Find the smallest possible value of the nonnegative number $\lambda$ such that the inequality $$\frac{a+b}{2}\geq\lambda \sqrt{ab}+(1-\lambda )\sqrt{\frac{a^2+b^2}{2}}$$ holds for all positive real numbers $a, b$.

Consider the polynomial f(x) = cx(x - 2) where $c$ is a positive real number. For any $n \in Z^+$, the notation $g_n(x)$ is a composite function $n$ times of $f$ and assume that the equation $g_n(x) = 0$ has all of the $2^n$ solutions are real numbers. 1. For $c = 5$, find in terms of $n$, the sum of all the solutions of $g_n(x)$, of which each multiple (if any) is counted only once. 2. Prove that $c\ge 1$.

A single section at a stadium can hold either $7$ adults or $11$ children. When $N$ sections are completely lled, an equal number of adults and children will be seated in them. What is the least possible value of $N$?

Find all rational numbers $a$,for which there exist infinitely many positive rational numbers $q$ such that the equation $[x^a].{x^a}=q$ has no solution in rational numbers.(A.Vasiliev)

Find all polynomials $W$ with real coefficients possessing the following property: if $x+y$ is a rational number, then $W(x)+W(y)$ is rational.

Find all functions $f: R \to R$ for which the image $f ([a, b])$ for all real $a \le b$ is (not necessarily closed!) interval of length $b - a$.

Find the real root of $x^5+5x^3+5x-1$. Hint: Let $x = u+k/u$.

Several rational numbers were written on the blackboard. Dima wrote off their fractional parts on paper. Then all the numbers on the board squared, and Dima wrote off another paper with fractional parts of the resulting numbers. It turned out that on Dima's papers were written the same sets of numbers (maybe in different order). Prove that the original numbers on the board were integers. (The fractional part of a number $x$ is such a number $\{x\}, 0 \le \{x\} <1$, that $x-\{x\}$ is an integer.)

Let $x, y$ and $z$ be consecutive integers such that \[\frac 1x+\frac 1y+\frac 1z >\frac{1}{45}.\] Find the maximum value of $x + y + z$.

In $10$ years the product of Melanie's age and Phil's age will be $400$ more than it is now. Find what the sum of Melanie's age and Phil's age will be $6$ years from now.

Let $ a,b,c\geq 0$ such that $ a\plus{}b\plus{}c\equal{}1$. Prove that: $ a^2\plus{}b^2\plus{}c^2\geq 4(ab\plus{}bc\plus{}ca)\minus{}1$

Find all functions $f :R \to R$ such that for all real numbers $x$ and $y$ $$f(x^3+y^3)=f(x^3)+3x^3f(x)f(y)+3f(x)(f(y))^2+y^6f(y)$$

[u]Round 9[/u] [b]p25. [/b]Define a sequence $\{a_n\}_{n \ge 1}$ of positive real numbers by $a_1 = 2$ and $a^2_n -2a_n +5 =4a_{n-1}$ for $n \ge 2$. Suppose $k$ is a positive real number such that $a_n <k$ for all positive integers $n$. Find the minimum possible value of $k$. [b]p26.[/b] Let $\vartriangle ABC$ be a triangle with $AB = 13$, $BC = 14$, and $C A = 15$. Suppose the incenter of $\vartriangle ABC$ is $I$ and the incircle is tangent to $BC$ and $AB$ at $D$ and $E$, respectively. Line $\ell$ passes through the midpoints of $BD$ and $BE$ and point $X$ is on $\ell$ such that $AX \parallel BC$. Find $X I$ . [b]p27.[/b] Let $x, y, z$ be positive real numbers such that $x y + yz +zx = 20$ and $x^2yz +x y^2z +x yz^2 = 100$. Additionally, let $s = \max (x y, yz,xz)$ and $m = \min(x, y, z)$. If $s$ is maximal, find $m$. [u]Round 10[/u] [b]p28.[/b] Let $\omega_1$ be a circle with center $O$ and radius $1$ that is internally tangent to a circle $\omega_2$ with radius $2$ at $T$ . Let $R$ be a point on $\omega_1$ and let $N$ be the projection of $R$ onto line $TO$. Suppose that $O$ lies on segment $NT$ and $\frac{RN}{NO} = \frac4 3$ . Additionally, let $S$ be a point on $\omega_2$ such that $T,R,S$ are collinear. Tangents are drawn from $S$ to $\omega_1$ and touch $\omega_1$ at $P$ and $Q$. The tangent to $\omega_1$ at $R$ intersects $PQ$ at $Z$. Find the area of triangle $\vartriangle ZRS$. [b]p29.[/b] Let $m$ and $n$ be positive integers such that $k =\frac{ m^2+n^2}{mn-1}$ is also a positive integer. Find the sum of all possible values of $k$. [b]p30.[/b] Let $f_k (x) = k \cdot \ min (x,1-x)$. Find the maximum value of $k \le 2$ for which the equation $f_k ( f_k ( f_k (x))) = x$ has fewer than $8$ solutions for $x$ with $0 \le x \le 1$. [u]Round 11[/u] In the following problems, $A$ is the answer to Problem $31$, $B$ is the answer to Problem $32$, and $C$ is the answer to Problem $33$. For this set, you should find the values of $A$,$B$, and $C$ and submit them as answers to problems $31$, $32$, and $33$, respectively. Although these answers depend on each other, each problem will be scored separately. [b]p31.[/b] Find $$A \cdot B \cdot C + \dfrac{1}{B+ \dfrac{1}{C +\dfrac{1}{B+\dfrac{1}{...}}}}$$ [b]p32.[/b] Let $D = 7 \cdot B \cdot C$. An ant begins at the bottom of a unit circle. Every turn, the ant moves a distance of $r$ units clockwise along the circle, where $r$ is picked uniformly at random from the interval $\left[ \frac{\pi}{2D} , \frac{\pi}{D} \right]$. Then, the entire unit circle is rotated $\frac{\pi}{4}$ radians counterclockwise. The ant wins the game if it doesn’t get crushed between the circle and the $x$-axis for the first two turns. Find the probability that the ant wins the game. [b]p33.[/b] Let $m$ and $n$ be the two-digit numbers consisting of the products of the digits and the sum of the digits of the integer $2016 \cdot B$, respectively. Find $\frac{n^2}{m^2 - mn}$. [u]Round 12[/u] [b]p34.[/b] There are five regular platonic solids: the tetrahedron, cube, octahedron, dodecahedron, and icosahedron. For each of these solids, define its adjacency angle to be the dihedral angle formed between two adjacent faces. Estimate the sum of the adjacency angles of all five solids, in degrees. If your estimate is $E$ and the correct answer is $A$, your score for this problem will be $\max \left(0, \lfloor 15 -\frac12 |A-E| \rfloor \right).$ [b]p35.[/b] Estimate the value of $$\log_{10} \left(\prod_{k|2016} k!\right), $$ where the product is taken over all positive divisors $k$ of $2016$. If your estimate is $E$ and the correct answer is $A$, your score for this problem will be $\max \left(0, \lceil 15 \cdot \min \left(\frac{E}{A}, 2- \frac{E}{A}\right) \rceil \right).$ [b]p36.[/b] Estimate the value of $\sqrt{2016}^{\sqrt[4]{2016}}$. If your estimate is $E$ and the correct answer is $A$, your score for this problem will be $\max \left(0, \lceil 15 \cdot \min \left(\frac{\ln E}{\ln A}, 2- \frac{\ln E}{\ln A}\right) \rceil \right).$ PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h3158461p28714996]here [/url] and 5-8 [url=https://artofproblemsolving.com/community/c3h3158474p28715078]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Given positive reals $a,b,c;$ show that we have \[\left(a+\frac 1b\right)\left(b+\frac 1c\right)\left(c+\frac 1a\right)\geq 8.\]

[b]Problem 1.[/b]Find all $ (x,y)$ such that: \[ \{\begin{matrix} \displaystyle\dfrac {1}{\sqrt {1 + 2x^2}} + \dfrac {1}{\sqrt {1 + 2y^2}} & = & \displaystyle\dfrac {2}{\sqrt {1 + 2xy}} \\ \sqrt {x(1 - 2x)} + \sqrt {y(1 - 2y)} & = & \displaystyle\dfrac {2}{9} \end{matrix}\; \]

You are initially given the number $n=1$. Each turn, you may choose any positive divisor $d\mid n$, and multiply $n$ by $d+1$. For instance, on the first turn, you must select $d=1$, giving $n=1\cdot(1+1)=2$ as your new value of $n$. On the next turn, you can select either $d=1$ or $2$, giving $n=2\cdot(1+1)=4$ or $n=2\cdot(2+1)=6$, respectively, and so on. Find an algorithm that, in at most $k$ steps, results in $n$ being divisible by the number $2021^{2021^{2021}} - 1$. An algorithm that completes in at most $k$ steps will be awarded: 1 pt for $k>2021^{2021^{2021}}$ 20 pts for $k=2021^{2021^{2021}}$ 50 pts for $k=10^{10^4}$ 75 pts for $k=10^{10}$ 90 pts for $k=10^5$ 95 pts for $k=6\cdot10^4$ 100 pts for $k=5\cdot10^4$

If $x,y$ are positive real numbers satisfying $x^3-xy+1=y^3$, find the minimum possible value of $y$.

Prove that for every natural number $ n \geq 2 $ the inequality holds $$ \log_n 2 \cdot \log_n 4 \cdot \log_n 6 \ldots \log_n (2n - 2) \leq 1.$$

[b]p1.[/b] What is $20\times 20 - 19\times 19$? [b]p2.[/b] Andover has a total of $1440$ students and teachers as well as a $1 : 5$ teacher-to-student ratio (for every teacher, there are exactly $5$ students). In addition, every student is either a boarding student or a day student, and $70\%$ of the students are boarding students. How many day students does Andover have? [b]p3.[/b] The time is $2:20$. If the acute angle between the hour hand and the minute hand of the clock measures $x$ degrees, find $x$. [img]https://cdn.artofproblemsolving.com/attachments/b/a/a18b089ae016b15580ec464c3e813d5cb57569.png[/img] [b]p4.[/b] Point $P$ is located on segment $AC$ of square $ABCD$ with side length $10$ such that $AP >CP$. If the area of quadrilateral $ABPD$ is $70$, what is the area of $\vartriangle PBD$? [b]p5.[/b] Andrew always sweetens his tea with sugar, and he likes a $1 : 7$ sugar-to-unsweetened tea ratio. One day, he makes a $100$ ml cup of unsweetened tea but realizes that he has run out of sugar. Andrew decides to borrow his sister's jug of pre-made SUPERSWEET tea, which has a $1 : 2$ sugar-to-unsweetened tea ratio. How much SUPERSWEET tea, in ml,does Andrew need to add to his unsweetened tea so that the resulting tea is his desired sweetness? [b]p6.[/b] Jeremy the architect has built a railroad track across the equator of his spherical home planet which has a radius of exactly $2020$ meters. He wants to raise the entire track $6$ meters off the ground, everywhere around the planet. In order to do this, he must buymore track, which comes from his supplier in bundles of $2$ meters. What is the minimum number of bundles he must purchase? Assume the railroad track was originally built on the ground. [b]p7.[/b] Mr. DoBa writes the numbers $1, 2, 3,..., 20$ on the board. Will then walks up to the board, chooses two of the numbers, and erases them from the board. Mr. DoBa remarks that the average of the remaining $18$ numbers is exactly $11$. What is the maximum possible value of the larger of the two numbers that Will erased? [b]p8.[/b] Nathan is thinking of a number. His number happens to be the smallest positive integer such that if Nathan doubles his number, the result is a perfect square, and if Nathan triples his number, the result is a perfect cube. What is Nathan's number? [b]p9.[/b] Let $S$ be the set of positive integers whose digits are in strictly increasing order when read from left to right. For example, $1$, $24$, and $369$ are all elements of $S$, while $20$ and $667$ are not. If the elements of $S$ are written in increasing order, what is the $100$th number written? [b]p10.[/b] Find the largest prime factor of the expression $2^{20} + 2^{16} + 2^{12} + 2^{8} + 2^{4} + 1$. [b]p11.[/b] Christina writes down all the numbers from $1$ to $2020$, inclusive, on a whiteboard. What is the sum of all the digits that she wrote down? [b]p12.[/b] Triangle $ABC$ has side lengths $AB = AC = 10$ and $BC = 16$. Let $M$ and $N$ be the midpoints of segments $BC$ and $CA$, respectively. There exists a point $P \ne A$ on segment $AM$ such that $2PN = PC$. What is the area of $\vartriangle PBC$? [b]p13.[/b] Consider the polynomial $$P(x) = x^4 + 3x^3 + 5x^2 + 7x + 9.$$ Let its four roots be $a, b, c, d$. Evaluate the expression $$(a + b + c)(a + b + d)(a + c + d)(b + c + d).$$ [b]p14.[/b] Consider the system of equations $$|y - 1| = 4 -|x - 1|$$ $$|y| =\sqrt{|k - x|}.$$ Find the largest $k$ for which this system has a solution for real values $x$ and $y$. [b]p16.[/b] Let $T_n = 1 + 2 + ... + n$ denote the $n$th triangular number. Find the number of positive integers $n$ less than $100$ such that $n$ and $T_n$ have the same number of positive integer factors. [b]p17.[/b] Let $ABCD$ be a square, and let $P$ be a point inside it such that $PA = 4$, $PB = 2$, and $PC = 2\sqrt2$. What is the area of $ABCD$? [b]p18.[/b] The Fibonacci sequence $\{F_n\}$ is defined as $F_0 = 0$, $F_1 = 1$, and $F_{n+2}= F_{n+1} + F_n$ for all integers $n \ge 0$. Let $$ S =\dfrac{1}{F_6 + \frac{1}{F_6}}+\dfrac{1}{F_8 + \frac{1}{F_8}}+\dfrac{1}{F_{10} +\frac{1}{F_{10}}}+\dfrac{1}{F_{12} + \frac{1}{F_{12}}}+ ... $$ Compute $420S$. [b]p19.[/b] Let $ABCD$ be a square with side length $5$. Point $P$ is located inside the square such that the distances from $P$ to $AB$ and $AD$ are $1$ and $2$ respectively. A point $T$ is selected uniformly at random inside $ABCD$. Let $p$ be the probability that quadrilaterals $APCT$ and $BPDT$ are both not self-intersecting and have areas that add to no more than $10$. If $p$ can be expressed in the form $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$, find $m + n$. Note: A quadrilateral is self-intersecting if any two of its edges cross. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

A sequence of real numbers $a_1,a_2,\ldots$ satisfies the relation $$a_n=-\max_{i+j=n}(a_i+a_j)\qquad\text{for all}\quad n>2017.$$ Prove that the sequence is bounded, i.e., there is a constant $M$ such that $|a_n|\leq M$ for all positive integers $n$.

Find all positive integers $ n\in\{1,2,3,\ldots,2009\}$ such that \[ 4n^6 \plus{} n^3 \plus{} 5\] is divisible by $ 7$.

Prove the inequality $$\sum_{k=1}^{n}\frac{\sin (k+1)x}{\sin kx}< 2\frac{\cos x}{\sin^2x}$$ where $0 < nx < \pi/2$, $n \in N$.

For each $n\in\mathbb N$, determine the sign of $n^6+5n^5\sin n+1$. For which $n\in\mathbb N$ does it hold that $\frac{n^2+5n\cos n+1}{n^6+5n^5\sin n+1}\ge10^{-4}$.

Determine all functions $f: \mathbb{R} \to \mathbb{R}$, such that $$f(xf(y)+1)=y+f(f(x)f(y))$$ for all $x, y \in \mathbb{R}$. (Theresia Eisenkölbl)

Determine the sum of absolute values for the complex roots of $ 20 x^8 \plus{} 7i x^7 \minus{}7ix \plus{} 20.$