Consider positive integers $a<b$ and the set $C\subset\{a,a+1,a+2,\dots ,b-2,b-1,b\}$. Suppose $C$ has more than $\frac{b-a+1}{2}$ elements. Prove that there are two elements $x,y\in C$ that satisfy $x+y=a+b$. [i] (From "Radu Păun" contest, Radu Miculescu)[/i]

Let $P$ be an arbitrary point inside triangle $ABC$, denote by $A_{1}$ (different from $P$) the second intersection of line $AP$ with the circumcircle of triangle $PBC$ and define $B_{1},C_{1}$ similarly. Prove that $\left(1 \plus{} 2\cdot\frac {PA}{PA_{1}}\right)\left(1 \plus{} 2\cdot\frac {PB}{PB_{1}}\right)\left(1 \plus{} 2\cdot\frac {PC}{PC_{1}}\right)\geq 8$.

The USAMTS tug-of-war team needs to pick a representative to send to the national tug-of-war convention. They don't care who they send, as long as they don't send the weakest person on the team. Their team consists of $20$ people, who each pull with a different constant strength. They want to design a tournament, with each round planned ahead of time, which at the end will allow them to pick a valid representative. Each round of the tournament is a $10$-on-$10$ tug-of-war match. A round may end in one side winning, or in a tie if the strengths of each side are matched. Show that they can choose a representative using a tournament with $10$ rounds.

Given natural numbers $a,b$ and function $f: \mathbb{N} \to \mathbb{N} $ such that for any natural number $n, f\left( n+a \right)$ is divided by $f\left( {\left[ {\sqrt n } \right] + b} \right)$. Prove that for any natural $n$ exist $n$ pairwise distinct and pairwise relatively prime natural numbers ${{a}_{1}}$, ${{a}_{2}}$, $\ldots$, ${{a}_{n}}$ such that the number $f\left( {{a}_{i+1}} \right)$ is divided by $f\left( {{a}_{i}} \right)$ for each $i=1,2, \dots ,n-1$ . (Here $[x]$ is the integer part of number $x$, that is, the largest integer not exceeding $x$.)

Triangle $ABC$ satisfies $\angle ABC=\angle ACB=78^\circ$. Points $D$ and $E$ lie on $AB,AC$ and satisfy $\angle BCD=24^\circ$ and $\angle CBE=51^\circ$. If $\angle BED=x^\circ$, find $x$.

Let $ u_1$, $ u_2$, $ \ldots$, $ u_{1987}$ be an arithmetic progression with $ u_1 \equal{} \frac {\pi}{1987}$ and the common difference $ \frac {\pi}{3974}$. Evaluate \[ S \equal{} \sum_{\epsilon_i\in\left\{ \minus{} 1, 1\right\}}\cos\left(\epsilon_1 u_1 \plus{} \epsilon_2 u_2 \plus{} \cdots \plus{} \epsilon_{1987} u_{1987}\right) \]

How many ways are there to rearrange the letters of CCARAVEN? [i]2017 CCA Math Bonanza Lightning Round #1.2[/i]

Square $ABCD$ is divided into $n^2$ equal small squares by lines parallel to its sides.A spider starts from $A$ and moving only rightward or upwards,tries to reach $C$.Every "movement" of the spider consists of $k$ steps rightward and $m$ steps upwards or $m$ steps rightward and $k$ steps upwards(it can follow any possible order for the steps of each "movement").The spider completes $l$ "movements" and afterwards it moves without limitation (it still moves rightwards and upwards only).If $n=m\cdot l$,find the number of the possible paths the spider can follow to reach $C$.Note that $n,m,k,l\in \mathbb{N^{*}}$ with $k<m$.

Let $n$ be a positive integer. Determine the maximum of the sum $x_1+\cdots+x_n$ where $x_1,\ldots,x_n$ are non-negative integers satisfying the condition \[x_1^3+\cdots+x_n^3\le7n.\]

Every two of $200$ points in space are connected by a segment, no two intersecting each other. Each segment is painted in one colour, and the total number of colours is $k$. Peter wants to paint each of the $200$ points in one of the colours used to paint the segments, so that no segment connects two points both in the same colour as the segment itself. Can Peter always do this if (a) k = 7; [i](4 points)[/i] (b) k = 10? [i](4 points)[/i]

Equilateral triangle $\vartriangle ABC$ is inscribed in circle $\Omega$, which has a radius of $1$. Let the midpoint of $BC$ be $M$. Line $AM$ intersects $\Omega$ again at point $D$. Let $\omega$ be the circle with diameter $MD$. Compute the radius of the circle that is tangent to BC on the same side of $BC$ as $\omega$, internally tangent to $\Omega$, and externally tangent to $\omega$.

Let $ABCD$ a convex quadrilateral with $[BC]$ and $[CD]$'s midpoint is $P$ and $N$ respectively. If $$[AP]+[AN]=d$$ Show that, area of the $ABCD$ is less then $\frac{1}{2}d^2$

$(BEL 4)$ Let $O$ be a point on a nondegenerate conic. A right angle with vertex $O$ intersects the conic at points $A$ and $B$. Prove that the line $AB$ passes through a fixed point located on the normal to the conic through the point $O.$

Let $R_1 =1$ and $R_{n+1}= 1+ n\slash R_n$ for $n\geq 1.$ Show that for $n\geq 1,$ $$ \sqrt{n} \leq R_n \leq \sqrt{n} +1.$$

[b]p1.[/b] How many lines of symmetry does a square have? [b]p2.[/b] Compute$ 1/2 + 1/6 + 1/12 + 1/4$. [b]p3.[/b] What is the maximum possible area of a rectangle with integer side lengths and perimeter $8$? [b]p4.[/b] Given that $1$ printer weighs $400000$ pennies, and $80$ pennies weighs $2$ books, what is the weight of a printer expressed in books? [b]p5.[/b] Given that two sides of a triangle are $28$ and $3$ and all three sides are integers, what is the sum of the possible lengths of the remaining side? [b]p6.[/b] What is half the sum of all positive integers between $1$ and $15$, inclusive, that have an even number of positive divisors? [b]p7.[/b] Austin the Snowman has a very big brain. His head has radius $3$, and the volume of his torso is one third of his head, and the volume of his legs combined is one third of his torso. If Austin's total volume is $a\pi$ where $a$ is an integer, what is $a$? [b]p8.[/b] Neethine the Kiwi says that she is the eye of the tiger, a fighter, and that everyone is gonna hear her roar. She is standing at point $(3, 3)$. Neeton the Cat is standing at $(11,18)$, the farthest he can stand from Neethine such that he can still hear her roar. Let the total area of the region that Neeton can stand in where he can hear Neethine's roar be $a\pi$ where $a$ is an integer. What is $a$? [b]p9.[/b] Consider $2018$ identical kiwis. These are to be divided between $5$ people, such that the first person gets $a_1$ kiwis, the second gets $a_2$ kiwis, and so forth, with $a_1 \le a_2 \le a_3 \le a_4 \le a_5$. How many tuples $(a_1, a_2, a_3, a_4, a_5)$ can be chosen such that they form an arithmetic sequence? [b]p10.[/b] On the standard $12$ hour clock, each number from $1$ to $12$ is replaced by the sum of its divisors. On this new clock, what is the number of degrees in the measure of the non-reflex angle between the hands of the clock at the time when the hour hand is between $7$ and $6$ while the minute hand is pointing at $15$? [b]p11.[/b] In equiangular hexagon $ABCDEF$, $AB = 7$, $BC = 3$, $CD = 8$, and $DE = 5$. The area of the hexagon is in the form $\frac{a\sqrt{b}}{c}$ with $b$ square free and $a$ and $c$ relatively prime. Find $a+b+c$ where $a, b,$ and $c$ are integers. [b]p12.[/b] Let $\frac{p}{q} = \frac15 + \frac{2}{5^2} + \frac{3}{5^3} + ...$ . Find $p + q$, where $p$ and $q$ are relatively prime positive integers. [b]p13.[/b] Two circles $F$ and $G$ with radius $10$ and $4$ respectively are externally tangent. A square $ABMC$ is inscribed in circle $F$ and equilateral triangle $MOP$ is inscribed in circle $G$ (they share vertex $M$). If the area of pentagon $ABOPC$ is equal to $a + b\sqrt{c}$, where $a$, $b$, $c$ are integers $c$ is square free, then find $a + b + c$. [b]p14.[/b] Consider the polynomial $P(x) = x^3 + 3x^2 + ax + 8$. Find the sum of all integer $a$ such that the sum of the squares of the roots of $P(x)$ divides the sum of the coecients of $P(x)$. [b]p15.[/b] Nithin and Antonio play a number game. At the beginning of the game, Nithin picks a prime $p$ that is less than $100$. Antonio then tries to find an integer $n$ such that $n^6 + 2n^5 + 2n^4 + n^3 + (n^2 + n + 1)^2$ is a multiple of $p$. If Antonio can find such a number n, then he wins, otherwise, he loses. Nithin doesn't know what he is doing, and he always picks his prime randomly while Antonio always plays optimally. The probability of Antonio winning is $a/b$ where $a$ and $b$ are relatively prime positive integers. Find$a + b$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

There are $n$ players in a round-robin ping-pong tournament (i.e. every two persons will play exactly one game). After some matches have been played, it is known that the total number of matches that have been played among any $n-2$ people is equal to $3^k$ (where $k$ is a fixed integer). Find the sum of all possible values of $n$.

Define $g(x)$ as the largest value of$ |y^2 - xy|$ for $y$ in $[0, 1]$. Find the minimum value of $g$ (for real $x$).

Prove that the polynomial $$(a-x)^6 -3a(a-x)^5 +\frac{5}{2} a^2 (a-x)^4 -\frac{1}{2} a^4 (a-x)^2 $$ takes only negative values for $0<x<a$.

Let $\mathbb{N}$ be the set of all positive integers. Find all functions $f : \mathbb{N} \to \mathbb{N}$ such that $$f^{f^{f(x)}(y)}(z)=x+y+z+1$$ for all $x,y,z \in \mathbb{N}$. [i]Proposed by William Wang.[/i]

Let $G$ be the barycenter of triangle $ABC$. Let $D$ be a point such that $AGDB$ is a parallelogram. Show that $BG \parallel CD$.

Let $S$ be the set of points on the rays forming the sides of a $120^{\circ}$ angle, and let $P$ be a fixed point inside the angle on the angle bisector. Consider all distinct equilateral triangles $PQR$ with $Q$ and $R$ in $S$. (Points $Q$ and $R$ may be on the same ray, and switching the names of $Q$ and $R$ does not create a distinct triangle.) There are [asy] draw((0,0)--(6,10.2), EndArrow); draw((0,0)--(6,-10.2), EndArrow); draw((0,0)--(6,0), dotted); dot((6,0)); label("P", (6,0), S); [/asy] $ \textbf{(A)}\ \text{exactly 2 such triangles} \\ \qquad\textbf{(B)}\ \text{exactly 3 such triangles} \\ \qquad\textbf{(C)}\ \text{exactly 7 such triangles} \\ \qquad\textbf{(D)}\ \text{exactly 15 such triangles} \\ \qquad\textbf{(E)}\ \text{more than 15 such triangles} $

Let $A = \{1,2,..., 100\}$ and $B$ is a subset of $A$ having $48$ elements. Show that $B$ has two distint elements $x$ and $y$ whose sum is divisible by $11$.

We say that a polygon $P$ is inscribed in another polygon $Q$ when all the vertices of $P$ belong to the perimeter of $Q$. We also say in this case that $Q$ is circumscribed to $P$. Given a triangle $T$, let $\ell$ be the largest side of a square inscribed in $T$ and $L$ is the shortest side of a square circumscribed to $T$ . Find the smallest possible value of the ratio $L/\ell$ .

Real numbers $a,b$ and $c$ satisfy the equalities $2015 (a + b + c) =1$ and $ab+bc+ca=2015 abc$. Determine the numeric value of the expression $E=a^{2015}+b^{2015}+c^{2015}.$

Solve in integers $20^x+13^y=2013^z$.