[u]Round 4[/u] [b]p10.[/b] How many divisors of $10^{11}$ have at least half as many divisors that $10^{11}$ has? [b]p11.[/b] Let $f(x, y) = \frac{x}{y}+\frac{y}{x}$ and $g(x, y) = \frac{x}{y}-\frac{y}{x} $. Then, if $\underbrace{f(f(... f(f(}_{2021 fs} f(f(1, 2), g(2,1)), 2), 2)... , 2), 2)$ can be expressed in the form $a + \frac{b}{c}$, where $a$, $b$,$c$ are nonnegative integers such that $b < c$ and $gcd(b,c) = 1$, find $a + b + \lceil (\log_2 (\log_2 c)\rceil $ [b]p12.[/b] Let $ABC$ be an equilateral triangle, and let$ DEF$ be an equilateral triangle such that $D$, $E$, and $F$ lie on $AB$, $BC$, and $CA$, respectively. Suppose that $AD$ and $BD$ are positive integers, and that $\frac{[DEF]}{[ABC]}=\frac{97}{196}$. The circumcircle of triangle $DEF$ meets $AB$, $BC$, and $CA$ again at $G$, $H$, and $I$, respectively. Find the side length of an equilateral triangle that has the same area as the hexagon with vertices $D, E, F, G, H$, and $I$. [u]Round 5 [/u] [b]p13.[/b] Point $X$ is on line segment $AB$ such that $AX = \frac25$ and $XB = \frac52$. Circle $\Omega$ has diameter $AB$ and circle $\omega$ has diameter $XB$. A ray perpendicular to $AB$ begins at $X$ and intersects $\Omega$ at a point $Y$. Let $Z$ be a point on $\omega$ such that $\angle YZX = 90^o$. If the area of triangle $XYZ$ can be expressed as $\frac{a}{b}$ for positive integers $a, b$ with $gcd(a, b) = 1$, find $a + b$. [b]p14.[/b] Andrew, Ben, and Clayton are discussing four different songs; for each song, each person either likes or dislikes that song, and each person likes at least one song and dislikes at least one song. As it turns out, Andrew and Ben don't like any of the same songs, but Clayton likes at least one song that Andrew likes and at least one song that Ben likes! How many possible ways could this have happened? [b]p15.[/b] Let triangle $ABC$ with circumcircle $\Omega$ satisfy $AB = 39$, $BC = 40$, and $CA = 25$. Let $P$ be a point on arc $BC$ not containing $A$, and let $Q$ and $R$ be the reflections of $P$ in $AB$ and $AC$, respectively. Let $AQ$ and $AR$ meet $\Omega$ again at $S$ and $T$, respectively. Given that the reflection of $QR$ over $BC$ is tangent to $\Omega$ , $ST$ can be expressed as $\frac{a}{b}$ for positive integers $a, b$ with $gcd(a,b)= 1$. Find $a + b$. PS. You should use hide for answers. Rounds 1-3 have been posted [url=https://artofproblemsolving.com/community/c4h3131401p28368159]here [/url] and 6-7 [url=https://artofproblemsolving.com/community/c4h3131434p28368604]here [/url],Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

A infinite sequence $\{ a_n \}_{n \ge 0}$ of real numbers satisfy $a_n \ge n^2$. Suppose that for each $i, j \ge 0$ there exist $k, l$ with $(i,j) \neq (k,l)$, $l - k = j - i$, and $a_l - a_k = a_j - a_i$. Prove that $a_n \ge (n + 2016)^2$ for some $n$.

[u]Round 1[/u] [b]p1.[/b] Alice and Bob compiled a list of movies that exactly one of them saw, then Cindy and Dale did the same. To their surprise, these two lists were identical. Prove that if Alice and Cindy list all movies that exactly one of them saw, this list will be identical to the one for Bob and Dale. [b]p2.[/b] Several whole rounds of cheese were stored in a pantry. One night some rats sneaked in and consumed $10$ of the rounds, each rat eating an equal portion. Some were satisfied, but $7$ greedy rats returned the next night to finish the remaining rounds. Their portions on the second night happened to be half as large as on the first night. How many rounds of cheese were initially in the pantry? [b]p3.[/b] You have $100$ pancakes, one with a single blueberry, one with two blueberries, one with three blueberries, and so on. The pancakes are stacked in a random order. Count the number of blueberries in the top pancake, and call that number N. Pick up the stack of the top N pancakes, and flip it upside down. Prove that if you repeat this counting-and-flipping process, the pancake with one blueberry will eventually end up at the top of the stack. [b]p4.[/b] There are two lemonade stands along the $4$-mile-long circular road that surrounds Sour Lake. $100$ children live in houses along the road. Every day, each child buys a glass of lemonade from the stand that is closest to her house, as long as she does not have to walk more than one mile along the road to get there. A stand's [u]advantage [/u] is the difference between the number of glasses it sells and the number of glasses its competitor sells. The stands are positioned such that neither stand can increase its advantage by moving to a new location, if the other stand stays still. What is the maximum number of kids who can't buy lemonade (because both stands are too far away)? [b]p5.[/b] Merlin uses several spells to move around his $64$-room castle. When Merlin casts a spell in a room, he ends up in a different room of the castle. Where he ends up only depends on the room where he cast the spell and which spell he cast. The castle has the following magic property: if a sequence of spells brings Merlin from some room $A$ back to room $A$, then from any other room $B$ in the castle, that same sequence brings Merlin back to room $B$. Prove that there are two different rooms $X$ and $Y$ and a sequence of spells that both takes Merlin from $X$ to $Y$ and from $Y$ to $X$. [u]Round 2[/u] [b]p6.[/b] Captains Hook, Line, and Sinker are deciding where to hide their treasure. It is currently buried at the $X$ in the map below, near the lairs of the three pirates. Each pirate would prefer that the treasure be located as close to his own lair as possible. You are allowed to propose a new location for the treasure to the pirates. If at least two out of the three pirates prefer the new location (because it moves closer to their own lairs), then the treasure will be moved there. Assuming the pirates’ lairs form an acute triangle, is it always possible to propose a sequence of new locations so that the treasure eventually ends up in your backyard (wherever that is)? [img]https://cdn.artofproblemsolving.com/attachments/c/c/a9e65624d97dec612ef06f8b30be5540cfc362.png[/img] [b]p7.[/b] Homer went on a Donut Diet for the month of May ($31$ days). He ate at least one donut every day of the month. However, over any stretch of $7$ consecutive days, he did not eat more than $13$ donuts. Prove that there was some stretch of consecutive days over which Homer ate exactly $30$ donuts. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $a_0$ be a positive real number and consider the general term sequence $a_n$ defined by $$a_n =a_{n-1} + \frac{1}{a_{n-1}} \,\,\, n=1,2,3,...$$ Prove that $a_{1998} > 63$.

Let $\{a_n\}^\infty_{n=0}$ be the sequence of integers such that $a_0=1$, $a_1=1$, $a_{n+2}=2a_{n+1}-2a_n$. Decide whether $$a_n=\sum_{k=0}^{\left\lfloor\frac n2\right\rfloor}\binom n{2k}(-1)^k.$$

Let be a natural number $ n\ge 2 $ and a real number $ r>1. $ Determine the natural numbers $ k $ having the property that the affixes of $ r^ke^{\pi ki/n} ,r^{k+1}e^{\pi (k+1)i/n} ,r^{k+n}e^{\pi (k+n)i/n} ,r^{k+n+1}e^{\pi (k+n+1) i/n} $ in the complex plane represent the vertices of a trapezoid.

Compute the sum of all real solutions $\alpha$ (in radians) to the equation \[ |\sin\alpha|=\left\lfloor \frac{\alpha}{20} \right\rfloor. \]

Determine the minimal and maximal values the expression $\frac{|a+b|+|b+c|+|c+a|}{|a|+|b|+|c|}$ can take, where $a,b,c$ are real numbers.

Find all natural numbers $n \ge 1$ such that there is a polynomial $p(x)$ with integer coefficients for which $p (1) = p (2) = 0$ and where $p (n)$ is a prime number .

The scores of this problem were: one time 17/20 (by the runner-up) one time 4/20 (by Andrei Negut) one time 1/20 (by the winner) the rest had zero... just to give an idea of the difficulty. Let $A_{i},B_{i},S_{i}$ ($i=1,2,3$) be invertible real $2\times 2$ matrices such that [list][*]not all $A_{i}$ have a common real eigenvector, [*]$A_{i}=S_{i}^{-1}B_{i}S_{i}$ for $i=1,2,3$, [*]$A_{1}A_{2}A_{3}=B_{1}B_{2}B_{3}=I$.[/list] Prove that there is an invertible $2\times 2$ matrix $S$ such that $A_{i}=S^{-1}B_{i}S$ for all $i=1,2,3$.

Prove that if $ a + b = 1 $, then $$ a^5 + b^5 \geq \frac{1}{16}$$

Prove that the equation $x^{2}+p|x| = qx - 1 $ has 4 distinct real solutions if and only if $p+|q|+2<0$ ($p$ and $q$ are two real parameters).

Fix integers $n\geq 2$ and $1\leq m\leq n-1$. Let $a_0, a_1, \dots, a_n$ be non-negative real numbers satisfying $a_0+a_1+\dots +a_n=1$. Prove that, if $\sum_{k=0}^n a_kx^k < x^m$ for some $0<x<1$, then $$\sum_{k=0}^{m-1}(m-k)a_k < \sum_{k=m+1}^n (k-m)a_k.$$

For every pair $(m, n)$ of positive integers, a positive real number $a_{m, n}$ is given. Assume that \[a_{m+1, n+1} = \frac{a_{m, n+1} a_{m+1, n} + 1}{a_{m, n}}\] for all positive integers $m$ and $n$. Suppose further that $a_{m, n}$ is an integer whenever $\min(m, n) \le 2$. Prove that $a_{m, n}$ is an integer for all positive integers $m$ and $n$.

If $x , y$ are real numbers such that $x^2 + xy + y^2 = 1$ , find the least and the greatest value( minimum and maximum) of the expression $K = x^3y + xy^3$ [u]Babis[/u] [b] Sorry !!! I forgot to write that these 4 problems( 4 topics) were [u]JUNIOR LEVEL[/u][/b]

Solve the equation $\{(x + 1)^3\} = x^3$, where $\{z\}$ is the fractional part of the number z, i.e. $\{z\} = z - [z]$.

Let $a_{ij}$ $i=1,2,3$; $j=1,2,3$ be real numbers such that $a_{ij}$ is positive for $i=j$ and negative for $i\neq j$. Prove the existence of positive real numbers $c_{1}$, $c_{2}$, $c_{3}$ such that the numbers \[a_{11}c_{1}+a_{12}c_{2}+a_{13}c_{3},\qquad a_{21}c_{1}+a_{22}c_{2}+a_{23}c_{3},\qquad a_{31}c_{1}+a_{32}c_{2}+a_{33}c_{3}\] are either all negative, all positive, or all zero. [i]Proposed by Kiran Kedlaya, USA[/i]

Consider the polynomial $P(x)=X^{2n}-X^{2n-1}+\dots-x+1$, where $n\in{N^*}$. Find the remainder of the division of polynomial $P(x^{2n+1})$ by $P(x)$.

What is the smallest natural number $n$ greater than $2012$ such that the polynomial $f(x) =(x^6 + x^4)^n - x^{4n} - x^6$ is divisible by $g(x) = x^4 + x^2 + 1$?

Let $k$ be a positive integer, and let $a,b$ and $c$ be positive real numbers. Show that \[a(1-a^k)+b(1-(a+b)^k)+c(1-(a+b+c)^k)<\frac{k}{k+1}.\] [i]* * *[/i]

Find all functions $f : \mathbb{R^{+}}\to \mathbb{R^{+}}$ such that $$f(x)^2=f(xy)+f(x+f(y))-1$$ for all $x, y\in \mathbb{R^{+}}$

Let $x$ be a real number. Suppose that there exist integers $a_0,a_1,\dots,a_n$, not all zero, such that \[\sum_{k=0}^n a_k\cos(kx)=\sum_{k=0}^na_k\sin(kx)=0.\] Characterize all possible values of $\cos x$. [i]Grisham Paimagam[/i]

Find all triples $ (x,y,z)$ of real numbers which satisfy the simultaneous equations \[ x \equal{} y^3 \plus{} y \minus{} 8\] \[y \equal{} z^3 \plus{} z \minus{} 8\] \[ z \equal{} x^3 \plus{} x \minus{} 8.\]

The real numbers $x$ and $y$ are such that for any distinct odd primes $p$ and $q$ the number $x^p + y^q$ is rational. Prove that $x$ and $y$ are rational numbers.

Determine the polynomial $P\in \mathbb{R}[x]$ for which there exists $n\in \mathbb{Z}_{>0}$ such that for all $x\in \mathbb{Q}$ we have: \[P\left(x+\frac1n\right)+P\left(x-\frac1n\right)=2P(x).\] [i]Dumitru Bușneag[/i]