Jacob and Aaron are playing a game in which Aaron is trying to guess the outcome of an unfair coin which shows heads $\tfrac{2}{3}$ of the time. Aaron randomly guesses ``heads'' $\tfrac{2}{3}$ of the time, and guesses ``tails'' the other $\tfrac{1}{3}$ of the time. If the probability that Aaron guesses correctly is $p$, compute $9000p$. [i]Proposed by Aaron Lin[/i]

Find the least prime factor of the number $\frac{2016^{2016}-3}{3}$.

The complex number w has positive imaginary part and satisfies $|w| = 5$. The triangle in the complex plane with vertices at $w, w^2,$ and $w^3$ has a right angle at $w$. Find the real part of $w^3$.

Amy and Bob play the game. At the beginning, Amy writes down a positive integer on the board. Then the players take moves in turn, Bob moves first. On any move of his, Bob replaces the number $n$ on the blackboard with a number of the form $n-a^2$, where $a$ is a positive integer. On any move of hers, Amy replaces the number $n$ on the blackboard with a number of the form $n^k$, where $k$ is a positive integer. Bob wins if the number on the board becomes zero. Can Amy prevent Bob’s win? [i]Maxim Didin, Russia[/i]

Prove that for every positive integer $n,$ the set $\{2,3,4,\ldots,3n+1\}$ can be partitioned into $n$ triples in such a way that the numbers from each triple are the lengths of the sides of some obtuse triangle. [i]Proposed by Canada[/i]

Let $n$ be an odd positive integer with $n > 1$ and let $a_1, a_2,... , a_n$ be positive integers such that gcd $(a_1, a_2,... , a_n) = 1$. Let $d$ = gcd $(a_1^n + a_1\cdot a_2 \cdot \cdot \cdot a_n, a_2^n + a_1\cdot a_2 \cdot \cdot \cdot a_n, ... , a_n^n + a_1\cdot a_2 \cdot \cdot \cdot a_n)$. Show that the possible values of $d$ are $d = 1, d = 2$

$2000$ numbers are considered: $11, 101, 1001, . . $. Prove that at least $99\%$ of these numbers are composite.

The natural numbers from 1 to 100 are arranged on a circle with the characteristic that each number is either larger as their two neighbours or smaller than their two neighbours. A pair of neighbouring numbers is called "good", if you cancel such a pair, the above property remains still valid. What is the smallest possible number of good pairs?

$120$ pirates distribute $119$ gold pieces among themselves. Then the captain checks if any pirate has $15$ or more gold pieces. If he finds the first one, he must give all his gold pieces to other pirates, whereby he may not give more than one gold piece to anyone. This control is repeated as long as there is any pirate with $15$ or more gold pieces. Does this process end after a lot of checks?

Mr. Rose gave a test to his two calculus classes. His first period class has $20$ students, and their average score on the test was $80$. His second period class has $30$ students, and their average score was $90$. What was the average score of all $50$ of his calculus students?

[b]p1.[/b] Two trains, $A$ and $B$, travel between cities $P$ and $Q$. On one occasion $A$ started from $P$ and $B$ from $Q$ at the same time and when they met $A$ had travelled $120$ miles more than $B$. It took $A$ four $(4)$ hours to complete the trip to $Q$ and B nine $(9)$ hours to reach $P$. Assuming each train travels at a constant speed, what is the distance from $P$ to $Q$? [b]p2.[/b] If $a$ and $b$ are integers, $b$ odd, prove that $x^2 + 2ax + 2b = 0$ has no rational roots. [b]p3.[/b] A diameter segment of a set of points in a plane is a segment joining two points of the set which is at least as long as any other segment joining two points of the set. Prove that any two diameter segments of a set of points in the plane must have a point in common. [b]p4.[/b] Find all positive integers $n$ for which $\frac{n(n^2 + n + 1) (n^2 + 2n + 2)}{2n + 1}$ is an integer. Prove that the set you exhibit is complete. [b]p5.[/b] $A, B, C, D$ are four points on a semicircle with diameter $AB = 1$. If the distances $\overline{AC}$, $\overline{BC}$, $\overline{AD}$, $\overline{BD}$ are all rational numbers, prove that $\overline{CD}$ is also rational. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $ABC$ be a triangle in which $BC<AC$. Let $M$ be the mid-point of $AB$, $AP$ be the altitude from $A$ on $BC$, and $BQ$ be the altitude from $B$ on to $AC$. Suppose that $QP$ produced meets $AB$ (extended) at $T$. If $H$ is the orthocenter of $ABC$, prove that $TH$ is perpendicular to $CM$.

Found all functions $f: \mathbb{R} \to \mathbb{R}$, such that for any $x,y \in \mathbb{R}$, \[f(x^2+xy+f(y))=f^2(x)+xf(y)+y.\]

As in the following diagram, square $ABCD$ and square $CEFG$ are placed side by side (i.e. $C$ is between $B$ and $E$ and $G$ is between $C$ and $D$). If $CE = 14$, $AB > 14$, compute the minimal area of $\triangle AEG$. [asy] size(120); defaultpen(linewidth(0.7)+fontsize(10)); pair D2(real x, real y) { pair P = (x,y); dot(P,linewidth(3)); return P; } int big = 30, small = 14; filldraw((0,big)--(big+small,0)--(big,small)--cycle, rgb(0.9,0.5,0.5)); draw(scale(big)*unitsquare); draw(shift(big,0)*scale(small)*unitsquare); label("$A$",D2(0,big),NW); label("$B$",D2(0,0),SW); label("$C$",D2(big,0),SW); label("$D$",D2(big,big),N); label("$E$",D2(big+small,0),SE); label("$F$",D2(big+small,small),NE); label("$G$",D2(big,small),NE); [/asy]

Let $x>1$ be a real number which is not an integer. For each $n\in\mathbb{N}$, let $a_n=\lfloor x^{n+1}\rfloor - x\lfloor x^n\rfloor$. Prove that the sequence $(a_n)$ is not periodic.

Let $ f$ be a finite real function of one variable. Let $ \overline{D}f$ and $ \underline{D}f$ be its upper and lower derivatives, respectively, that is, \[ \overline{D}f\equal{}\limsup_{{h,k\rightarrow 0}_{{h,k \geq 0}_{h\plus{}k>0}}} \frac{f(x\plus{}h)\minus{}f(x\minus{}k)}{h\plus{}k}\] , \[ \underline{D}f\equal{}\liminf_{{h,k\rightarrow 0}_{{h,k \geq 0}_{h\plus{}k>0}}} \frac{f(x\plus{}h)\minus{}f(x\minus{}k)}{h\plus{}k}.\] Show that $ \overline{D}f$ and $ \underline{D}f$ are Borel-measurable functions. [A. Csaszar]