color]
2017 Harvard-MIT Mathematics Tournament, 36
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) Does $\sum_{i=1}^{p-1}\frac{1}{i}\equiv 0\pmod{p^2}$ for all odd prime numbers $p$? (Note that $\frac{1}{i}$ denotes the number such that $i\cdot\frac{1}{i}\equiv 1\pmod{p^2}$)
(b) Do there exist $2017$ positive perfect cubes that sum to a perfect cube?
(c) Does there exist a right triangle with rational side lengths and area $5$?
(d) A [i]magic square[/i] is a $3\times 3$ grid of numbers, all of whose rows, columns, and major diagonals sum to the same value. Does there exist a magic square whose entries are all [color = red]different[/color] prime numbers?
(e) Is $\prod_{p} \frac{p^2+1}{p^2-1} = \frac{2^2+1}{2^2-1}\cdot\frac{3^2+1}{3^2-1}\cdot\frac{5^2+1}{5^2-1}\cdot\frac{7^2+1}{7^2-1}\cdot\dots$ a rational number?
(f) Do there exist infinite number of pairs of [i]distinct[/i] integers $(a,b)$ such that $a$ and $b$ have the same set of prime divisors, and $a+1$ and $b+1$ have the same set of prime divisors?
[color = red]The USAYNO disclaimer is only included in problem 33. I have included it here for convenience.[/color]
[color = red]A clarification was issued for problem 36(d) during the test. I have included it above.[/color]