Jane has $4$ pears, $5$ bananas, $3$ lemons, $1$ orange, and $6$ apples. If she uses one of each fruit to make a fruit smoothie, what is the total number of fruits that she has left?

Given two points $P,Q$ of the plane, denote $P+Q$ the midpoint of (possibly degenerate) segment $PQ$ and $P\cdot Q$ the image of $P$ in rotation around the origin $Q$ under $+90^\circ.$ a) Are these operations commutative? b) Given two distinct points $A,B$ the equation \[Y\cdot X=(A\cdot X)+B\] defines a map $X\mapsto Y.$ Determine what the mapping is. c) Construct all fixed points of the map from b).

Prove that $\frac{5^{125}-1}{5^{25}-1}$ is a composite number.

Prove that for any $ n$, there is a subset $ \{a_1,\dots,a_n\}$ of $ \mathbb N$ such that for each subset $ S$ of $ \{1,\dots,n\}$, $ \sum_{i\in S}a_i$ has the same set of prime divisors.

Two fair octahedral dice, each with the numbers $1$ through $8$ on their faces, are rolled. Let $N$ be the remainder when the product of the numbers showing on the two dice is divided by $8$. Find the expected value of $N$.

Let $\omega$ be the circumcircle of triangle $ABC$ with $AB<AC$. Let $I$ be its incenter and let $M$ be the midpoint of $BC$. The foot of the perpendicular from $M$ to $AI$ is $H$. The lines $MH, BI, AB$ form a triangle $T_b$ and the lines $MH, CI, AC$ form a triangle $T_c$. The circumcircle of $T_b$ meets $\omega$ at $B'$ and the circumcircle of $T_c$ meets $\omega$ at $C'$. Prove that $B', H, C'$ are collinear.

Determine the largest constant $K\geq 0$ such that $$\frac{a^a(b^2+c^2)}{(a^a-1)^2}+\frac{b^b(c^2+a^2)}{(b^b-1)^2}+\frac{c^c(a^2+b^2)}{(c^c-1)^2}\geq K\left (\frac{a+b+c}{abc-1}\right)^2$$ holds for all positive real numbers $a,b,c$ such that $ab+bc+ca=abc$. [i]Proposed by Orif Ibrogimov (Czech Technical University of Prague).[/i]

$a, b, c$ are real, and the sum of any two is greater than the third. Show that $\frac{2(a + b + c)(a^2 + b^2 + c^2)}{3} > a^3 + b^3 + c^3 + abc$.

Let $A=\{-2562,-2561,...,2561,2562\}$. Prove that for any bijection (1-1, onto function) $f:A\to A$, $$\sum_{k=1}^{2562}\left\lvert f(k)-f(-k)\right\rvert\text{ is maximized if and only if } f(k)f(-k)<0\text{ for any } k=1,2,...,2562.$$

Let $S$ be the set of positive integers not divisible by $p^4$ for all primes $p$. Anastasia and Bananastasia play a game. At the beginning, Anastasia writes down the positive integer $N$ on the board. Then the players take moves in turn; Bananastasia moves first. On any move of his, Bananastasia replaces the number $n$ on the blackboard with a number of the form $n-a$, where $a\in S$ is a positive integer. On any move of hers, Anastasia replaces the number $n$ on the blackboard with a number of the form $n^k$, where $k$ is a positive integer. Bananastasia wins if the number on the board becomes zero. Compute the second-smallest possible value of $N$ for which Anastasia can prevent Bananastasia from winning. [i]Proposed by Brandon Wang and Vincent Huang[/i]

In a convex quadrilateral $ABCD$, ${\angle}ABD=65^\circ$,${\angle}CBD=35^\circ$, ${\angle}ADC=130^\circ$ and $BC=AB$.Find the angles of $ABCD$.

A solid pyramid $TABCD$, with a quadrilateral base $ABCD$ is to be coloured on each of the five faces such that no two faces with a common edge will have the same colour. If five different colours are available, what is the number of ways to colour the pyramid?

A four-digit integer in base $10$ is [i]friendly[/i] if its digits are four consecutive digits in any order. A four-digit integer is [i]shy[/i] if there exist two adjacent digits in its representation that differ by $1.$ Compute the number of four-digit integers that are both friendly and shy.

Find all positive integer solutions $(x, y, z)$ of the equation $1 + 2^x \cdot 7^y=z^2$.

Let the sum of positive reals $a,b,c$ be equal to 1. Prove an inequality \[ \sqrt{{ab}\over {c+ab}}+\sqrt{{bc}\over {a+bc}}+\sqrt{{ac}\over {b+ac}}\le 3/2 \]. [b]proposed by Fedor Petrov[/b]

We call a tiling of an $m\times$ n rectangle with arabos (see figure below) [i]regular[/i] if there is no sub-rectangle which is tiled with arabos. Prove that if for some $m$ and $n$ there exists a [i]regular[/i] tiling of the $m\times n$ rectangle then there exists a [i]regular[/i] tiling also for the $2m \times 2n$ rectangle. [img]https://cdn.artofproblemsolving.com/attachments/1/1/2ab41cc5107a21760392253ed52d9e4ecb22d1.png[/img]

Prove that there do not exist distinct prime numbers $p$ and $q$ and a positive integer $n$ satisfying the equation $p^{q-1}- q^{p-1}=4n^2$

Let $a,b$ be positive real numbers.Prove that $(1+a)^{8}+(1+b)^{8}\geq 128ab(a+b)^{2}$.

Compare the numbers: $P = 888...888 \times 333 .. 333$ ($2010$ digits of $8$ and $2010$ digits of $3$) and $Q = 444...444\times 666...6667$ ($2010$ digits of $4$ and $2009$ digits of $6$) (A): $P = Q$, (B): $P > Q$, (C): $P < Q$.

Given a square $ABCD$, points $E$ and $F$ are taken inside the segments $BC$ and $CD$ so that $\angle EAF = 45^o$. The lines $AE$ and $AF$ intersect the circle circumscribed to the square at points $G$ and $H$ respectively. Prove that lines $EF$ and $GH$ are parallel.

Nathan starts with the number $0$, and randomly adds either $1$ or $2$ with equal probability until his number reaches or exceeds $2018$. What is the probability his number ends up being exactly $2018$?

Given that $g(n) = \frac{1}{{2 + \frac{1}{{3 + \frac{1}{{... + \frac{1}{{n - 1}}}}}}}}$ and $k(n) = \frac{1}{{2 + \frac{1}{{3 + \frac{1}{{... + \frac{1}{{n - 1 + \frac{1}{n}}}}}}}}}$, for natural $n$. Prove that $\left| {g(n) - k(n)} \right| \le \frac{1}{{(n - 1)!n!}}$.

Teacher has 100 weights with masses $1$ g, $2$ g, $\dots$, $100$ g. He wants to give 30 weights to Petya and 30 weights to Vasya so that no 11 Petya's weights have the same total mass as some 12 Vasya's weights, and no 11 Vasya's weights have the same total mass as some 12 Petya's weights. Can the teacher do that?

Smallville is populated by unmarried men and women, some of them are acquainted. Two city’s matchmakers are aware of all acquaintances. Once, one of matchmakers claimed: “I could arrange that every brunette man would marry a woman he was acquainted with”. The other matchmaker claimed “I could arrange that every blonde woman would marry a man she was acquainted with”. An amateur mathematician overheard their conversation and said “Then both arrangements could be done at the same time! ” Is he right?

Call a positive integer [i]emphatic[/i] if it can be written in the form $a^2+b!$, where $a$ and $b$ are positive integers. Prove that there are infinitely many positive integers $n$ such that $n$, $n+1$, and $n+2$ are all [i]emphatic[/i]. [i]Allen Wang[/i]