There are 300 apples, any two of which differ in weight by no more than three times. Prove that they can be arranged into bags of four apples each so that any two bags differ in weight by no more than than one and a half times.

A rectangle is drawn on checkered paper, the sides of which form angles of $45^o$ with the grid lines, and the vertices do not lie on the grid lines. Can an odd number of grid lines intersect each side of a rectangle?

Let $a,b,c$ be sides of a triangle. Show that $a^3+b^3+3abc>c^3$.

Let $A=\{1,2,3,\ldots,40\}$. Find the least positive integer $k$ for which it is possible to partition $A$ into $k$ disjoint subsets with the property that if $a,b,c$ (not necessarily distinct) are in the same subset, then $a\ne b+c$.

On a plane, two equilateral triangles (of side length $1$) share a side, and a circle is drawn with the common side as a diameter. Find the area of the set of all points that lie inside exactly one of these shapes. [i]Proposed by Howard Halim[/i]

Find all prime numbers whose sixth power does not give remainder $1$ when dividing by $504$

Let $ABC$ be an acute triangle and let $D$ be the foot of the perpendicular from $A$ to side $BC$. Let $P$ be a point on segment $AD$. Lines $BP$ and $CP$ intersect sides $AC$ and $AB$ at $E$ and $F$, respectively. Let $J$ and $K$ be the feet of the peroendiculars from $E$ and $F$ to $AD$, respectively. Show that $\frac{FK}{KD}=\frac{EJ}{JD}$.

Consider the endomorphism ring of an Abelian torsion-free (resp. torsion) group $ G$. Prove that this ring is Neumann-regular if and only if $ G$ is a discrete direct sum of groups isomorphic to the additive group of the rationals (resp. ,a discrete direct sum of cyclic groups of prime order). (A ring $ R$ is called Neumann-regular if for every $ \alpha \in R$ there exists a $ \beta \in R$ such that $ \alpha \beta \alpha\equal{}\alpha$.) [i]E. Freid[/i]

Let $A$ and $B$ be two sets of non-negative integers, define $A+B$ as the set of the values obtained when we sum any (one) element of the set $A$ with any (one) element of the set $B$. For instance, if $A=\{2,3\}$ and $B=\{0,1,2,5\}$ so $A+B=\{2,3,4,5,7,8\}$. Determine the least integer $k$ such that there is a pair of sets $A$ and $B$ of non-negative integers with $k$ and $2k$ elements, respectively, and $A+B=\{0,1,2,\dots, 2019,2020\}$

In a word of more than $10$ letters, any two consecutive letters are different. Prove that one can change places of two consecutive letters so that the resulting word is not [i]periodic[/i], that is, cannot be divided into equal subwords.

For any two nonnegative integers $n$ and $k$ satisfying $n\geq k$, we define the number $c(n,k)$ as follows: - $c\left(n,0\right)=c\left(n,n\right)=1$ for all $n\geq 0$; - $c\left(n+1,k\right)=2^{k}c\left(n,k\right)+c\left(n,k-1\right)$ for $n\geq k\geq 1$. Prove that $c\left(n,k\right)=c\left(n,n-k\right)$ for all $n\geq k\geq 0$.

Given are $4004$ distinct points, which lie in the interior of a convex polygon of area $1$. Prove that there exists a convex polygon of area $\frac{1}{2003}$, included in the given polygon, such that it does not contain any of the given points in its interior.

$(\text{a})$ Do there exist 14 consecutive positive integers each of which is divisible by one or more primes $p$ from the interval $2\le p \le 11$? $(\text{b})$ Do there exist 21 consecutive positive integers each of which is divisible by one or more primes $p$ from the interval $2\le p \le 13$?

Let there be an infinite sequence $ a_{k} $ with $ k\geq 1 $ defined by: $ a_{k+2} = a_{k} + 14 $ and $ a_{1} = 12 $ , $ a_{2} = 24 $. [b]a)[/b] Does $2012$ belong to the sequence? [b]b)[/b] Prove that the sequence doesn't contain perfect squares.

Is it possible to cut a square into five squares?

Given a circle $\Omega$ tangent to side $AB$ of angle $\angle BAC$ and lying outside this angle. We consider circles $w$ inscribed in angle $BAC$. The internal tangent of $\Omega$ and $w$, different from $AB$, touches $w$ at a point $K$. Let $L$ be the point of tangency of $w$ and $AC$. Prove that all such lines $KL$ pass through a fixed point without depending on the choice of circle $w$.

A teacher wants her $N$ students to know each other, so she creates various clubs of three people, so that each student can participate in several clubs. The clubs are formed in such a way that if $A$ and $B$ are two people, then there is a single club such that $A$ and $B$ are two of its three members. [b](1)[/b] Show that there is no way for the teacher to form the clubs if $N = 11$. [b](2)[/b] Show that the teacher can do it if $N = 9$.

Let $A$ be the sum of seven $7\text{’s}$. Let $B$ be the sum of seven $A\text{’s}$. What is $B$?

Let $p>7$ be a prime and let $A$ be subset of $\{0,1, \ldots, p-1\}$ with size at least $\frac{p-1}{2}$. Show that for each integer $r$, there exist $a, b, c, d \in A$, not necessarily distinct, such that $ab-cd \equiv r \pmod p$.

Let $n$ be a positive integer and let $p$ be a prime number. Prove that if $a$, $b$, $c$ are integers (not necessarily positive) satisfying the equations \[ a^n + pb = b^n + pc = c^n + pa\] then $a = b = c$. [i]Proposed by Angelo Di Pasquale, Australia[/i]

Let $B$ and $C$ be two fixed points on a circle centered at $O$ that are not diametrically opposed. Let $A$ be a variable point on the circle distinct from $B$ and $C$ and not belonging to the perpendicular bisector of $BC$. Let $H$ be the orthocenter of $\triangle ABC$, and $M$ and $N$ be the midpoints of the segments $BC$ and $AH$, respectively. The line $AM$ intersects the circle again at $D$, and finally, $NM$ and $OD$ intersect at $P$. Determine the locus of points $P$ as $A$ moves around the circle.

In the game of Colonel Blotto, you have 100 troops to distribute among 10 castles. Submit a 10-tuple $(x_1, x_2, \dots x_{10})$ of nonnegative integers such that $x_1 + x_2 + \dots + x_{10} = 100$, where each $x_i$ represent the number of troops you want to send to castle $i$. Your troop distribution will be matched up against each opponent's and you will win 10 points for each castle that you send more troops to (if you send the same number, you get 5 points, and if you send fewer, you get none). Your aim is to score the most points possible averaged over all opponents. For example, if team $A$ submits $(90,10,0,\dots,0)$, team B submits $(11,11,11,11,11,11,11,11,11,1)$, and team C submits $(10,10,10,\dots 10)$, then team A will win 10 points against team B and 15 points against team C, while team B wins 90 points against team C. Team A averages 12.5 points, team B averages 90 points, and team C averages 47.5 points. [i]2017 CCA Math Bonanza Lightning Round #5.4[/i]

For an arbitrary positive integer $m$, not divisible by $3$, consider the permutation $x \mapsto 3x \pmod{m}$ on the set $\{ 1,2,\dotsc ,m-1\}$. This permutation can be decomposed into disjointed cycles; for instance, for $m=10$ the cycles are $(1\mapsto 3\to 9,\mapsto 7,\mapsto 1)$, $(2\mapsto 6\mapsto 8\mapsto 4\mapsto 2)$ and $(5\mapsto 5)$. For which integers $m$ is the number of cycles odd?

Let $ABC$ be an acute triangle with circumcircle $\omega$. The altitudes $AD$, $BE$ and $CF$ of the triangle $ABC$ intersect at point $H$. A point $K$ is chosen on the line $EF$ such that $KH\parallel BC$. Prove that the reflection of $H$ in $KD$ lies on $\omega$.

Consider fractions $\frac{a}{b}$ where $a$ and $b$ are positive integers. (a) Prove that for every positive integer $n$, there exists such a fraction $\frac{a}{b}$ such that $\sqrt{n} \le \frac{a}{b} \le \sqrt{n+1}$ and $b \le \sqrt{n}+1$. (b) Show that there are infinitely many positive integers $n$ such that no such fraction $\frac{a}{b}$ satisfies $\sqrt{n} \le \frac{a}{b} \le \sqrt{n+1}$ and $b \le \sqrt{n}$.