Given an integer $n\ge3$ consider positive integers $x_1,\ldots,x_n$ such that $x_1<x_2<\cdots<x_n<2x_1$. If $p$ is a prime and $r$ is a positive integer such that $p^r$ divides the product $x_1\cdots x_n$, prove that $$\frac{x_1\cdots x_n}{p^r}>n!.$$

Let $\mathbb{R}^+$ denote the set of positive real numbers. Find all functions $f : \mathbb{R}^+\to\mathbb{R}^+$ that satisfy \[ \Big(1+yf(x)\Big)\Big(1-yf(x+y)\Big)=1\] for all $x,y\in\mathbb{R}^+$.

[b]p1.[/b] Let $2^k$ be the largest power of $2$ dividing $30! = 30 \cdot 29 \cdot 28 ... 2 \cdot 1$. Find $k$. [b]p2.[/b] Let $d(n)$ be the total number of digits needed to write all the numbers from $1$ to $n$ in base $10$, for example, $d(5) = 5$ and $d(20) = 31$. Find $d(2012)$. [b]p3.[/b] Jim and TongTong play a game. Jim flips $10$ coins and TongTong flips $11$ coins, whoever gets the most heads wins. If they get the same number of heads, there is a tie. What is the probability that TongTong wins? [b]p4.[/b] There are a certain number of potatoes in a pile. When separated into mounds of three, two remain. When divided into mounds of four, three remain. When divided into mounds of five, one remain. It is clear there are at least $150$ potatoes in the pile. What is the least number of potatoes there can be in the pile? [b]p5.[/b] Call an ordered triple of sets $(A, B, C)$ nice if $|A \cap B| = |B \cap C| = |C \cap A| = 2$ and $|A \cap B \cap C| = 0$. How many ordered triples of subsets of $\{1, 2, · · · , 9\}$ are nice? [b]p6.[/b] Brett has an $ n \times n \times n$ cube (where $n$ is an integer) which he dips into blue paint. He then cuts the cube into a bunch of $ 1 \times 1 \times 1$ cubes, and notices that the number of un-painted cubes (which is positive) evenly divides the number of painted cubes. What is the largest possible side length of Brett’s original cube? Note that $\lfloor x\rfloor$ denotes the largest integer less than or equal to $x$. [b]p7.[/b] Choose two real numbers $x$ and $y$ uniformly at random from the interval $[0, 1]$. What is the probability that $x$ is closer to $1/4$ than $y$ is to $1/2$? [b]p8. [/b] In triangle $ABC$, we have $\angle BAC = 20^o$ and $AB = AC$. $D$ is a point on segment $AB$ such that $AD = BC$. What is $\angle ADC$, in degree. [b]p9.[/b] Let $a, b, c, d$ be real numbers such that $ab + c + d = 2012$, $bc + d + a = 2010$, $cd + a + b = 2013$, $da + b + c = 2009$. Find $d$. [b]p10. [/b]Let $\theta \in [0, 2\pi)$ such that $\cos \theta = 2/3$. Find $\sum_{n=0}^{\infty}\frac{1}{2^n}\cos(n \theta)$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

(a) Determine the minimal value of $\displaystyle\left(x+\dfrac{1}{y}\right)\left(x+\dfrac{1}{y}-2018\right)+\left(y+\dfrac{1}{x}\right)\left(y+\dfrac{1}{x}-2018\right), $ where $x$ and $y$ vary over the positive reals. (b) Determine the minimal value of $\displaystyle\left(x+\dfrac{1}{y}\right)\left(x+\dfrac{1}{y}+2018\right)+\left(y+\dfrac{1}{x}\right)\left(y+\dfrac{1}{x}+2018\right), $ where $x$ and $y$ vary over the positive reals.

The function $f: Z \to Z$ satisfies the following conditions: 1) $f(f(n))=n$ for all integers $n$ 2) $f(f(n+2)+2) = n$ for all integers $n$ 3) $f(0)=1$. Find the value of $f(1995)$ and $f(-1994)$.

Let $P(x)$ be a polynomial of odd degree. Prove that the equation $P(P(x)) = 0$ has at least as many different real roots as the equation $P(x) = 0$ [hide=original wording]Пусть P(x) — многочлен нечетной степени. Докажите, что уравнение P(P(x)) = 0 имеет не меньше различных действительных корней, чем уравнение P(x) = 0[/hide]

Suppose $f,g \in \mathbb{R}[x]$ are non constant polynomials. Suppose neither of $f,g$ is the square of a real polynomial but $f(g(x))$ is. Prove that $g(f(x))$ is not the square of a real polynomial.

A sequence of reals $a_1, a_2, \cdots$ satisfies for all $m>1$, $$a_{m+1}a_{m-1}=a_m^2-a_1^2$$ Prove that for all $m>n>1$, the sequence satisfies the equation $$a_{m+n}a_{m-n}=a_m^2-a_n^2$$ [i]Proposed by Ivan Chan Kai Chin[/i]

If $ n$ runs through all the positive integers, then $ f(n) \equal{} \left[n \plus{} \sqrt {\frac {n}{3}} \plus{} \frac {1}{2} \right]$ runs through all positive integers skipping the terms of the sequence $ a_n \equal{} 3 \cdot n^2 \minus{} 2 \cdot n.$

Let $a_1,a_2,\cdots,a_{41}\in\mathbb{R},$ such that $a_{41}=a_1, \sum_{i=1}^{40}a_i=0,$ and for any $i=1,2,\cdots,40, |a_i-a_{i+1}|\leq 1.$ Determine the greatest possible value of $(1)a_{10}+a_{20}+a_{30}+a_{40};$ $(2)a_{10}\cdot a_{20}+a_{30}\cdot a_{40}.$

Let $P(x)$ be a polynomial such that, when divided by $x - 2$, the remainder is $3$ and, when divided by $x - 3$, the remainder is $2$. If, when divided by $(x - 2)(x - 3)$, the remainder is $ax + b$, find $a^2 + b^2$.

Let $\mathcal{X}$ be the collection of all non-empty subsets (not necessarily finite) of the positive integer set $\mathbb{N}$. Determine all functions $f: \mathcal{X} \to \mathbb{R}^+$ satisfying the following properties: (i) For all $S$, $T \in \mathcal{X}$ with $S\subseteq T$, there holds $f(T) \le f(S)$. (ii) For all $S$, $T \in \mathcal{X}$, there hold \[f(S) + f(T) \le f(S + T),\quad f(S)f(T) = f(S\cdot T), \] where $S + T = \{s + t\mid s\in S, t\in T\}$ and $S \cdot T = \{s\cdot t\mid s\in S, t\in T\}$. [i]Proposed by Li4, Untro368, and Ming Hsiao.[/i]

Let $a, b, c, d$ be four positive reals such that $abc+abd+acd+bcd = 1$. Determine all possible values for $$(ab + cd)(ac + bd)(ad + bc).$$ [i]Proposed by usjl and YaWNeeT[/i]

Let $P$ be a polynomial with integer coefficients. There exist integers $a$ and $b$ such that $P(a) \cdot P(b)=-(a-b)^2$. Prove that $P(a)+P(b)=0$.

The equation $x^2 + bx + c = 0$ has two different roots $x_1$ and $x_2$. It is also known that the numbers $b$, $x_1$, $c$, $x_2$ in the indicated order form an arithmetic progression. Find the difference of this progression.

[b]p1.[/b] Jack made $3$ quarts of fruit drink from orange and apple juice. His drink contains $45\%$ of orange juice. Nick prefers more orange juice in the drink. How much orange juice should he add to the drink to obtain a drink composed of $60\%$ of orange juice? [b]p2.[/b] A square is tiled by smaller squares as shown in the figure. Find the area of the black square in the middle if the perimeter of the big square $ABCD$ is $40$ cm. [img]https://cdn.artofproblemsolving.com/attachments/8/c/d54925cba07f63ec8578048f46e1e730cb8df3.png[/img] [b]p3.[/b] For one particular number $a > 0$ the function f satisfies the equality $f(x + a) =\frac{1 + f(x)}{1 - f(x)}$ for all $x$. Show that $f$ is a periodic function. (A function $f$ is periodic with the period $T$ if $f(x + T) = f(x)$ for any $x$.) [b]p4.[/b] If $a, b, c, x, y, z$ are numbers so that $\frac{x}{a}+\frac{y}{b}+\frac{z}{c}= 1$ and $\frac{a}{x}+\frac{b}{y}+\frac{c}{z}= 0$. Show that $\frac{x^2}{a^2} +\frac{y^2}{b^2} +\frac{z^2}{c^2} = 1$ [b]p5.[/b] Is it possible that a four-digit number $AABB$ is a perfect square? (Same letters denote the same digits). [b]p6.[/b] A finite number of arcs of a circle are painted black (see figure). The total length of these arcs is less than $\frac15$ of the circumference. Show that it is possible to inscribe a square in the circle so that all vertices of the square are in the unpainted portion of the circle. [img]https://cdn.artofproblemsolving.com/attachments/2/c/bdfa61917a47f3de5dd3684627792a9ebf05d5.png[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find all values of $a$ such that $x^6 - 6x^5 + 12x^4 + ax^3 + 12x^2 - 6x +1$ is nonnegative for all real $x$.

Let $x_1, x_2, …, x_n$ and $y_1, y_2, …,y_n$ be positive reals such that $$x_1 + x_2 +.. + x_n \ge y_i \ge x^2_i$$ for all $i = 1, 2, …, n$. Prove that $$\frac{x_1}{x_1y_1 + x_2}+ + \frac{x_2}{x_2y_2 + x_3} + ...+ \frac{x_n}{x_ny_n + x_1}> \frac{1}{2n}.$$

Let $f(x) = x-\tfrac1{x}$, and define $f^1(x) = f(x)$ and $f^n(x) = f(f^{n-1}(x))$ for $n\ge2$. For each $n$, there is a minimal degree $d_n$ such that there exist polynomials $p$ and $q$ with $f^n(x) = \tfrac{p(x)}{q(x)}$ and the degree of $q$ is equal to $d_n$. Find $d_n$.

Consider functions $\, f: [0,1] \rightarrow \mathbb{R} \,$ which satisfy (i) $f(x) \geq 0 \,$ for all $\, x \,$ in $\, [0,1],$ (ii) $f(1) = 1,$ (iii) $f(x) + f(y) \leq f(x+y)\,$ whenever $\, x, \, y, \,$ and $\, x + y \,$ are all in $\, [0,1]$. Find, with proof, the smallest constant $\, c \,$ such that \[ f(x) \leq cx \] for every function $\, f \,$ satisfying (i)-(iii) and every $\, x \,$ in $\, [0,1]$.

Prove that $\sqrt[3]{5}$ is irrational.

For given positive integers $n$ and $N$, let $P_n$ be the set of all polynomials $f(x)=a_0+a_1x+\cdots+a_nx^n$ with integer coefficients such that: [list] (a) $|a_j| \le N$ for $j = 0,1, \cdots ,n$; (b) The set $\{ j \mid a_j = N\}$ has at most two elements. [/list] Find the number of elements of the set $\{f(2N) \mid f(x) \in P_n\}$.

[i]20 problems for 20 minutes.[/i] [b]p1.[/b] Determine how many digits the number $10^{10}$ has. [b]p2.[/b] Let $ABC$ be a triangle with $\angle ABC = 60^o$ and $\angle BCA = 70^o$. Compute $\angle CAB$ in degrees. [b]p3.[/b] Given that $x : y = 2012 : 2$ and $y : z = 1 : 2013$, compute $x : z$. Express your answer as a common fraction. [b]p4.[/b] Determine the smallest perfect square greater than $2400$. [b]p5.[/b] At $12:34$ and $12:43$, the time contains four consecutive digits. Find the next time after 12:43 that the time contains four consecutive digits on a 24-hour digital clock. [b]p6.[/b] Given that $ \sqrt{3^a \cdot 9^a \cdot 3^a} = 81^2$, compute $a$. [b]p7.[/b] Find the number of positive integers less than $8888$ that have a tens digit of $4$ and a units digit of $2$. [b]p8.[/b] Find the sum of the distinct prime divisors of $1 + 2012 + 2013 + 2011 \cdot 2013$. [b]p9.[/b] Albert wants to make $2\times 3$ wallet sized prints for his grandmother. Find the maximum possible number of prints Albert can make using one $4 \times 7$ sheet of paper. [b]p10.[/b] Let $ABC$ be an equilateral triangle, and let $D$ be a point inside $ABC$. Let $E$ be a point such that $ADE$ is an equilateral triangle and suppose that segments $DE$ and $AB$ intersect at point $F$. Given that $\angle CAD = 15^o$, compute $\angle DFB$ in degrees. [b]p11.[/b] A palindrome is a number that reads the same forwards and backwards; for example, $1221$ is a palindrome. An almost-palindrome is a number that is not a palindrome but whose first and last digits are equal; for example, $1231$ and $1311$ are an almost-palindromes, but $1221$ is not. Compute the number of $4$-digit almost-palindromes. [b]p12.[/b] Determine the smallest positive integer $n$ such that the sum of the digits of $11^n$ is not $2^n$. [b]p13.[/b] Determine the minimum number of breaks needed to divide an $8\times 4$ bar of chocolate into $1\times 1 $pieces. (When a bar is broken into pieces, it is permitted to rotate some of the pieces, stack some of the pieces, and break any set of pieces along a vertical plane simultaneously.) [b]p14.[/b] A particle starts moving on the number line at a time $t = 0$. Its position on the number line, as a function of time, is $$x = (t-2012)^2 -2012(t-2012)-2013.$$ Find the number of positive integer values of $t$ at which time the particle lies in the negative half of the number line (strictly to the left of $0$). [b]p15.[/b] Let $A$ be a vertex of a unit cube and let $B$,$C$, and $D$ be the vertices adjacent to A. The tetrahedron $ABCD$ is cut off the cube. Determine the surface area of the remaining solid. [b]p16.[/b] In equilateral triangle $ABC$, points $P$ and $R$ lie on segment $AB$, points $I$ and $M$ lie on segment $BC$, and points $E$ and $S$ lie on segment $CA$ such that $PRIMES$ is a equiangular hexagon. Given that $AB = 11$, $PS = 2$, $RI = 3$, and $ME = 5$, compute the area of hexagon $PRIMES$. [b]p17.[/b] Find the smallest odd positive integer with an odd number of positive integer factors, an odd number of distinct prime factors, and an odd number of perfect square factors. [b]p18.[/b] Fresh Mann thinks that the expressions $2\sqrt{x^2 -4} $and $2(\sqrt{x^2} -\sqrt4)$ are equivalent to each other, but the two expressions are not equal to each other for most real numbers $x$. Find all real numbers $x$ such that $2\sqrt{x^2 -4} = 2(\sqrt{x^2} -\sqrt4)$. [b]p19.[/b] Let $m$ be the positive integer such that a $3 \times 3$ chessboard can be tiled by at most $m$ pairwise incongruent rectangles with integer side lengths. If rotations and reflections of tilings are considered distinct, suppose that there are $n$ ways to tile the chessboard with $m$ pairwise incongruent rectangles with integer side lengths. Find the product $mn$. [b]p20.[/b] Let $ABC$ be a triangle with $AB = 4$, $BC = 5$, and $CA = 6$. A triangle $XY Z$ is said to be friendly if it intersects triangle $ABC$ and it is a translation of triangle $ABC$. Let $S$ be the set of points in the plane that are inside some friendly triangle. Compute the ratio of the area of $S$ to the area of triangle $ABC$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $\{a_i\}_{i=0}^\infty$ be a sequence of real numbers such that \[\sum_{n=1}^\infty\dfrac {x^n}{1-x^n}=a_0+a_1x+a_2x^2+a_3x^3+\cdots\] for all $|x|<1$. Find $a_{1000}$. [i]Proposed by David Altizio[/i]

Find all positive real numbers $(x,y,z)$ such that: $$x = \frac{1}{y^2+y-1}$$ $$y = \frac{1}{z^2+z-1}$$ $$z = \frac{1}{x^2+x-1}$$