If $ m \equal{} \frac {cab}{a \minus{} b}$, then $ b$ equals: $ \textbf{(A)}\ \frac {m(a \minus{} b)}{ca} \qquad\textbf{(B)}\ \frac {cab \minus{} ma}{ \minus{} m} \qquad\textbf{(C)}\ \frac {1}{1 \plus{} c} \qquad\textbf{(D)}\ \frac {ma}{m \plus{} ca}$ $ \textbf{(E)}\ \frac {m \plus{} ca}{ma}$

Shauna takes $5$ tests, each worth a maximum of a $100$ points. Her scores on the first three tests were $76$, $94$, and $87$. In order to average an $81$ on all five tests, what is the lowest score she could earn on one of the two tests? $\textbf{(A) } 48 \qquad\textbf{(B) } 52 \qquad\textbf{(C) } 66 \qquad\textbf{(D) } 70 \qquad\textbf{(E) } 74$

Let $ABC$ be a triangle with area $S$, and let $P$ be a point in the plane. Prove that $AP+BP+CP\geq 2\sqrt[4]{3}\sqrt{S}$.

Find all elements $n \in A = \{2,3,...,2016\} \subset \mathbb{N}$ such that: every number $m \in A$ smaller than $n$, and coprime with $n$, must be a prime number

Determine all functions $f : \mathbb{R} \to \mathbb{R}$ that satisfy the following two properties. (i) The Riemann integral $\int_a^b f(t) \mathrm dt$ exists for all real numbers $a < b$. (ii) For every real number $x$ and every integer $n \ge 1$ we have \[ f(x) = \frac{n}{2} \int_{x-\frac{1}{n}}^{x+\frac{1}{n}} f(t) \mathrm dt. \]

Let $GL(n, K)$ be a linear group over the field K with a topology induced by a non-Archimedean absolute value of the field K. Prove that if the matrix $M \in GL (n, K)$ is contained by some compact subgroup of $GL(n, K)$, then all eigenvalues of M have absolute value 1.

Sherry starts at the number 1. Whenever she's at 1, she moves one step up (to 2). Whenever she's at a number strictly between 1 and 10, she moves one step up or one step down, each with probability $\frac{1}{2}$. When she reaches 10, she stops. What is the expected number (average number) of steps that Sherry will take?

Problem Shortlist BMO 2017 Let $ a $,$ b$,$ c$, be positive real numbers such that $abc= 1 $. Prove that $$\frac{1}{a^{5}+b^{5}+c^{2}}+\frac{1}{b^{5}+c^{5}+a^{2}}+\frac{1}{c^{5}+b^{5}+b^{2}}\leq 1 . $$

Let $S$ be the sum of the base 10 logarithms of all the proper divisors of 1000000. What is the integer nearest to $S$?

We say that a positive integer is [i]quad-divi[/i] if it is divisible by the sum of the squares of its digits, and also none of its digits is equal to zero. a) Find a quad-divi number such that the sum of its digits is $24$. b) Find a quad-divi number such that the sum of its digits is $1001$.

[b]a) [/b]Is the number $ 1111\cdots11$ (with $ 2010$ ones) a prime number? [b]b)[/b] Prove that every prime factor of $ 1111\cdots11$ (with $ 2011$ ones) is of the form $ 4022j\plus{}1$ where $ j$ is a natural number.

Let $ T$ be a triangulation of an $ n$-dimensional sphere, and to each vertex of $ T$ let us assign a nonzero vector of a linear space $ V$. Show that if $ T$ has an $ n$-dimensional simplex such that the vectors assigned to the vertices of this simplex are linearly independent, then another such simplex must also exist. [i]L. Lovasz[/i]

Let $p, q$, and $r$ be prime numbers such that $2pqr + p + q + r = 2020$. Find $pq + qr + rp$.

For a positive integer $n$, define $f(n)=\sum_{i=0}^{\infty}\frac{\gcd(i,n)}{2^i}$ and let $g:\mathbb N\rightarrow \mathbb Q$ be a function such that $\sum_{d\mid n}g(d)=f(n)$ for all positive integers $n$. Given that $g(12321)=\frac{p}{q}$ for relatively prime integers $p$ and $q$, find $v_2(p)$. [i]Proposed by Michael Ren[/i]

The point $P$ is chosen inside or on the equilateral triangle $ABC$ of side length $1$. The reflection of $P$ with respect to $AB$ is $K$, the reflection of $K$ about $BC$ is $M$, and the reflection of $M$ with respect to $AC$ is $N$. What is the maximum length of $NP$? $\textbf{(A)}\ 2\sqrt 3\qquad\textbf{(B)}\ \sqrt 3\qquad\textbf{(C)}\ \frac{\sqrt 3}{2} \qquad\textbf{(D)}\ 3\qquad\textbf{(E)}\ 1$

Let $ a$, $ b$, $ c$ be positive real numbers. Prove the inequality \[ \frac{1}{a\left(b+1\right)}+\frac{1}{b\left(c+1\right)}+\frac{1}{c\left(a+1\right)}\geq \frac{3}{1+abc}. \]

What is the smallest positive integer $k$ such that $k(3^3 + 4^3 + 5^3) = a^n$ for some positive integers $a$ and $n$, with $n > 1$?

Let $p,q$ be two distinct odd primes. Calculate $\displaystyle \sum_{j=1}^{\frac{p-1}{2}}\left \lfloor \frac{qj}{p}\right \rfloor +\sum_{j=1}^{\frac{q-1}{2}}\left \lfloor \frac{pj}{q}\right\rfloor$.

The product of the lengths of the two congruent sides of an obtuse isosceles triangle is equal to the product of the base and twice the triangle’s height to the base. What is the measure, in degrees, of the vertex angle of this triangle? $\textbf{(A)}\ 105 \qquad\textbf{(B)}\ 120 \qquad\textbf{(C)}\ 135 \qquad\textbf{(D)}\ 150 \qquad\textbf{(E)}\ 165$

Of $ 28$ students taking at least one subject the number taking Mathematics and English only equals the number taking Mathematics only. No student takes English only or History only, and six students take Mathematics and History, but not English. The number taking English and History only is five times the number taking all three subjects. If the number taking all three subjects is even and non-zero, the number taking English and Mathematics only is: $ \textbf{(A)}\ 5 \qquad \textbf{(B)}\ 6 \qquad \textbf{(C)}\ 7 \qquad \textbf{(D)}\ 8 \qquad \textbf{(E)}\ 9$

Give a geometric series $(a_n)$ with common ratio of $q$, let $b_1=a_1+a_2+a_3,b_2=a_4+a_5+a_6,\cdots,b_n=a_{3n}+a_{3n+1}+a_{3n+2}$, then sequence $(b_n)$ $\text{(A)}$ is an arithmetic sequence $\text{(B)}$ is a geometric series with common ratio of $q$ $\text{(C)}$ is a geometric series with common ratio of $q^3$ $\text{(D)}$ is neither an arithmetic sequence nor a geometric series

Let $n$ be a positive integer. Tasty and Stacy are given a circular necklace with $3n$ sapphire beads and $3n$ turquoise beads, such that no three consecutive beads have the same color. They play a cooperative game where they alternate turns removing three consecutive beads, subject to the following conditions: [list] [*]Tasty must remove three consecutive beads which are turquoise, sapphire, and turquoise, in that order, on each of his turns. [*]Stacy must remove three consecutive beads which are sapphire, turquoise, and sapphire, in that order, on each of her turns. [/list] They win if all the beads are removed in $2n$ turns. Prove that if they can win with Tasty going first, they can also win with Stacy going first. [i]Yannick Yao[/i]

Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that $f(f(x)f(1-x))=f(x)$ and $f(f(x))=1-f(x)$, for all real $x$.

A circle $k$ with center $M$ and radius $r$ is given. Find the locus of the incenters of all obtuse-angled triangles inscribed in $k$.

If $x + y = 306$ and $\frac{x}{y} = \frac{7}{10}$, compute $y - x$.