The numbers $a, b$ and $c$ are such that $a^2 + b^2 + c^2 = 1$. Prove that $$a^4 + b^4 + c^4 + 2(ab^2 + bc^2 + ca^2)^2\le 1. $$ At what $a, b$ and $c$ does inequality turn into equality?

Given [i]Fibonacci[/i] sequence $(F_n),$ and a positive integer $m$, denote $k(m)$ by the smallest positive integer satisfying $F_{n+k(m)}\equiv F_n(\bmod m),$ for all natural numbers $n$, $p$ is an odd prime such that $p \equiv \pm 1(\bmod 5)$. Prove that: a) ${F_{p + 1}} \equiv 0(\bmod p).$ b) $k(p)|2p+2.$ c) $k(p)$ is divisible by $4.$

Let $0<f(1)<f(2)<f(3)<\ldots$ a sequence with all its terms positive$.$ The $n-th$ positive integer which doesn't belong to the sequence is $f(f(n))+1.$ Find $f(240).$

[b]p1.[/b] Give a fake proof that $0 = 1$ on the back of this page. The most convincing answer to this question at this test site will receive a point. [b]p2.[/b] It is often said that once you assume something false, anything can be derived from it. You may assume for this question that $0 = 1$, but you can only use other statements if they are generally accepted as true or if your prove them from this assumption and other generally acceptable mathematical statements. With this in mind, on the back of this page prove that every number is the same number. [b]p3.[/b] Suppose you write out all integers between $1$ and $1000$ inclusive. (The list would look something like $1$, $2$, $3$, $...$ , $10$, $11$, $...$ , $999$, $1000$.) Which digit occurs least frequently? [b]p4.[/b] Pick a real number between $0$ and $1$ inclusive. If your response is $r$ and the standard deviation of all responses at this site to this question is $\sigma$, you will receive $r(1 - (r - \sigma)^2)$ points. [b]p5.[/b] Find the sum of all possible values of $x$ that satisfy $243^{x+1} = 81^{x^2+2x}$. [b]p6.[/b] How many times during the day are the hour and minute hands of a clock aligned? [b]p7.[/b] A group of $N + 1$ students are at a math competition. All of them are wearing a single hat on their head. $N$ of the hats are red; one is blue. Anyone wearing a red hat can steal the blue hat, but in the process that person’s red hat disappears. In fact, someone can only steal the blue hat if they are wearing a red hat. After stealing it, they would wear the blue hat. Everyone prefers the blue hat over a red hat, but they would rather have a red hat than no hat at all. Assuming that everyone is perfectly rational, find the largest prime $N$ such that nobody will ever steal the blue hat. [b]p8.[/b] On the back of this page, prove there is no function f$(x)$ for which there exists a (finite degree) polynomial $p(x)$ such that $f(x) = p(x)(x + 3) + 8$ and $f(3x) = 2f(x)$. [b]p9.[/b] Given a cyclic quadrilateral $YALE$ with $Y A = 2$, $AL = 10$, $LE = 11$, $EY = 5$, what is the area of $YALE$? [b]p10.[/b] About how many pencils are made in the U.S. every year? If your answer to this question is $p$, and our (good) estimate is $\rho$, then you will receive $\max(0, 1 -\frac 12 | \log_{10}(p) - \log_{10}(\rho)|)$ points. [b]p11.[/b] The largest prime factor of $520, 302, 325$ has $5$ digits. What is this prime factor? [b]p12.[/b] The previous question was on the individual round from last year. It was one of the least frequently correctly answered questions. The first step to solving the problem and spotting the pattern is to divide $520, 302, 325$ by an appropriate integer. Unfortunately, when solving the problem many people divide it by $n$ instead, and then they fail to see the pattern. What is $n$? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $a,b$ be positive integers satisfying $$\sqrt{\dfrac{ab}{2b^2-a}}=\dfrac{a+2b}{4b}$$. Find $|10(a-5)(b-15)|+8$.

Find all functions $f : R \to R$ that satisfy the functional equation $$f(x + f(xy)) = f(x)(1 + y).$$

Given $x,y>0$ such that $x^2y^2+2x^3y=1$. Find the minimum value of sum $x+y$

Solve the equation: $\log_{x} 2 \cdot \log_{2x} 2 = \log_{4x} 2$.

Given a circle and $2006$ points lying on this circle. Albatross colors these $2006$ points in $17$ colors. After that, Frankinfueter joins some of the points by chords such that the endpoints of each chord have the same color and two different chords have no common points (not even a common endpoint). Hereby, Frankinfueter intends to draw as many chords as possible, while Albatross is trying to hinder him as much as he can. What is the maximal number of chords Frankinfueter will always be able to draw?

Find all polynomials $p(x)$ satisfying the condition: $$p(x^2-2x)=p(x-2)^2.$$

Let $ n$ be a natural number equal or greater than 3 . A sequence of non-negative numbers $ (c_0,c_1,\ldots,c_n)$ satisfies the condition: $ c_{p}c_{s}\plus{}c_{r}c_{t}\equal{} c_{p\plus{}r}c_{r\plus{}s}$ for all non-negative $ p,q,r,s$ such that $ p\plus{}q\plus{}r\plus{}s\equal{}n$. Determine all possible values of $ c_2$ when $ c_1\equal{}1$.

Initially, a non-constant polynomial $S(x)$ with real coefficients is written down on a board. Whenever the board contains a polynomial $P(x)$, not necessarily alone, one can write down on the board any polynomial of the form $P(C + x)$ or $C + P(x)$ where $C$ is a real constant. Moreover, if the board contains two (not necessarily distinct) polynomials $P(x)$ and $Q(x)$, one can write $P(Q(x))$ and $P(x) + Q(x)$ down on the board. No polynomial is ever erased from the board. Given two sets of real numbers, $A = \{ a_1, a_2, \dots, a_n \}$ and $B = \{ b_1, \dots, b_n \}$, a polynomial $f(x)$ with real coefficients is $(A,B)$-[i]nice[/i] if $f(A) = B$, where $f(A) = \{ f(a_i) : i = 1, 2, \dots, n \}$. Determine all polynomials $S(x)$ that can initially be written down on the board such that, for any two finite sets $A$ and $B$ of real numbers, with $|A| = |B|$, one can produce an $(A,B)$-[i]nice[/i] polynomial in a finite number of steps. [i]Proposed by Navid Safaei, Iran[/i]

[u]Round 1[/u] [b]p1.[/b] Alex is on a week-long mining quest. Each morning, she mines at least $1$ and at most $10$ diamonds and adds them to her treasure chest (which already contains some diamonds). Every night she counts the total number of diamonds in her collection and finds that it is divisible by either $22$ or $25$. Show that she miscounted. [b]p2.[/b] Hermione set out a row of $11$ Bertie Bott’s Every Flavor Beans for Ron to try. There are $5$ chocolateflavored beans that Ron likes and $6$ beans flavored like earwax, which he finds disgusting. All beans look the same, and Hermione tells Ron that a chocolate bean always has another chocolate bean next to it. What is the smallest number of beans that Ron must taste to guarantee he finds a chocolate one? [b]p3.[/b] There are $101$ pirates on a pirate ship: the captain and $100$ crew. Each pirate, including the captain, starts with $1$ gold coin. The captain makes proposals for redistributing the coins, and the crew vote on these proposals. The captain does not vote. For every proposal, each crew member greedily votes “yes” if he gains coins as a result of the proposal, “no” if he loses coins, and passes otherwise. If strictly more crew members vote “yes” than “no,” the proposal takes effect. The captain can make any number of proposals, one after the other. What is the largest number of coins the captain can accumulate? [b]p4.[/b] There are $100$ food trucks in a circle and $10$ gnomes who sample their menus. For the first course, all the gnomes eat at different trucks. For each course after the first, gnome #$1$ moves $1$ truck left or right and eats there; gnome #$2$ moves $2$ trucks left or right and eats there; ... gnome #$10$ moves $10$ trucks left or right and eats there. All gnomes move at the same time. After some number of courses, each food truck had served at least one gnome. Show that at least one gnome ate at some food truck twice. [b]p5.[/b] The town of Lumenville has $100$ houses and is preparing for the math festival. The Tesla wiring company lays lengths of power wire in straight lines between the houses so that power flows between any two houses, possibly by passing through other houses.The Edison lighting company hangs strings of lights in straight lines between pairs of houses so that each house is connected by a string to exactly one other. Show that however the houses are arranged, the Edison company can always hang their strings of lights so that the total length of the strings is no more than the total length of the power wires the Tesla company used. [img]https://cdn.artofproblemsolving.com/attachments/9/2/763de9f4138b4dc552247e9316175036c649b6.png[/img] [u]Round 2[/u] [b]p6.[/b] What is the largest number of zeros that could appear at the end of $1^n + 2^n + 3^n + 4^n$, where n can be any positive integer? [b]p7.[/b] A tennis academy has $2023$ members. For every group of 1011 people, there is a person outside of the group who played a match against everyone in it. Show there is someone who has played against all $2022$ other members. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Real constants $ a, b, c$ are such that there is exactly one square all of whose vertices lie on the cubic curve $ y \equal{} x^3 \plus{} ax^2 \plus{} bx \plus{} c.$ Prove that the square has sides of length $ \sqrt[4]{72}.$

[b]p1.[/b] We define $a \oplus b = \frac{ab}{a+b}$. Compute $(3 \oplus 5) \oplus (5 \oplus 4)$. [b]p2.[/b] Let $ABCD$ be a quadrilateral with $\angle A = 45^o$ and $\angle B = 45^o$. If $BC = 5\sqrt2$, $AD = 6\sqrt2$, and $AB = 18$, find the length of side $CD$. [b]p3.[/b] A positive real number $x$ satisfies the equation $x^2 + x + 1 + \frac{1}{x }+\frac{1}{x^2} = 10$. Find the sum of all possible values of $x + 1 + \frac{1}{x}$. [b]p4.[/b] David writes $6$ positive integers on the board (not necessarily distinct) from least to greatest. The mean of the first three numbers is $3$, the median of the first four numbers is $4$, the unique mode of the first five numbers is $5$, and the range of all 6 numbers is $6$. Find the maximum possible value of the product of David’s $6$ integers. [b]p5.[/b] Let $ABCD$ be a convex quadrilateral such that $\angle A = \angle B = 120^o$ and $\angle C = \angle D = 60^o$. There exists a circle with center $I$ which is tangent to all four sides of $ABCD$. If $IA \cdot IB \cdot IC \cdot ID = 240$, find the area of quadrilateral $ABCD$. [b]p6.[/b] The letters $EXETERMATH$ are placed into cells on an annulus as shown below. How many ways are there to color each cell of the annulus with red, blue, green, or yellow such that each letter is always colored the same color and adjacent cells are always colored differently? [img]https://cdn.artofproblemsolving.com/attachments/3/5/b470a771a5279a7746c06996f2bb5487c33ecc.png[/img] [b]p7.[/b] Let $ABCD$ be a square, and let $\omega$ be a quarter circle centered at $A$ passing through points $B$ and $D$. Points $E$ and $F$ lie on sides $BC$ and $CD$ respectively. Line $EF$ intersects $\omega$ at two points, $G$ and $H$. Given that $EG = 2$, $GH = 16$ and $HF = 9$, find the length of side $AB$. [b]p8.[/b] Let x be equal to $\frac{2022! + 2021!}{2020! + 2019! + 2018!}$ . Find the closest integer to $2\sqrt{x}$. [b]p9.[/b] For how many ordered pairs of positive integers $(m, n)$ is the absolute difference between $lcm(m, n)$ and $gcd(m, n)$ equal to $2023$? [b]p10.[/b] There are $2023$ distinguishable frogs sitting on a number line with one frog sitting on $i$ for all integers $i$ between $-1011$ and $1011$, inclusive. Each minute, every frog randomly jumps either one unit left or one unit right with equal probability. After $1011$ minutes, over all possible arrangements of the frogs, what is the average number of frogs sitting on the number $0$? [b]p11.[/b] Albert has a calculator initially displaying $0$ with two buttons: the first button increases the number on the display by one, and the second button returns the square root of the number on the display. Each second, he presses one of the two buttons at random with equal probability. What is the probability that Albert’s calculator will display the number $6$ at some point? [b]p12.[/b] For a positive integer $k \ge 2$, let $f(k)$ be the number of positive integers $n$ such that n divides $(n-1)!+k$. Find $$f(2) + f(3) + f(4) + f(5) + ... + f(100).$$ [b]p13.[/b] Mr. Atf has nine towers shaped like rectangular prisms. Each tower has a $1$ by $1$ base. The first tower as height $1$, the next has height $2$, up until the ninth tower, which has height $9$. Mr. Atf randomly arranges these $9$ towers on his table so that their square bases form a $3$ by $3$ square on the surface of his table. Over all possible solids Mr. Atf could make, what is the average surface area of the solid? [b]p14.[/b] Let $ABCD$ be a cyclic quadrilateral whose diagonals are perpendicular. Let $E$ be the intersection of $AC$ and $BD$, and let the feet of the altitudes from $E$ to the sides $AB$, $BC$, $CD$, $DA$ be $W, X, Y , Z$ respectively. Given that $EW = 2EY$ and $EW \cdot EX \cdot EY \cdot EZ = 36$, find the minimum possible value of $\frac{1}{[EAB]} +\frac{1}{[EBC]}+\frac{1}{[ECD]} +\frac{1}{[EDA]}$. The notation $[XY Z]$ denotes the area of triangle $XY Z$. [b]p15.[/b] Given that $x^2 - xy + y^2 = (x + y)^3$, $y^2 - yz + z^2 = (y + z)^3$, and $z^2 - zx + x^2 = (z + x)^3$ for complex numbers $x, y, z$, find the product of all distinct possible nonzero values of $x + y + z$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

$p$ is a polynomial with integer coefficients and for every natural $n$ we have $p(n)>n$. $x_k $ is a sequence that: $x_1=1, x_{i+1}=p(x_i)$ for every $N$ one of $x_i$ is divisible by $N.$ Prove that $p(x)=x+1$

Let $a,b,c,d \in R^+$ with $abcd=1$. Prove that $$\left(\frac{1+ab}{1+a}\right)^{2008}+\left(\frac{1+bc}{1+b}\right)^{2008}+\left(\frac{1+cd }{1+c}\right)^{2008}+\left(\frac{1+da}{1+d}\right)^{2008} \geq 4$$ [i](dektep)[/i]

Find the greatest value of $C$ for which, for any $x, y, z,u$, and such that for $0\le x\le y \le z\le u$, holds the inequality $$(x + y +z + u)^2 \ge Cyz .$$

Can the graphs of a polynomial of degree 20 and the function $\displaystyle y={1\over x^{40}}$ have exactly 30 points of intersection? [i]Proposed by K. Kokhas[/i]

Given that \[\frac{a-b}{c-d}=2\quad\text{and}\quad\frac{a-c}{b-d}=3\] for certain real numbers $a,b,c,d$, determine the value of \[\frac{a-d}{b-c}.\]

Let $P(x)$ and $Q(x)$ be two polynomials with integer coefficients, such that no nonconstant polynomial with rational coefficients divides both $P(x)$ and $Q(x).$ Suppose that for every positive integer $n$ the integers $P(n)$ and $Q(n)$ are positive, and $2^{Q(n)}-1$ divides $3^{P(n)}-1.$ Prove that $Q(x)$ is a constant polynomial. [i]Proposed by Oleksiy Klurman, Ukraine[/i]

Consider the set $A = \{1, 2, 3, ..., 2n - 1\}$, where $n \ge 2$ is a positive integer. We remove from the set $A$ at least $n - 1$ elements such that: • if $a \in A$ has been removed, and $2a \in A$, then $2a$ has also been removed, • if $a, b \in A (a \ne b)$ have been removed and $a + b \in A$, then $a + b$ has also been removed. Which numbers have to be removed such that the sum of the remaining numbers is maximum?

[b]p16.[/b] When not writing power rounds, Eric likes to climb trees. The strength in his arms as a function of time is $s(t) = t^3 - 3t^2$. His climbing velocity as a function of the strength in his arms is $v(s) = s^5 + 9s^4 + 19s^3 - 9s^2 - 20s$. At how many (possibly negative) points in time is Eric stationary? [b]p17[/b]. Consider a triangle $\vartriangle ABC$ with angles $\angle ACB = 60^o$, $\angle ABC = 45^o$. The circumcircle around $\vartriangle ABH$, where $H$ is the orthocenter of $\vartriangle ABC$, intersects $BC$ for a second time in point $P$, and the center of that circumcircle is $O_c$. The line $PH$ intersects $AC$ in point $Q$, and $N$ is center of the circumcircle around $\vartriangle AQP$. Find $\angle NO_cP$. [b]p18.[/b] If $x, y$ are positive real numbers and $xy^3 = \frac{16}{9}$ , what is the minimum possible value of $3x + y$? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $\mathbb{Z} _{>0}$ be the set of positive integers. Find all functions $f: \mathbb{Z} _{>0}\rightarrow \mathbb{Z} _{>0}$ such that \[ m^2 + f(n) \mid mf(m) +n \] for all positive integers $m$ and $n$.

Let be two (not necessarily distinct) roots of two rational polynoms (respectively) that are irreducible over the rationals. Prove that these polynoms have the same degree if the sum of those two roots is rational. [i]Bogdan Enescu[/i]