A Haiku is a Japanese poem of seventeen syllables, in three lines of five, seven, and five. Ada has been told To write down five haikus plus Two more every hour. Such that she needs to Write down five in the first hour Seven, nine, so on. Ada has so far Forty haikus and writes down Seven every hour. At which hour after She begins will she not have Enough haikus done? [i]Proposed by Ada Tsui[/i]

Which of the following is closest to the product $ (.48017)(.48017)(.48017) $? $ \text{(A)}\ 0.011\qquad\text{(B)}\ 0.110\qquad\text{(C)}\ 1.10\qquad\text{(D)}\ 11.0\qquad\text{(E)}\ 110 $

How many integers $ x$ satisfy the equation \[ (x^2 \minus{} x \minus{} 1)^{x \plus{} 2} \equal{} 1 \]$ \textbf{(A)}\ 2 \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 5 \qquad \textbf{(E)}\ \text{none of these}$

Thee tiles are marked $ X$ and two other tiles are marked $ O$. The five tiles are randomly arranged in a row. What is the probability that the arrangement reads $ XOXOX$? $ \textbf{(A)}\ \frac{1}{12}\qquad \textbf{(B)}\ \frac{1}{10}\qquad \textbf{(C)}\ \frac{1}{6}\qquad \textbf{(D)}\ \frac{1}{4}\qquad \textbf{(E)}\ \frac{1}{3}$

The function $f: \mathbb{N}\to\mathbb{N}_{0}$ satisfies for all $m,n\in\mathbb{N}$: \[f(m+n)-f(m)-f(n)=0\text{ or }1, \; f(2)=0, \; f(3)>0, \; \text{ and }f(9999)=3333.\] Determine $f(1982)$.

We recursively define a set of [i]goody pairs[/i] of words on the alphabet $\{a,b\}$ as follows: - $(a,b)$ is a goody pair; - $(\alpha, \beta) \not= (a,b)$ is a goody pair if and only if there is a goody pair $(u,v)$ such that $(\alpha, \beta) = (uv,v)$ or $(\alpha, \beta) = (u,uv)$ Show that if $(\alpha, \beta)$ is a good pair then there exists a palindrome $\gamma$ (possibly empty) such that $\alpha\beta = a \gamma b$

Below is pictured a regular seven-pointed star. Find the measure of angle $a$ in radians. [asy] size(150); draw(unitcircle, white); pair A = dir(180/7); pair B = dir(540/7); pair C = dir(900/7); pair D = dir(180); pair E = dir(-900/7); pair F = dir(-540/7); pair G = dir (-180/7); draw(A--D); draw(B--E); draw(C--F); draw(D--G); draw(E--A); draw(F--B); draw(G--C); label((-0.1,0.5), "$a$"); [/asy]

We say that the string $d_kd_{k-1} \cdots d_1d_0$ represents a number $n$ in base $-2$ if each $d_i$ is either $0$ or $1$, and $n = d_k(-2)^k + d_{k-1}(-2)^{k-1} + \cdots + d_1(-2) + d_0$. For example, $110_{-2}$ represents the number $2$. What string represents $2016$ in base $-2$?

For arbitrary reals $ x$, $ y$ and $ z$ prove the following inequality: $ x^{2} \plus{} y^{2} \plus{} z^{2} \minus{} xy \minus{} yz \minus{} zx \geq \max \{\frac {3(x \minus{} y)^{2}}{4} , \frac {3(y \minus{} z)^{2}}{4} , \frac {3(y \minus{} z)^{2}}{4} \}$

Let $ABCDE$ be an equiangular convex pentagon of perimeter $1$. The pairwise intersections of the lines that extend the side of the pentagon determine a five-pointed star polygon. Let $s$ be the perimeter of the star. What is the difference between the maximum and minimum possible perimeter of $s$? $ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ \frac{1}{2} \qquad\textbf{(C)}\ \frac{\sqrt{5}-1}{2} \qquad\textbf{(D)}\ \frac{\sqrt{5}+1}{2} \qquad\textbf{(E)}\ \sqrt{5} $

Let $M$, $N$ and $P$ be midpoints of sides $BC, AC$ and $AB$, respectively, and let $O$ be circumcenter of acute-angled triangle $ABC$. Circumcircles of triangles $BOC$ and $MNP$ intersect at two different points $X$ and $Y$ inside of triangle $ABC$. Prove that \[\angle BAX=\angle CAY.\]

Consider the arithmetic sequence $8, 21,34,47,....$ a) Prove that this sequence contains infinitely many integers written only with digit $9$. b) How many such integers less than $2010^{2010}$ are in the se­quence?

The natural numbers from $1$ up to $300$ are evenly located around a circle. We say that such an ordering is [i]alternate [/i ]if each number is less than its two neighbors or is greater than its two neighbors. We will call a pair of neighboring numbers a [i]good [/i] pair if, by removing that pair from the circumference, the remaining numbers form an alternate ordering. Determine the least possible number of good pairs in which there can be an alternate ordering of the numbers from $1$ at $300$ inclusive.

A deck of $ 2n\plus{}1$ cards consists of a joker and, for each number between 1 and $ n$ inclusive, two cards marked with that number. The $ 2n\plus{}1$ cards are placed in a row, with the joker in the middle. For each $ k$ with $ 1 \leq k \leq n,$ the two cards numbered $ k$ have exactly $ k\minus{}1$ cards between them. Determine all the values of $ n$ not exceeding 10 for which this arrangement is possible. For which values of $ n$ is it impossible?

Last digit of $2019^{2019}$ is

Let $A$ be a point and let k be a circle through $A$. Let $B$ and $C$ be two more points on $k$. Let $X$ be the intersection of the bisector of $\angle ABC$ with $k$. Let $Y$ be the reflection of $A$ wrt point $X$, and $D$ the intersection of the straight line $YC$ with $k$. Prove that point $D$ is independent of the choice of $B$ and $C$ on the circle $k$.

The average age of the participants in a mathematics competition (gymnasts and high school students) increases by exactly one month if three high school age students $18$ years each are included in the competition or if three gymnasts aged $12$ years each are excluded from the competition. How many participants were initially in the contest?

A circle centered with $O$. $C$ is a stable point in circle. A chord $[AB]$, parallel to $OC$.Prove that, $$[AC]^2+[BC]^2$$ is stable.

An arithmetic sequence of five terms is considered $good$ if it contains 19 and 20. For example, $18.5,19.0,19.5,20.0,20.5$ is a $good$ sequence. For every $good$ sequence, the sum of its terms is totalled. What is the total sum of all $good$ sequences?

In a sport competition, $m$ teams have participated. We know that each two teams have competed exactly one time and the result is winning a team and losing the other team (i.e. there is no equal result). Prove that there exists a team $x$ such that for each team $y,$ either $x$ wins $y$ or there exists a team $z$ for which $x$ wins $z$ and $z$ wins $y.$ [i][i.e. prove that in every tournament there exists a king.][/i]

Let $x$ and $y$ be two real numbers such that $2 \sin x \sin y + 3 \cos y + 6 \cos x \sin y = 7$. Find $\tan^2 x + 2 \tan^2 y$.

Determine all pairs of polynomials $f,g\in\mathbb{C}[x]$ with complex coefficients such that the following equalities hold for all $x\in\mathbb{C}$: $f(f(x))-g(g(x))=1+i$ $f(g(x))-g(f(x))=1-i$

Vasya has $n{}$ candies of several types, where $n>145$. It is known that for any group of at least 145 candies, there is a type of candy which appears exactly 10 times. Find the largest possible value of $n{}$. [i]Proposed by A. Antropov[/i]

Three identical square sheets of paper each with side length $6{ }$ are stacked on top of each other. The middle sheet is rotated clockwise $30^\circ$ about its center and the top sheet is rotated clockwise $60^\circ$ about its center, resulting in the $24$-sided polygon shown in the figure below. The area of this polygon can be expressed in the form $a-b\sqrt{c}$, where $a$, $b$, and $c$ are positive integers, and $c$ is not divisible by the square of any prime. What is $a+b+c?$ [asy] size(160); defaultpen(linewidth(1.1)); path square = (1,1)--(1,-1)--(-1,-1)--(-1,1)--cycle; filldraw(square,white); filldraw(rotate(30)*square,white); filldraw(rotate(60)*square,white); dot((0,0),linewidth(7)); [/asy] $\textbf{(A)}\: 75\qquad\textbf{(B)} \: 93\qquad\textbf{(C)} \: 96\qquad\textbf{(D)} \: 129\qquad\textbf{(E)} \: 147$

Nico picks $13$ pairwise distinct $3-$digit positive integers. Ian then selects several of these 13 numbers, the ones he wants, and using only once each selected number and some of the operations addition, subtraction, multiplication and division ($+,-,\times ,:$) must get an expression whose value is greater than $3$ and less than $4$. If he succeeds, Ian wins; otherwise, Nico wins. Which of the two has a winning strategy?