Let $a$, $b$, $c$ be positive real numbers for which \[ \frac{5}{a} = b+c, \quad \frac{10}{b} = c+a, \quad \text{and} \quad \frac{13}{c} = a+b. \] If $a+b+c = \frac mn$ for relatively prime positive integers $m$ and $n$, compute $m+n$. [i]Proposed by Evan Chen[/i]

You drop a 7 cm long piece of mechanical pencil lead on the floor. A bully takes the lead and breaks it at a random point into two pieces. A piece of lead is unusable if it is 2 cm or shorter. If the expected value of the number of usable pieces afterwards is $\frac{m}n$ for relatively prime positive integers $m$ and $n$, compute $100m + n$. [i]Proposed by Aaron Lin[/i]

When flipped, coin A shows heads $\textstyle\frac{1}{3}$ of the time, coin B shows heads $\textstyle\frac{1}{2}$ of the time, and coin C shows heads $\textstyle\frac{2}{3}$ of the time. Anna selects one of the coins at random and flips it four times, yielding three heads and one tail. The probability that Anna flipped coin A can be expressed as $\textstyle\frac{p}{q}$ for relatively prime positive integers $p$ and $q$. Compute $p + q$. [i]Proposed by Eugene Chen[/i]

Let $ n\in N^*$. A permutation $ (a_1,a_2,...,a_n)$ of the numbers $ (1,2,...,n)$ is called [i]quadratic [/i] iff at least one of the numbers $ a_1,a_1\plus{}a_2,...,a_1\plus{}a_2\plus{}a\plus{}...\plus{}a_n$ is a perfect square. Find the greatest natural number $ n\leq 2003$, such that every permutation of $ (1,2,...,n)$ is quadratic.

28. [b][15][/b] Find the shortest distance between the lines $\frac{x+2}{2}=\frac{y-1}{3}=\frac{z}{1}$ and $\frac{x-3}{-1}=\frac{y}{1}=\frac{z+1}{2}$ 29. [b][15][/b] Find the largest real number $k$ such that there exists a sequence of positive reals ${a_i}$ for which $\sum_{n=1}^{\infty}a_n$ converges but $\sum_{n=1}^{\infty}\frac{\sqrt{a_n}}{n^k}$ does not. 30. [b][15][/b] Find the largest integer $n$ such that the following holds: there exists a set of $n$ points in the plane such that, for any choice of three of them, some two are unit distance apart. 31. [b][17][/b] Two random points are chosen on a segment and the segment is divided at each of these two points. Of the three segments obtained, find the probability that the largest segment is more than three times longer than the smallest segment. 32. [b][17][/b] Find the sum of all positive integers $n\le 2015$ that can be expressed in the form $\left\lceil{\frac{x}{2}}\right \rceil +y+xy$, where $x$ and $y$ are positive integers. 33. [b][17][/b] How many ways are there to place four points in the plane such that the set of pairwise distances between the points consists of exactly $2$ elements? (Two configurations are the same if one can be obtained from the other via rotation and scaling.) 34. [b][20][/b] Let $n$ be the second smallest integer that can be written as the sum of two positive cubes in two different ways. Compute $n$. If your guess is $a$, you will receive $\max(25-5\cdot \max(\frac{a}{n},\frac{n}{a}),0)$, rounded up. 35. [b][20][/b] Let $n$ be the smallest positive integer such that any positive integer can be expressed as the sum of $n$ integer 2015th powers. Find $n$. If your answer is $a$, your score will be $\max(20-\frac{1}{5}|\log _{10} \frac{a}{n}|,0)$, rounded up. 36. [b][20][/b] Consider the following seven false conjectures with absurdly high counterexamples. Pick any subset of them, and list their labels in order of their smallest counterexample (the smallest $n$ for which the conjecture is false) from smallest to largest. For example, if you believe that the below list is already ordered by counterexample size, you should write ”PECRSGA”. - [b]P.[/b] (Polya’s conjecture) For any integer $n$, at least half of the natural numbers below $n$ have an odd number of prime factors. - [b]E.[/b] (Euler’s conjecture) There is no perfect cube $n$ that can be written as the sum of three positive cubes. - [b]C.[/b] (Cyclotomic) The polynomial with minimal degree whose roots are the primitive $n$th roots of unity has all coefficients equal to $-1$, $0$, or $1$. - [b]R.[/b] (Prime race) For any integer $n$, there are more primes below $n$ equal to $2(\mod 3)$ than there are equal to $1 (\mod 3)$. - [b]S.[/b] (Seventeen conjecture) For any integer $n$, $n^{17} + 9$ and $(n + 1)^{17} + 9$ are relatively prime. - [b]G.[/b] (Goldbach’s (other) conjecture) Any odd composite integer $n$ can be written as the sum of a prime and twice a square. - [b]A.[/b] (Average square) Let $a_1 = 1$ and $a_{k+1}=\frac{1+a_1^2+a_2^2+...+a_k^2}{k}$. Then $a_n$ is an integer for any n. If your answer is a list of $4\le n\le 7$ labels in the correct order, your score will be $(n-2)(n-3)$. Otherwise, your score will be $0$.

Find the number of rationals $\frac{m}{n}$ such that (i) $0 < \frac{m}{n} < 1$; (ii) $m$ and $n$ are relatively prime; (iii) $mn = 25!$.

Points $A$ and $B$ are the endpoints of a diameter of a circle with center $C$. Points $D$ and $E$ lie on the same diameter so that $C$ bisects segment $\overline{DE}$. Let $F$ be a randomly chosen point within the circle. The probability that $\triangle DEF$ has a perimeter less than the length of the diameter of the circle is $\tfrac{17}{128}$. There are relatively prime positive integers m and n so that the ratio of $DE$ to $AB$ is $\tfrac{m}{n}.$ Find $m + n$.

Given that $\sum_{k=1}^{35}\sin 5k=\tan \frac mn,$ where angles are measured in degrees, and $m$ and $n$ are relatively prime positive integers that satisfy $\frac mn<90,$ find $m+n.$

Let $k,n\ge 1$ be relatively prime integers. All positive integers not greater than $k+n$ are written in some order on the blackboard. We can swap two numbers that differ by $k$ or $n$ as many times as we want. Prove that it is possible to obtain the order $1,2,\dots,k+n-1, k+n$.

Let $n$ be a positive integer, and set $S$ be the set of all integers in $\{1,2,\dots,n\}$ which are relatively prime to $n$. Set $S_1 = S \cap \left(0, \frac n3 \right]$, $S_2 = S \cap \left( \frac n3, \frac {2n}3 \right]$, $S_3 = S \cap \left( \frac{2n}{3}, n \right]$. If the cardinality of $S$ is a multiple of $3$, prove that $S_1$, $S_2$, $S_3$ have the same cardinality.

15 positive integers, all less than 1998(and no one equal to 1), are relatively prime (no pair has a common factor > 1). Show that at least one of them must be prime.

There are nonzero real numbers $a$ and $b$ so that the roots of $x^2 + ax + b$ are $3a$ and $3b$. There are relatively prime positive integers $m$ and $n$ so that $a - b = \tfrac{m}{n}$. Find $m + n$.

Let $a$ be an odd integer. Prove that $a^{2^m}+2^{2^m}$ and $a^{2^n}+2^{2^n}$ are relatively prime for all positive integers $n$ and $m$ with $n\not= m$.

Triangle $ABC$ has a right angle at $C$, $AC = 3$ and $BC = 4$. Triangle $ABD$ has a right angle at $A$ and $AD = 12$. Points $C$ and $D$ are on opposite sides of $\overline{AB}$. The line through $D$ parallel to $\overline{AC}$ meets $\overline{CB}$ extended at $E$. If $\frac{DE}{DB} = \frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, then $m + n = $ [asy] size(170); defaultpen(fontsize(10pt)+linewidth(.8pt)); pair C=origin, A=(0,3), B=(4,0), D=(7.2,12.6), E=(7.2,0); draw(A--C--B--A--D--B--E--D); label("$A$",A,W); label("$B$",B,S); label("$C$",C,SW); label("$D$",D,NE); label("$E$",E,SE); [/asy] $ \textbf{(A)}\ 25\qquad\textbf{(B)}\ 128\qquad\textbf{(C)}\ 153\qquad\textbf{(D)}\ 243\qquad\textbf{(E)}\ 256 $

PUMaCDonalds, a newly-opened fast food restaurant, has 5 menu items. If the first 4 customers each choose one menu item at random, the probability that the 4th customer orders a previously unordered item is $m/n$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Find the number of functions $ f: \mathbb N\rightarrow\mathbb N$ which satisfying: (i) $ f(1) \equal{} 1$ (ii) $ f(n)f(n \plus{} 2) \equal{} f^2(n \plus{} 1) \plus{} 1997$ for every natural numbers n.

Find all triples $(l, m, n)$ of distinct positive integers satisfying \[{\gcd(l, m)}^{2}= l+m, \;{\gcd(m, n)}^{2}= m+n, \; \text{and}\;\;{\gcd(n, l)}^{2}= n+l.\]

Find the remainder when $19^{92}$ is divided by 92.

Find all pairs of relatively prime integers $(x, y)$ that satisfy equality $2 (x^3 - x) = 5 (y^3 - y)$.

For every positive integer $ n$, $ b(n)$ denote the number of positive divisors of $ n$ and $ p(n)$ denote the sum of all positive divisors of $ n$. For example, $ b(14)\equal{}4$ and $ p(14)\equal{}24$. Let $ k$ be a positive integer greater than $ 1$. (a) Prove that there are infinitely many positive integers $ n$ which satisfy $ b(n)\equal{}k^2\minus{}k\plus{}1$. (b) Prove that there are finitely many positive integers $ n$ which satisfy $ p(n)\equal{}k^2\minus{}k\plus{}1$.

Find all pairs of positive integers $(m,n)$ for which there exist relatively prime integers $a$ and $b$ greater than $1$ such that $$\frac{a^m+b^m}{a^n+b^n}$$ is an integer.

Find all positive integer $ m$ if there exists prime number $ p$ such that $ n^m\minus{}m$ can not be divided by $ p$ for any integer $ n$.

By a [i]pure repeating decimal[/i] (in base $10$), we mean a decimal $0.\overline{a_1\cdots a_k}$ which repeats in blocks of $k$ digits beginning at the decimal point. An example is $.243243243\cdots = \tfrac{9}{37}$. By a [i]mixed repeating decimal[/i] we mean a decimal $0.b_1\cdots b_m\overline{a_1\cdots a_k}$ which eventually repeats, but which cannot be reduced to a pure repeating decimal. An example is $.011363636\cdots = \tfrac{1}{88}$. Prove that if a mixed repeating decimal is written as a fraction $\tfrac pq$ in lowest terms, then the denominator $q$ is divisible by $2$ or $5$ or both.

find all pairs of relatively prime natural numbers $ (m,n) $ in such a way that there exists non constant polynomial f satisfying \[ gcd(a+b+1, mf(a)+nf(b) > 1 \] for every natural numbers $ a $ and $ b $

Let $m_1,m_2,...,m_{2013} > 1$ be 2013 pairwise relatively prime positive integers and $A_1,A_2,...,A_{2013}$ be 2013 (possibly empty) sets with $A_i\subseteq \{1,2,...,m_i-1\}$ for $i=1,2,...,2013$. Prove that there is a positive integer $N$ such that \[ N \le \left( 2\left\lvert A_1 \right\rvert + 1 \right)\left( 2\left\lvert A_2 \right\rvert + 1 \right)\cdots\left( 2\left\lvert A_{2013} \right\rvert + 1 \right) \] and for each $i = 1, 2, ..., 2013$, there does [i]not[/i] exist $a \in A_i$ such that $m_i$ divides $N-a$. [i]Proposed by Victor Wang[/i]