Let $n$ be a fixed natural number. Find all natural numbers $ m$ for which \[\frac{1}{a^n}+\frac{1}{b^n}\ge a^m+b^m\] is satisfied for every two positive numbers $ a$ and $ b$ with sum equal to $2$.

p1. Bahri lives quite close to the clock gadang in the city of Bukit Tinggi West Sumatra. Bahri has an antique clock. On Monday $4$th March $2013$ at $10.00$ am, Bahri antique clock is two minutes late in comparison with Clock Tower. A day later, the antique clock was four minutes late compared to the Clock Tower. March $6$, $2013$ the clock is late six minutes compared to Jam Gadang. The following days Bahri observed that his antique clock exhibited the same pattern of delay. On what day and what date in $2014$ the antique Bahri clock (hand short and long hands) point to the same number as the Clock Tower? p2. In one season, the Indonesian Football League is participated by $20$ teams football. Each team competes with every other team twice. The result of each match is $3$ if you win, $ 1$ if you draw, and $0$ if you lose. Every week there are $10$ matches involving all teams. The winner of the competition is the team that gets the highest total score. At the end what week is the fastest possible, the winner of the competition on is the season certain? p3. Look at the following picture. The quadrilateral $ABCD$ is a cyclic. Given that $CF$ is perpendicular to $AF$, $CE$ is perpendicular to $BD$, and $CG$ is perpendicular to $AB$. Is the following statements true? Write down your reasons. $$\frac{BD}{CE}=\frac{AB}{CG}+ \frac{AD}{CF}$$ [img]https://cdn.artofproblemsolving.com/attachments/b/0/dbd97b4c72bc4ebd45ed6fa213610d62f29459.png[/img] p4. Suppose $M=2014^{2014}$. If the sum of all the numbers (digits) that make up the number $M$ equals $A$ and the sum of all the digits that make up the number $A$ equals $B$, then find the sum of all the numbers that make up $B$. p5. Find all positive integers $n < 200$ so that $n^2 + (n + 1)^2$ is square of an integer.

a) Let $x$ and $y$ be positive numbers such that $x + y = 2$. Show that $\frac{1}{x}+\frac{1}{y} \le \frac{1}{x^2}+\frac{1}{y^2}$ b) Let $x,y$ and $z$ be positive numbers such that $x + y +z= 2$. Show that $\frac{1}{x}+\frac{1}{y} +\frac{1}{z} +\frac{9}{4} \le \frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}$ .

Let $ABC$ be a triangle with $AB = 5$, $BC = 7$, and $CA = 8$, and let $D$ be a point on arc $\widehat{BC}$ of its circumcircle $\Omega$. Suppose that the angle bisectors of $\angle ADB$ and $\angle ADC$ meet $AB$ and $AC$ at $E$ and $F$, respectively, and that $EF$ and $BC$ meet at $G$. Line $GD$ meets $\Omega$ at $T$. If the maximum possible value of $AT^2$ can be expressed as $\frac{a}{b}$ for positive integers $a, b$ with $\gcd (a,b) = 1$, find $a + b$. [i]Proposed by Andrew Wu[/i]

Fuzzy likes isosceles trapezoids. He can choose lengths from $1, 2, \dots, 8$, where he may choose any amount of each length. He takes a multiset of three integers from $1, \dots, 8$. From this multiset, one length will become a base length, one will become a diagonal length, and one will become a leg length. He uses each element as either a diagonal, leg, or base length exactly once. Fuzzy is happy if he can use these lengths to make an isosceles trapezoid such that the undecided base has nonzero rational length. How many multiset choices can he make? (Multisets are unordered) [i]Proposed by Timothy Qian[/i]

A [i]standard $n$-sided die[/i] has $n$ sides labeled $1$ to $n.$ Luis, Luke, and Sean play a game in which they roll a fair standard $4$-sided die, a fair standard $6$-sided die, and a fair standard $8$-sided die, respectively. They lose the game if Luis's roll is less than Luke's roll, and Luke's roll is less than Sean's roll. Compute the probability that they lose the game.

Let $\omega$ be the circumcircle of a triangle $ABC$. Denote by $M$ and $N$ the midpoints of the sides $AB$ and $AC$, respectively, and denote by $T$ the midpoint of the arc $BC$ of $\omega$ not containing $A$. The circumcircles of the triangles $AMT$ and $ANT$ intersect the perpendicular bisectors of $AC$ and $AB$ at points $X$ and $Y$, respectively; assume that $X$ and $Y$ lie inside the triangle $ABC$. The lines $MN$ and $XY$ intersect at $K$. Prove that $KA=KT$.

Triangle $ABC$ with incircle $(I)$ touches the sides $AB, BC, AC$ at $F, D, E$, res. $I_b, I_c$ are $B$- and $C-$ excenters of $ABC$. $P, Q$ are midpoints of $I_bE, I_cF$. $(PAC)\cap AB=\{ A, R\}$, $(QAB)\cap AC=\{ A,S\}$. a. Prove that $PR, QS, AI$ are concurrent. b. $DE, DF$ cut $I_bI_c$ at $K, J$, res. $EJ\cap FK=\{ M\}$. $PE, QF$ cut $(PAC), (QAB)$ at $X, Y$ res. Prove that $BY, CX, AM$ are concurrent.

Determine all values of parameter $\alpha\in [0,2\pi]$ such that the equation $$(2\cos\alpha-1)x^2+4x+4\cos\alpha+2=0$$ has 1) a positive root $x_1$, 2) if a second root $x_2$ exists and if $x_2\neq x_1$, the $x_2\leq 0$.

The figure $ABCDE$ is a convex pentagon. Find the sum $\angle DAC + \angle EBD +\angle ACE +\angle BDA + \angle CEB$?

A triangular table with $n$ rows and $n$ columns is given, the $i$-th row ends with a field in the $v$-th column. In each field of the table, some of the numbers $1,2,..., n$ are written such that for each $k \in {1, 2,..., n}$ all the numbers $1,2,..., n$ occur in the union of the $k$-th row and the $k$-th column. Prove that for odd $n$, each of the numbers $1,2,..., n$ is written in the last box of a row. [img]https://cdn.artofproblemsolving.com/attachments/f/9/2aed55628edb1505c7de27c152127b04d8d991.png[/img]

On the midline of the isosceles trapezoid $ABCD$ ($BC \parallel AD$) find the point $K$, for which the sum of the angles $\angle DAK + \angle BCK$ will be the smallest.

Investigate the boundary of the domain of stability ($\max \text{Re }\lambda_j < 0$) in the space of coefficients of the equation $\dddot{x} + a\ddot{x} + b\dot{x} + cx = 0$.

Let $\mathbb{Z}_{\ge 0}$ denote the set of nonnegative integers. Define a function $f:\mathbb{Z}_{\ge 0} \to\mathbb{Z}$ with $f\left(0\right)=1$ and \[ f\left(n\right)=512^{\left\lfloor n/10 \right\rfloor}f\left(\left\lfloor n/10 \right\rfloor\right)\] for all $n \ge 1$. Determine the number of nonnegative integers $n$ such that the hexadecimal (base $16$) representation of $f\left(n\right)$ contains no more than $2500$ digits. [i]Proposed by Tristan Shin[/i]

Suppose that k and x are positive integers such that $$\frac{k}{2}=\left( \sqrt{1 +\frac{\sqrt3}{2}}\right)^x+\left( \sqrt{1 -\frac{\sqrt3}{2}}\right)^x.$$ Find the sum of all possible values of $k$

Alice and Bob take $a$ and $b$ candies respectively, where $0\leq a,b\leq3$, from a pile of $6$ identical candies. They draw the candies one at a time, but one person may draw multiple candies in a row. For example, if $a=2$ and $b=3$, a possible order of drawing could be Alice, Bob, Bob, Alice, Bob. In how many ways (considering order of drawing and values of $a$ and $b$) can this happen? [i]2017 CCA Math Bonanza Team Round #6[/i]

At Kanye Crest Academy, employees get paid in CCA Math Bananas$^{\text{TM}}$. At the end of $2018$, Professor Shian Bray was given a $10\%$ pay raise from his salary at the end of $2017$. However, inflation caused the worth of a CCA Math Banana$^{\text{TM}}$ to decrease by $1\%$. If Prof. Bray's salary at the end of $2017$ was worth one million dollars, how much (in dollars) was Prof. Bray's salary worth at the end of $2018$? Assume that the value of the dollar has not changed. [i]2019 CCA Math Bonanza Lightning Round #1.2[/i]

[b]p1.[/b] One out of $12$ coins is counterfeited. It is known that its weight differs from the weight of a valid coin but it is unknown whether it is lighter or heavier. How to detect the counterfeited coin with the help of four trials using only a two-pan balance without weights? [b]p2.[/b] Below a $3$-digit number $c d e$ is multiplied by a $2$-digit number $a b$ . Find all solutions $a, b, c, d, e, f, g$ if it is known that they represent distinct digits. $\begin{tabular}{ccccc} & & c & d & e \\ x & & & a & b \\ \hline & & f & e & g \\ + & c & d & e & \\ \hline & b & b & c & g \\ \end{tabular}$ [b]p3.[/b] Find all integer $n$ such that $\frac{n + 1}{2n - 1}$is an integer. [b]p4[/b]. There are several straight lines on the plane which split the plane in several pieces. Is it possible to paint the plane in brown and green such that each piece is painted one color and no pieces having a common side are painted the same color? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

$(FRA 5)$ Let $\alpha(n)$ be the number of pairs $(x, y)$ of integers such that $x+y = n, 0 \le y \le x$, and let $\beta(n)$ be the number of triples $(x, y, z)$ such that$ x + y + z = n$ and $0 \le z \le y \le x.$ Find a simple relation between $\alpha(n)$ and the integer part of the number $\frac{n+2}{2}$ and the relation among $\beta(n), \beta(n -3)$ and $\alpha(n).$ Then evaluate $\beta(n)$ as a function of the residue of $n$ modulo $6$. What can be said about $\beta(n)$ and $1+\frac{n(n+6)}{12}$? And what about $\frac{(n+3)^2}{6}$? Find the number of triples $(x, y, z)$ with the property $x+ y+ z \le n, 0 \le z \le y \le x$ as a function of the residue of $n$ modulo $6.$What can be said about the relation between this number and the number $\frac{(n+6)(2n^2+9n+12)}{72}$?

Prove that $$x^4 + 2x^2 + 2x + 2$$ is not the product of two polynomials $x^2 + ax + b$ and $x^2 + cx + d$ in which $a$, $b$, $c$, $d$ are integers.

We record one number every workday: Monday, Tuesday, Wednesday, Thursday, and Friday. On the first Monday the number we record is ten and a half. On every Tuesday and every Thursday the number we record is one third of what it was on the previous workday. On every Monday, Wednesday, and Friday the number we record is double what it was on the previous workday. This will go on forever. What is the sum of all of the numbers we will record?

A very well known family of mathematicians has three children called [i]Antonia, Bernhard[/i] and [i]Christian[/i]. Each evening one of the children has to do the dishes. One day, their dad decided to construct of plan that says which child has to do the dishes at which day for the following $55$ days. Let $x$ be the number of possible such plans in which Antonia has to do the dishes on three consecutive days at least once. Furthermore, let $y$ be the number of such plans in which there are three consecutive days in which Antonia does the dishes on the first, Bernhard on the second and Christian on the third day. Determine, whether $x$ and $y$ are different and if so, then decide which of those is larger.

In trapezoid $ABCD$, diagonal $AC$ is the bisector of angle $A$. Point $K$ is the midpoint of diagonal $AC$. It is known that $DC = DK$. Find the ratio of the bases $AD: BC$.

A set system $ (S,L)$ is called a Steiner triple system, if $ L\neq\emptyset$, any pair $ x,y\in S$, $ x\neq y$ of points lie on a unique line $ \ell\in L$, and every line $ \ell\in L$ contains exactly three points. Let $ (S,L)$ be a Steiner triple system, and let us denote by $ xy$ the thrid point on a line determined by the points $ x\neq y$. Let $ A$ be a group whose factor by its center $ C(A)$ is of prime power order. Let $ f,h: S\to A$ be maps, such that $ C(A)$ contains the range of $ f$, and the range of $ h$ generates $ A$. Show, that if \[ f(x) \equal{} h(x)h(y)h(x)h(xy)\] holds for all pairs $ x\neq y$ of points, then $ A$ is commutative, and there exists an element $ k\in A$, such that $ f(x) \equal{} kh(x)$ for all $ x\in S$.

Show that there is no function $f : \mathbb N \to \mathbb N$ satisfying $f(f(n)) = n + 1$ for each positive integer $n.$