Let the sequence $u_n$ be defined by $u_0=0$ and $u_{2n}=u_n$, $u_{2n+1}=1-u_n$ for each $n\in\mathbb N_0$. (a) Calculate $u_{1990}$. (b) Find the number of indices $n\le1990$ for which $u_n=0$. (c) Let $p$ be a natural number and $N=(2^p-1)^2$. Find $u_N$.

Let $ABCDEFG$ be a regular heptagon. If $AC = m$ and $AD = n$, prove that $AB =\frac{mn}{m+n}$.

Richard comes across an infinite row of magic hats, $H_1, H_2, \dots$ each of which may contain a dollar bill with probabilities $p_1, p_2, \dots$. If Richard draws a dollar bill from $H_i$, then $p_{i+1} = p_i$, and if not, $p_{i+1}=\frac{1}{2}p_i$. If $p_1 = \frac{1}{2}$ and $E$ is the expected amount of money Richard makes from all the hats, compute $\lfloor 100E \rfloor$. [i]Proposed by Alex Li[/i]

Prove that $$\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c} \ge \dfrac{a}{b}+\dfrac{b}{c}+\dfrac{c}{a}+2(a+b+c)$$ for the all $a,b,c$ positive real numbers satisfying $a^2+b^2+c^2+2abc \le 1$.

Let $a,b,c$ be positive integers which satisfy the condition: $16(a+b+c)\geq \frac{1}{a}+\frac{1}{b}+\frac{1}{c}$. Prove that \[\sum_{cyc} \left( \frac{1}{a+b+\sqrt{2a+2c}} \right)^{3}\leq \frac{8}{9}\]

Find four positive integers each not exceeding $70000$ and each having more than $100$ divisors.

Seven lattice points form a convex heptagon with all sides having distinct lengths. Find the minimum possible value of the sum of the squares of the sides of the heptagon.

The set of $ x$-values satisfying the inequality $ 2 \leq |x\minus{}1| \leq 5$ is: $ \textbf{(A)}\ \minus{}4 \leq x \leq \minus{}1 \text{ or } 3 \leq x \leq 6 \qquad \textbf{(B)}\ 3 \leq x \leq 6 \text{ or } \minus{}6 \leq x \leq \minus{}3 \qquad \textbf{(C)}\ x \leq \minus{}1 \text{ or } x \geq 3 \qquad \textbf{(D)}\ \minus{}1 \leq x \leq 3 \qquad \textbf{(E)}\ \minus{}4 \leq x \leq 6$

$3$ players take turns drawing lines that connect vertices of a regular $n$-gon. No player may draw a line that intersects another line at a point other than a vertex of the $n-$gon. The last player able to draw a line wins. For how many $n$ in the range $4\le n \le 100$ does the first player have a winning strategy?

Let $f(0), f(1), f(2), \dots$ be a sequence satisfying \[ f(0) = 0 \quad \text{and} \quad f(n) = n - f(f(n-1)) \] for $n=1,2,3,\dots$. Give a formula for $f(n)$ such that its value can be immediately computed using $n$ without having to compute the previous terms.

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 $B = (-1, 0)$ and $C = (1, 0)$ be fixed points on the coordinate plane. A nonempty, bounded subset $S$ of the plane is said to be [i]nice[/i] if $\text{(i)}$ there is a point $T$ in $S$ such that for every point $Q$ in $S$, the segment $TQ$ lies entirely in $S$; and $\text{(ii)}$ for any triangle $P_1P_2P_3$, there exists a unique point $A$ in $S$ and a permutation $\sigma$ of the indices $\{1, 2, 3\}$ for which triangles $ABC$ and $P_{\sigma(1)}P_{\sigma(2)}P_{\sigma(3)}$ are similar. Prove that there exist two distinct nice subsets $S$ and $S'$ of the set $\{(x, y) : x \geq 0, y \geq 0\}$ such that if $A \in S$ and $A' \in S'$ are the unique choices of points in $\text{(ii)}$, then the product $BA \cdot BA'$ is a constant independent of the triangle $P_1P_2P_3$.

Find the smallest positive integer $n$ such that we can write numbers $1,2,\dots ,n$ in a 18*18 board such that: i)each number appears at least once ii)In each row or column,there are no two numbers having difference 0 or 1

Let $\mathbb N$ denote the set of positive integers, and for a function $f$, let $f^k(n)$ denote the function $f$ applied $k$ times. Call a function $f : \mathbb N \to \mathbb N$ [i]saturated[/i] if \[ f^{f^{f(n)}(n)}(n) = n \] for every positive integer $n$. Find all positive integers $m$ for which the following holds: every saturated function $f$ satisfies $f^{2014}(m) = m$. [i]Proposed by Evan Chen[/i]

In triangle $ABC, \angle C= 60^o, \angle A= 45^o$. Let $M$ be the midpoint of $BC, H$ be the orthocenter of triangle $ABC$. Prove that line $MH$ passes through the midpoint of arc $AB$ of the circumcircle of triangle $ABC$.

Suppose $$2+\frac{1}{1+\frac{1}{2+\frac{2}{3+x}}}=\frac{144}{53}.$$ What is the value of $x?$ $\textbf{(A) }\frac34 \qquad \textbf{(B) }\frac78 \qquad \textbf{(C) }\frac{14}{15} \qquad \textbf{(D) }\frac{37}{38} \qquad \textbf{(E) }\frac{52}{53}$