$P$ is a monic polynomial of odd degree greater than one such that there exists a function $f : \mathbb{R} \rightarrow \mathbb{N}$ such that for each $x \in \mathbb{R}$ ,\[f(P(x))=P(f(x))\] (a) Prove that there are a finite number of natural numbers in range of $f$. (b) Prove that if $f$ is not constant then the equation $P(x)-x=0$ has at least two real solutions. (c) For each natural $n>1$ prove that there exists a function $f : \mathbb{R} \rightarrow \mathbb{N}$ and a monic polynomial of odd degree greater than one $P$ such that for each $x \in \mathbb{R}$ ,\[f(P(x))=P(f(x))\] and range of $f$ contains exactly $n$ different numbers. Time allowed for this problem was 105 minutes.

Find all pariwise distinct real numbers $x,y,z$ such that $\left\{\frac{x-y}{y-z},\frac{y-z}{z-x},\frac{z-x}{x-y} \right\} = \{x,y,z\}$. (It means, those three fractions make a permutation of $x, y$, and $z$.)

Suppose that $a^{\log_{b}c}+b^{\log_{c}a}=m$. Find the value of $c^{\log_{b}a}+a^{\log_{c}b}$ .

Find all functions $f: \mathbb{Q}^+ \to \mathbb{Q}^+$ such that $$\dfrac{f(x)f(y)}{f(xy)} = \dfrac{\left( \sqrt{f(x)} + \sqrt{f(y)} \right)^2}{f(x+y)}$$ holds for all positive rational numbers $x, y$.

Find all functions $f:\mathbb{Q}\to\mathbb{Q}$ such that $$f(xy)+f(x+y)=f(x)f(y)+f(x)+f(y)$$ for all $x,y\in\mathbb{Q}$.

Given a real number $q$, $1 < q < 2$ define a sequence $ \{x_n\}$ as follows: for any positive integer $n$, let \[x_n=a_0+a_1 \cdot 2+ a_2 \cdot 2^2 + \cdots + a_k \cdot 2^k \qquad (a_i \in \{0,1\}, i = 0,1, \cdots m k)\] be its binary representation, define \[x_k= a_0 +a_1 \cdot q + a_2 \cdot q^2 + \cdots +a_k \cdot q^k.\] Prove that for any positive integer $n$, there exists a positive integer $m$ such that $x_n < x_m \leq x_n+1$.

[u]Round 9[/u] [b]p25.[/b] For how many nonempty subsets of $\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16\}$ is the sum of the elements divisble by $32$? [b]p26.[/b] America declared independence in $1776$. Take the sum of the cubes of the digits of $1776$ and let that equal $S_1$. Sum the cubes of the digits of $S_1$ to get $S_2$. Repeat this process $1776$ times. What is $S_{1776}$? [b]p27.[/b] Every Golden Grahams box contains a randomly colored toy car, which is one of four colors. What is the expected number of boxes you have to buy in order to obtain one car of each color? [u]Round 10[/u] [b]p28.[/b] Let $B$ be the answer to Question $29$ and $C$ be the answer to Question $30$. What is the sum of the square roots of $B$ and $C$? [b]p29.[/b] Let $A$ be the answer to Question $28$ and $C$ be the answer to Question $30$. What is the sum of the sums of the digits of $A$ and $C$? [b]p30.[/b] Let $A$ be the answer to Question $28$ and $B$ be the answer to Question $29$. What is $A + B$? [u]Round 11[/u] [b]p31.[/b] If $x + \frac{1}{x} = 4$, find $x^6 + \frac{1}{x^6}$. [b]p32.[/b] Given a positive integer $n$ and a prime $p$, there is are unique nonnegative integers $a$ and $b$ such that $n = p^b \cdot a$ and $gcd (a, p) = 1$. Let $v_p(n)$ denote this uniquely determined $a$. Let $S$ denote the set of the first 20 primes. Find $\sum_{ p \in S} v_p \left(1 + \sum^{100}_{i=0} p^i \right)$. [b]p33. [/b] Find the maximum value of n such that $n+ \sqrt{(n - 1) +\sqrt{(n - 2) + ... +\sqrt{1}}} < 49$ (Note: there would be $n - 1$ square roots and $n$ total terms). [u]Round 12[/u] [b]p34.[/b] Give two numbers $a$ and $b$ such that $2015^a < 2015! < 2015^b$. If you are incorrect you get $-5$ points; if you do not answer you get $0$ points; otherwise you get $\max \{20-0.02(|b - a| - 1), 0\}$ points, rounded down to the nearest integer. [b]p35.[/b] Twin primes are prime numbers whose difference is $2$. Let $(a, b)$ be the $91717$-th pair of twin primes, with $a < b$. Let $k = a^b$, and suppose that $j$ is the number of digits in the base $10$ representation of $k$. What is $j^5$? If the correct answer is $n$ and you say $m$, you will receive $\max \left(20 - | \log \left(| \frac{m}{n} |\right), 0 \right)$ points, rounded down to the nearest integer. [b]p36.[/b] Write down any positive integer. Let the sum of the valid submissions (i.e. positive integer submissions) for all teams be $S$. One team will be chosen randomly, according to the following distribution: if your team's submission is $n$, you will be chosen with probability $\frac{n}{S}$ . The amount of points that the chosen team will win is the greatest integer not exceeding $\min \{K, \frac{ 10000}{S} \}$. $K$ is a predetermined secret value. PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h3157009p28696627]here [/url] and 5-8 [url=https://artofproblemsolving.com/community/c3h3157013p28696685]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Show that, if $a,b,...,n$ are distinct natural numbers, none divisible by any primes greater than $3$, then $$\frac{1}{a}+\frac{1}{b}+...+ \frac{1}{n}< 3$$

Prove that if the numbers $a, b, c$ are related by the relation $\frac{1}{a}+ \frac{1}{b}+ \frac{1}{c}= \frac{1}{a+b+c}$ then the sum of some two of them is equal to zero.

Let $ a,b,c,d $ be four real numbers such that $ |ax^3+bx^2+cx+d|\le 1,\forall x\in [0,1] . $ Prove that $ |dx^2+cx^2+bx+a|\le 9/2,\forall x\in [0,1] . $ [i]Lavinia Savu[/i]

Let $ f(x)$ be a $ n \minus{}$degree polynomial all of whose coefficients are equal to $ \pm 1$, and having $ x \equal{} 1$ as its $ m$ multiple root. If $ m\ge 2^k (k\ge 2,k\in N)$, then $ n\ge 2^{k \plus{} 1} \minus{} 1.$

[b]p1.[/b] The length of the side $AB$ of the trapezoid with bases $AD$ and $BC$ is equal to the sum of lengths $|AD|+|BC|$. Prove that bisectors of angles $A$ and $B$ do intersect at a point of the side $CD$. [b]p2.[/b] Polynomials $P(x) = x^4 + ax^3 + bx^2 + cx + 1$ and $Q(x) = x^4 + cx^3 + bx^2 + ax + 1$ have two common roots. Find these common roots of both polynomials. [b]p3.[/b] A girl has a box with $1000$ candies. Outside the box there is an infinite number of chocolates and muffins. A girl may replace: $\bullet$ two candies in the box with one chocolate bar, $\bullet$ two muffins in the box with one chocolate bar, $\bullet$ two chocolate bars in the box with one candy and one muffin, $\bullet$ one candy and one chocolate bar in the box with one muffin, $\bullet$ one muffin and one chocolate bar in the box with one candy. Is it possible that after some time it remains only one object in the box? [b]p4.[/b] There are $9$ straight lines drawn in the plane. Some of them are parallel some of them intersect each other. No three lines do intersect at one point. Is it possible to have exactly $17$ intersection points? [b]p5.[/b] It is known that $x$ is a real number such that $x+\frac{1}{x}$ is an integer. Prove that $x^n+\frac{1}{x^n}$ is an integer for any positive integer $n$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $x,y,z$ be real numbers such that $0<x,y,z<1$. Find the minimum value of: $$\frac {xyz(x+y+z)+(xy+yz+zx)(1-xyz)}{xyz\sqrt {1-xyz}}$$

[b]p1.[/b] Anne purchased yesterday at WalMart in Puerto Rico $6$ identical notebooks, $8$ identical pens and $7$ identical erasers. Anne remembers that each eraser costs $73$ cents. She did not buy anything else. Anne told her mother that she spent $12$ dollars and $76$ cents at Walmart. Can she be right? Note that in Puerto Rico there is no sales tax. [b]p2.[/b] Two men ski one after the other first in a flat field and then uphill. In the field the men run with the same velocity $12$ kilometers/hour. Uphill their velocity drops to $8$ kilometers/hour. When both skiers enter the uphill trail segment the distance between them is $300$ meters less than the initial distance in the field. What was the initial distance between skiers? (There are $1000$ meters in 1 kilometer.) [b]p3.[/b] In the equality $** + **** = ****$ all the digits are replaced by $*$. Restore the equality if it is known that any numbers in the equality does not change if we write all its digits in the opposite order. [b]p4.[/b] If a polyleg has even number of legs he always tells truth. If he has an odd number of legs he always lies. Once a green polyleg told a dark-blue polyleg ”- I have $8$ legs. And you have only $6$ legs!” The offended dark-blue polyleg replied ”-It is me who has $8$ legs, and you have only $7$ legs!” A violet polyleg added ”-The dark-blue polyleg indeed has $8$ legs. But I have $9$ legs!” Then a stripped polyleg started: ”-None of you has $8$ legs. Only I have 8 legs!” Which polyleg has exactly $8$ legs? [b]p5.[/b] Cut the figure shown below in two equal pieces. (Both the area and the form of the pieces must be the same.) [img]https://cdn.artofproblemsolving.com/attachments/e/4/778678c1e8748e213ffc94ba71b1f3cc26c028.png[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $ I$ be an ideal of the ring $\mathbb{Z}\left[x\right]$ of all polynomials with integer coefficients such that a) the elements of $ I$ do not have a common divisor of degree greater than $ 0$, and b) $ I$ contains of a polynomial with constant term $ 1$. Prove that $ I$ contains the polynomial $ 1 + x + x^2 + ... + x^{r-1}$ for some natural number $ r$. [i]Gy. Szekeres[/i]

Find all functions $f : \mathbb{R} \to \mathbb{R}$ that satisfy \[f(x^2 + y) + f(f(x) - y) = 2f(f(x)) + 2y^2\quad\text{ for all }x, y \in \mathbb{R}.\]

Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ that satisfy the following conditions: a. $x+f(y+f(x))=y+f(x+f(y)) \quad \forall x,y \in \mathbb{R}$ b. The set $I=\left\{\frac{f(x)-f(y)}{x-y}\mid x,y\in \mathbb{R},x\neq y \right\}$ is an interval. [i]Proposed by Navid Safaei[/i]

[u]Round 1[/u] [b]p1.[/b] Jom and Terry both flip a fair coin. What is the probability both coins show the same side? [b]p2.[/b] Under the same standard air pressure, when measured in Fahrenheit, water boils at $212^o$ F and freezes at $32^o$ F. At thesame standard air pressure, when measured in Delisle, water boils at $0$ D and freezes at $150$ D. If x is today’s temperature in Fahrenheit and y is today’s temperature expressed in Delisle, we have $y = ax + b$. What is the value of $a + b$? (Ignore units.) [b]p3.[/b] What are the last two digits of $5^1 + 5^2 + 5^3 + · · · + 5^{10} + 5^{11}$? [u]Round 2[/u] [b]p4.[/b] Compute the average of the magnitudes of the solutions to the equation $2x^4 + 6x^3 + 18x^2 + 54x + 162 = 0$. [b]p5.[/b] How many integers between $1$ and $1000000$ inclusive are both squares and cubes? [b]p6.[/b] Simon has a deck of $48$ cards. There are $12$ cards of each of the following $4$ suits: fire, water, earth, and air. Simon randomly selects one card from the deck, looks at the card, returns the selected card to the deck, and shuffles the deck. He repeats the process until he selects an air card. What is the probability that the process ends without Simon selecting a fire or a water card? [u]Round 3[/u] [b]p7.[/b] Ally, Beth, and Christine are playing soccer, and Ally has the ball. Each player has a decision: to pass the ball to a teammate or to shoot it. When a player has the ball, they have a probability $p$ of shooting, and $1 - p$ of passing the ball. If they pass the ball, it will go to one of the other two teammates with equal probability. Throughout the game, $p$ is constant. Once the ball has been shot, the game is over. What is the maximum value of $p$ that makes Christine’s total probability of shooting the ball $\frac{3}{20}$ ? [b]p8.[/b] If $x$ and $y$ are real numbers, then what is the minimum possible value of the expression $3x^2 - 12xy + 14y^2$ given that $x - y = 3$? [b]p9.[/b] Let $ABC$ be an equilateral triangle, let $D$ be the reflection of the incenter of triangle $ABC$ over segment $AB$, and let $E$ be the reflection of the incenter of triangle $ABD$ over segment $AD$. Suppose the circumcircle $\Omega$ of triangle $ADE$ intersects segment $AB$ again at $X$. If the length of $AB$ is $1$, find the length of $AX$. [u]Round 4[/u] [b]p10.[/b] Elaine has $c$ cats. If she divides $c$ by $5$, she has a remainder of $3$. If she divides $c$ by $7$, she has a remainder of $5$. If she divides $c$ by $9$, she has a remainder of $7$. What is the minimum value $c$ can be? [b]p11.[/b] Your friend Donny offers to play one of the following games with you. In the first game, he flips a fair coin and if it is heads, then you win. In the second game, he rolls a $10$-sided die (its faces are numbered from $1$ to $10$) $x$ times. If, within those $x$ rolls, the number $10$ appears, then you win. Assuming that you like winning, what is the highest value of $x$ where you would prefer to play the coin-flipping game over the die-rolling game? [b]p12.[/b] Let be the set $X = \{0, 1, 2, ..., 100\}$. A subset of $X$, called $N$, is defined as the set that contains every element $x$ of $X$ such that for any positive integer $n$, there exists a positive integer $k$ such that n can be expressed in the form $n = x^{a_1}+x^{a_2}+...+x^{a_k}$ , for some integers $a_1, a_2, ..., a_k$ that satisfy $0 \le a_1 \le a_2 \le ... \le a_k$. What is the sum of the elements in $N$? PS. You should use hide for answers. Rounds 5-7 have be posted [url=https://artofproblemsolving.com/community/c4h2782880p24446580]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Consider the set $\mathbb Q^2$ of points in $\mathbb R^2$, both of whose coordinates are rational. [b](a)[/b] Prove that the union of segments with vertices from $\mathbb Q^2$ is the entire set $\mathbb R^2$. [b](b)[/b] Is the convex hull of $\mathbb Q^2$ (i.e., the smallest convex set in $\mathbb R^2$ that contains $\mathbb Q^2$) equal to $\mathbb R^2$ ?

$\begin{cases} x^3-ax^2+b^3=0 \\x^3-bx^2+c^3=0 \\ x^3-cx^2+a^3=0 \end{cases}$ Prove that system hasn`t solutions if $a,b,c$ are different.

Find all functions $f:\mathbb{R}\to \mathbb{R}$ such that $$f(x)+f(yf(x)+f(y))=f(x+2f(y))+xy$$for all $x,y\in \mathbb{R}$. [i]Proposed by Adrian Beker[/i]

Given $1991$ distinct positive real numbers, the product of any ten of these numbers is always greater than $1$. Prove that the product of all $1991$ numbers is also greater than $1$.

Define the sequence $A_1, A_2, A_3, \dots$ by $A_1 = 1$ and for $n=1,2,3,\dots$ $$A_{n+1}=\frac{A_n+2}{A_n +1}.$$ Define the sequences $B_1, B_2, B_3,\dots$ by $B_1=1$ and for $n=1,2,3,\dots$ $$B_{n+1}=\frac{B_n^2 +2}{2B_n}.$$ Prove that $B_{n+1}=A_{2^n}$ for all non-negative integers $n$.

Let $M$ be the set of positive integers which do not have a prime divisor greater than 3. For any infinite family of subsets of $M$, say $A_1,A_2,\ldots $, prove that there exist $i\ne j$ such that for each $x\in A_i$ there exists some $y\in A_j $ such that $y\mid x$.

Consider all possible functions defined for $x = 1, 2, ..., M$ and taking values $​​y = 1, 2, ..., n$. We denote the set of such functions by $T.$ By $T_0$ we denote the subset of $T$ consisting of functions whose value changes exactly by $ 1$ (in one direction or another) when the argument changes by $1$. Prove that if $M\ge 2n-4$, then among the functions from of the set $T$, there is a function that coincides at least at one point with any function from $T_0$. Specify at least one such function. Prove that if $M <2n-4$, then there is no such function.