Let $m,n$ be positive integers of distinct parities and such that $m<n<5m$. Show that there exists a partition with two element subsets of the set $\{ 1,2,3,\ldots ,4mn\}$ such that the sum of numbers in each set is a perfect square. [i]Dinu Șerbănescu[/i]

On a plane is given a finite set of points. Prove that the points can be covered by open squares $Q_1,Q_2,...,Q_n$ such that $1 \le\frac{N_j}{S_j} \le 4$ for $j = 1,...,n,$ where $N_j$ is the number of points from the set inside square $Q_j$ and $S_j$ is the area of $Q_j$.

We put four numbers $1,2, 3,4$ around a circle in order. One starts at the number $1$ and every step, he moves to an adjacent number on either side. How many ways he can move such that sum of the numbers he visits in his path (including the starting number) is equal to $21$?

If $a_1<a_2<\cdots<a_n$ be real numbers, prove that: \[ a_1a_2^4+a_2a_3^4+\cdots+a_{n-1}a_n^4+a_na_1^4\geq a_2a_1^4+a_3a_2^4+\cdots+a_na_{n-1}^4+a_1a_n^4. \]

Prove that the cube of any positive integer greater than 1 can be represented as a difference of the squares of two positive integers.

You are given two diameters $AB$ and $CD$ of circle $\Omega$ with radius $1$. A circle is drawn in one of the smaller sectors formed such that it is tangent to $AB$ at $E$, tangent to $CD$ at $F$, and tangent to $\Omega$ at $P$. Lines $PE$ and $PF$ intersect $\Omega$ again at $X$ and $Y$ . What is the length of $XY$ , given that $AC = \frac23$ ?

Compute the largest value of $r$ such that three non-overlapping circles of radius $r$ can be inscribed in a unit square.

Ana writes an infinite list of numbers using the following procedure. The first number of the list is a positive integer $a$ chosen by Ana. From there, each number in the list is obtained by calculating the sum of all the integers from $1$ to the last number written. For example, if $a = 3$, Ana's list starts as $3, 6, 21, 231, \dots$ because $1 + 2 + 3 = 6$, $1 + 2 + 3 + 4 + 5 + 6 = 21$ and $1 + 2 + 3 + \dots + 21 = 231$. Is it possible for all the numbers in Ana's list to be even?

How many different values can $ \angle ABC$ take, where $ A,B,C$ are distinct vertices of a cube?

Let $f:(-1,1)\to \mathbb{R}$ be a twice differentiable function such that $$2f’(x)+xf''(x)\geqslant 1 \quad \text{ for } x\in (-1,1).$$ Prove that $$\int_{-1}^{1}xf(x)dx\geqslant \frac{1}{3}.$$ [i]Proposed by Orif Ibrogimov, ETH Zurich and National University of Uzbekistan and Karim Rakhimov, Scuola Normale Superiore and National University of Uzbekistan[/i]

[b]p1.[/b] Two players play the following game. On the lowest left square of an $8 \times 8$ chessboard there is a rook (castle). The first player is allowed to move the rook up or to the right by an arbitrary number of squares. The second layer is also allowed to move the rook up or to the right by an arbitrary number of squares. Then the first player is allowed to do this again, and so on. The one who moves the rook to the upper right square wins. Who has a winning strategy? [b]p2.[/b] Find the smallest positive whole number that ends with $17$, is divisible by $17$, and the sum of its digits is $17$. [b]p3.[/b] Three consecutive $2$-digit numbers are written next to each other. It turns out that the resulting $6$-digit number is divisible by $17$. Find all such numbers. [b]p4.[/b] Let $ABCD$ be a convex quadrilateral (a quadrilateral $ABCD$ is called convex if the diagonals $AC$ and $BD$ intersect). Suppose that $\angle CBD = \angle CAB$ and $\angle ACD = \angle BDA$ . Prove that $\angle ABC = \angle ADC$. [b]p5.[/b] A circle of radius $1$ is cut into four equal arcs, which are then arranged to make the shape shown on the picture. What is its area? [img]https://cdn.artofproblemsolving.com/attachments/f/3/49c3fe8b218ab0a5378ecc635b797a912723f9.png[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Determine all real numbers a for which there exists positive reals $x_{1}, \ldots, x_{5}$ which satisfy the relations $ \sum_{k=1}^{5} kx_{k}=a,$ $ \sum_{k=1}^{5} k^{3}x_{k}=a^{2},$ $ \sum_{k=1}^{5} k^{5}x_{k}=a^{3}.$

Given a sequence of prime numbers $p_1, p_2,\cdots$ , with the following property: $p_{n+2}$ is the largest prime divisor of $p_n+p_{n+1}+2018$ Show that the set $\{p_i\}_{i\in \mathbb{N}}$ is finite.

Circle $\omega$ is tangent to sides $AB$,$AC$ of triangle $ABC$ at $D$,$E$ respectively, such that $D\neq B$, $E\neq C$ and $BD+CE<BC$. $F$,$G$ lies on $BC$ such that $BF=BD$, $CG=CE$. Let $DG$ and $EF$ meet at $K$. $L$ lies on minor arc $DE$ of $\omega$, such that the tangent of $L$ to $\omega$ is parallel to $BC$. Prove that the incenter of $\triangle ABC$ lies on $KL$.

Let $O$ be the circumcenter of triangle $ABC$. A line through $O$ intersects the sides $CA$ and $CB$ at points $D$ and $E$ respectively, and meets the circumcircle of $ABO$ again at point $P \neq O$ inside the triangle. A point $Q$ on side $AB$ is such that $\frac{AQ}{QB}=\frac{DP}{PE}$. Prove that $\angle APQ = 2\angle CAP$. [i]Proposed by Dusan Djukic[/i]

Find out the maximum value of the numbers of edges of a solid regular octahedron that we can see from a point out of the regular octahedron.(We define we can see an edge $AB$ of the regular octahedron from point $P$ outside if and only if the intersection of non degenerate triangle $PAB$ and the solid regular octahedron is exactly edge $AB$.

A circle with radius $r$ is contained within the region bounded by a circle with radius $R$. The area bounded by the larger circle is $a/b$ times the area of the region outside the smaller circle and inside the larger circle. Then $R:r$ equals: $\textbf{(A) }\sqrt a:\sqrt b\qquad \textbf{(B) }\sqrt a:\sqrt{a-b}\qquad \textbf{(C) }\sqrt b:\sqrt{a-b}\qquad$ $\textbf{(D) }a:\sqrt{a-b}\qquad \textbf{(E) }b:\sqrt{a-b}$

The real numbers $x_1, x_2, ... , x_n$ belong to the interval $(0,1)$ and satisfy $x_1 + x_2 + ... + x_n = m + r$, where $m$ is an integer and $r \in [0,1)$. Show that $x_1 ^2 + x_2 ^2 + ... + x_n ^2 \leq m + r^2$.

Let $x,y,z$ be the non-negative real numbers satisfying $xy+yz+zx\leq 1$. Prove that: $$1-xy-yz-zx\leq (6-2\sqrt{6})(1-\min\{x,y,z\}).$$

Submit a positive integer less than or equal to $15$. Your goal is to submit a number that is close to the number of teams submitting it. If you submit $N$ and the total number of teams at the competition (including your own team) who submit $N$ is $T$, your score will be $\frac{2}{0.5|N-T|+1}$. [i]2020 CCA Math Bonanza Lightning Round #5.4[/i]

Let $m,n$ are positive integers. a)Prove that $(m,n)=2\sum_{k=0}^{m-1}[\frac{kn}{m}]+m+n-mn$. b)If $m,n\geq 2$, prove that $\sum_{k=0}^{m-1}[\frac{kn}{m}]=\sum_{k=0}^{n-1}[\frac{km}{n}]$.

Let $S = \{1,2,..., 2017\}$. Find the maximal $n$ with the property that there exist $n$ distinct subsets of $S$ such that for no two subsets their union equals $S$. Proposed by Gerhard Woeginger

Let $I$ be the incenter and $O$ be the circumcenter of triangle $ABC,$ where $\angle A < \angle B < \angle C.$ Points $P$ and $Q$ are such that $AIOP$ and $BIOQ$ are isosceles trapezoids ($AI \parallel OP,$ $BI \parallel OQ$). Prove that $CP = CQ.$ [i]Proposed by Volodymyr Brayman and Matthew Kurskyi[/i]

We define the following sequences: • Sequence $A$ has $a_n = n$. • Sequence $B$ has $b_n = a_n$ when $a_n \not\equiv 0$ (mod 3) and $b_n = 0$ otherwise. • Sequence $C$ has $c_n =\sum_{i=1}^{n} b_i$ .• Sequence $D$ has $d_n = c_n$ when $c_n \not\equiv 0$ (mod 3) and $d_n = 0$ otherwise. • Sequence $E$ has $e_n =\sum_{i=1}^{n}d_i$ Prove that the terms of sequence E are exactly the perfect cubes.

Felix is playing a card-flipping game. $n$ face-down cards are randomly colored, each with equal probability of being black or red. Felix starts at the $1$st card. When Felix is at the $k$th card, he guesses its color and then flips it over. For $k < n$, if he guesses correctly, he moves onto the $(k + 1)$-th card. If he guesses incorrectly, he gains $k$ penalty points, the cards are replaced with newly randomized face-down cards, and he moves back to card $1$ to continue guessing. If Felix guesses the $n$th card correctly, the game ends. What is the expected number of penalty points Felix earns by the end of the game?