Prove that if $ x,y$ are nonnegative real numbers with $ x\plus{}y\equal{}2$, then: $ x^2 y^2 (x^2\plus{}y^2) \le 2$.

Determine the smallest possible real constant $C$ such that the inequality $$|x^3 + y^3 + z^3 + 1| \leq C|x^5 + y^5 + z^5 + 1|$$ holds for all real numbers $x, y, z$ satisfying $x + y + z = -1$.

There are 300 apples, any two of which differ in weight by no more than twice. Prove that they can be arranged in packages of two apples so that any two packages differ in weight by no more than one and a half times.

[b]p1.[/b] What is the area of a triangle with side lengths $ 6$, $ 8$, and $10$? [b]p2.[/b] Let $f(n) = \sqrt{n}$. If $f(f(f(n))) = 2$, compute $n$. [b]p3.[/b] Anton is buying AguaFina water bottles. Each bottle costs $14 $dollars, and Anton buys at least one water bottle. The number of dollars that Anton spends on AguaFina water bottles is a multiple of $10$. What is the least number of water bottles he can buy? [b]p4.[/b] Alex flips $3$ fair coins in a row. The probability that the first and last flips are the same can be expressed in the form $m/n$ for relatively prime positive integers $m$ and $n$. Compute $m + n$. [b]p5.[/b] How many prime numbers $p$ satisfy the property that $p^2 - 1$ is not a multiple of $6$? [b]p6.[/b] In right triangle $\vartriangle ABC$ with $AB = 5$, $BC = 12$, and $CA = 13$, point $D$ lies on $\overline{CA}$ such that $AD = BD$. The length of $CD$ can then be expressed in the form $m/n$ for relatively prime positive integers $m$ and $n$. Compute $m + n$. [b]p7.[/b] Vivienne is deciding on what courses to take for Spring $2021$, and she must choose from four math courses, three computer science courses, and five English courses. Vivienne decides that she will take one English course and two additional courses that are either computer science or math. How many choices does Vivienne have? [b]p8.[/b] Square $ABCD$ has side length $2$. Square $ACEF$ is drawn such that $B$ lies inside square $ACEF$. Compute the area of pentagon $AFECD$. [b]p9.[/b] At the Boba Math Tournament, the Blackberry Milk Team has answered $4$ out of the first $10$ questions on the Boba Round correctly. If they answer all $p$ remaining questions correctly, they will have answered exactly $\frac{9p}{5}\%$ of the questions correctly in total. How many questions are on the Boba Round? [b]p10.[/b] The sum of two positive integers is $2021$ less than their product. If one of them is a perfect square, compute the sum of the two numbers. [b]p11.[/b] Points $E$ and $F$ lie on edges $\overline{BC}$ and $\overline{DA}$ of unit square $ABCD$, respectively, such that $BE =\frac13$ and $DF =\frac13$ . Line segments $\overline{AE}$ and $\overline{BF}$ intersect at point $G$. The area of triangle $EFG$ can be written in the form $m/n$ , where $m$ and $n$ are relatively prime positive integers. Compute $m+n$. [b]p12.[/b] Compute the number of positive integers $n \le 2020$ for which $n^{k+1}$ is a factor of $(1+2+3+· · ·+n)^k$ for some positive integer $k$. [b]p13.[/b] How many permutations of $123456$ are divisible by their last digit? For instance, $123456$ is divisible by $6$, but $561234$ is not divisible by $4$. [b]p14.[/b] Compute the sum of all possible integer values for $n$ such that $n^2 - 2n - 120$ is a positive prime number. [b]p15. [/b]Triangle $\vartriangle ABC$ has $AB =\sqrt{10}$, $BC =\sqrt{17}$, and $CA =\sqrt{41}$. The area of $\vartriangle ABC$ can be expressed in the form $m/n$ for relatively prime positive integers $m$ and $n$. Compute $m + n$. [b]p16.[/b] Let $f(x) = \frac{1 + x^3 + x^{10}}{1 + x^{10}}$ . Compute $f(-20) + f(-19) + f(-18) + ...+ f(20)$. [b]p17.[/b] Leanne and Jing Jing are walking around the $xy$-plane. In one step, Leanne can move from any point $(x, y)$ to $(x + 1, y)$ or $(x, y + 1)$ and Jing Jing can move from $(x, y)$ to $(x - 2, y + 5)$ or $(x + 3, y - 1)$. The number of ways that Leanne can move from $(0, 0)$ to $(20, 20)$ is equal to the number of ways that Jing Jing can move from $(0, 0)$ to $(a, b)$, where a and b are positive integers. Compute the minimum possible value of $a + b$. [b]p18.[/b] Compute the number positive integers $1 < k < 2021$ such that the equation $x +\sqrt{kx} = kx +\sqrt{x}$ has a positive rational solution for $x$. [b]p19.[/b] In triangle $\vartriangle ABC$, point $D$ lies on $\overline{BC}$ with $\overline{AD} \perp \overline{BC}$. If $BD = 3AD$, and the area of $\vartriangle ABC$ is $15$, then the minimum value of $AC^2$ is of the form $p\sqrt{q} - r$, where $p, q$, and $r$ are positive integers and $q$ is not divisible by the square of any prime number. Compute $p + q + r$. [b]p20. [/b]Suppose the decimal representation of $\frac{1}{n}$ is in the form $0.p_1p_2...p_j\overline{d_1d_2...d_k}$, where $p_1, ... , p_j$ , $d_1,... , d_k$ are decimal digits, and $j$ and $k$ are the smallest possible nonnegative integers (i.e. it’s possible for $j = 0$ or $k = 0$). We define the [i]preperiod [/i]of $\frac{1}{n}$ to be $j$ and the [i]period [/i]of $\frac{1}{n}$ to be $k$. For example, $\frac16 = 0.16666...$ has preperiod $1$ and period $1$, $\frac17 = 0.\overline{142857}$ has preperiod $0$ and period $6$, and $\frac14 = 0.25$ has preperiod $2$ and period $0$. What is the smallest positive integer $n$ such that the sum of the preperiod and period of $\frac{1}{n}$ is $ 8$? PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $a, b$, and $c$ be positive real numbers. Prove: $\frac{a}{b}+ \frac{b}{c}+ \frac{c}{a}\le \frac{a^2}{b^2} + \frac{b^2}{c^2} + \frac{c^2}{a^2}$ .

Given a positive integer $n$. (a) If $P$ is a polynomial of degree $n$ where $P(x) \in \mathbb{Z}$ for every $x \in \mathbb{Z}$, prove that for every $a,b \in \mathbb{Z}$ where $P(a) \neq P(b)$, \[\text{lcm}(1, 2, \ldots, n) \ge \left| \dfrac{a-b}{P(a) - P(b)} \right|\] (b) Find one $P$ (for each $n$) such that the equality case above is achieved for some $a,b \in \mathbb{Z}$.

The real numbers $x,y,z, m, n$ are positive, such that $m + n \ge 2$. Prove that $x\sqrt{yz(x + my)(x + nz)} + y\sqrt{xz(y + mx)(y + nz)} + z\sqrt{xy(z + mx)(x + ny) }\le \frac{3(m + n)}{8} (x + y)(y + z)(z + x)$

For a positive integer $n$ and a nonzero digit $d$, let $f(n, d)$ be the smallest positive integer $k$, such that $kn$ starts with $d$. What is the maximal value of $f(n, d)$, over all positive integers $n$ and nonzero digits $d$?

[b]7.1. / 6.5[/b] Prove that out of any six people there will always be three pairs of acquaintances or three pairs of strangers. [b]7.2[/b] Given a circle $O$ and a square $K$, as well as a line $L$. Construct a segment of given length parallel to $L$ and such that its ends lie on $O$ and $K$ respectively [b]7.3[/b] The three-digit number $\overline{abc}$ is divisible by $37$. Prove that the sum of the numbers $\overline{bca}$ and $\overline{cab}$ is also divisible by $37$.[b] (typo corrected)[/b] [b]7.4.[/b] Point $C$ is the midpoint of segment $AB$. On an arbitrary ray drawn from point $C$ and not lying on line $AB$, three consecutive points $P$, $M$ and $Q$ so that $PM=MQ$. Prove that $AP+BQ>2CM$. [img]https://cdn.artofproblemsolving.com/attachments/f/3/a8031007f5afc31a8b5cef98dd025474ac0351.png[/img] [b]7.5.[/b] Given $2n+1$ different objects. Prove that you can choose an odd number of objects from them in as many ways as an even number. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c3983442_1961_leningrad_math_olympiad]here[/url].

Find all pairs $(x,y)$ of real numbers that satisfy the system \begin{align*} x \cdot \sqrt{1-y^2} &=\frac14 \left(\sqrt3+1 \right), \\ y \cdot \sqrt{1-x^2} &= \frac14 \left( \sqrt3 -1 \right). \end{align*}

The expression \[ (x \plus{} y \plus{} z)^{2006} \plus{} (x \minus{} y \minus{} z)^{2006} \]is simplified by expanding it and combining like terms. How many terms are in the simplified expression? $ \textbf{(A) } 6018 \qquad \textbf{(B) } 671,676 \qquad \textbf{(C) } 1,007,514 \qquad \textbf{(D) } 1,008,016 \qquad \textbf{(E) } 2,015,028$

The polynomial $W(x)=x^2+ax+b$ with integer coefficients has the following property: for every prime number $p$ there is an integer $k$ such that both $W(k)$ and $W(k+1)$ are divisible by $p$. Show that there is an integer $m$ such that $W(m)=W(m+1)=0$.

Let the base $2$ representation of $x\in[0;1)$ be $x=\sum_{i=0}^\infty \frac{x_i}{2^{i+1}}$. (If $x$ is dyadically rational, i.e. $x\in\left\{\frac{k}{2^n}\,:\, k,n\in\mathbb{Z}\right\}$, then we choose the finite representation.) Define function $f_n:[0;1)\to\mathbb{Z}$ by $$f_n(x)=\sum_{j=0}^{n-1}(-1)^{\sum_{i=0}^j x_i}.$$Does there exist a function $\varphi:[0;\infty)\to[0;\infty)$ such that $\lim_{x\to\infty} \varphi(x)=\infty$ and $$\sup_{n\in\mathbb{N}}\int_0^1 \varphi(|f_n(x)|)\mathrm{d}x<\infty\, ?$$

For a sequence of integers $a_1,a_2,a_3,...$ with $0<a_1<a_2<a_3<...$ applies: $$a_n=4a_{n-1}-a_{n-2} \,\,\, for \,\,\, n > 2$$ It is further given that $a_4 = 194$. Calculate $a_5$.

The fraction $$\left(\frac{\frac13+1}{3} +\frac{1+ \frac13}{3} \right) / \left(\frac{3}{\frac{1}{3+1}+\frac{ 1}{1+3}}\right)$$ can be written as $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

Depending on the real number \( a \), find all polynomials \( P(x) \) with real coefficients such that \[ (x^3 - ax^2 + 1)P(x) = (x^3 + ax^2 + 1)P(x-1) \] for every real number \( x \).

Let $z_1$ and $z_2$ be the zeros of the polynomial $f(x) = x^2 + 6x + 11$. Compute $(1 + z^2_1z_2)(1 + z_1z_2^2)$.

Given an integer $ m\geq 2$, and two real numbers $ a,b$ with $ a > 0$ and $ b\neq 0$. The sequence $ \{x_n\}$ is such that $ x_1 \equal{} b$ and $ x_{n \plus{} 1} \equal{} ax^{m}_{n} \plus{} b$, $ n \equal{} 1,2,...$. Prove that (1)when $ b < 0$ and m is even, the sequence is bounded if and only if $ ab^{m \minus{} 1}\geq \minus{} 2$; (2)when $ b < 0$ and m is odd, or when $ b > 0$ the sequence is bounded if and only if $ ab^{m \minus{} 1}\geq\frac {(m \minus{} 1)^{m \minus{} 1}}{m^m}$.

Thirty numbers are arranged on a circle in such a way that each number equals the absolute difference of its two neighbors. Given that the sum of the numbers is $2000$, determine the numbers.

Let $f: Z^+ \to Z^+ \cup \{0\}$ a function that meets the following conditions: a) $f (a b) = f (a) + f (b)$, b) $f (a) = 0$ provided that the digits of the unit of $a$ are $7$, c) $f (10) = 0$. Find $f (2016).$

The roots of the equation $x^2 + (3a + b)x + a^2 + 2b^2 = 0$ are $x_1$ and $x_2$ with $x_1 \ne x_2$. Determine the values of $a$ and $b$ so that the roots of the equation $ x^2 - 2a(3a + 2b)x + 5a^2b^2 + 4b^4 = 0$ let $x^2_1$ and $x^2_2$.

Determine all real constants $t$ such that whenever $a$, $b$ and $c$ are the lengths of sides of a triangle, then so are $a^2+bct$, $b^2+cat$, $c^2+abt$.

The absolute values of all roots of the quadratic equation $x^2+Ax+B = 0$ and $x^2+Cx+D = 0$ are less then $1$. Prove that so are absolute values of the roots of the quadratic equation $x^2 + \frac{A + C}{2} x + \frac{B + D}{2} = 0$.

Solve the equation $(x^3-1000)^{1/2}=(x^2+100)^{1/3}$

A weird calculator has a numerical display and only two buttons, $\boxed{D\sharp}$ and $\boxed{D\flat}$. The first button doubles the displayed number and then adds $1$. The second button doubles the displayed number and then subtracts $1$. For example, if the display is showing $5$, then pressing the $\boxed{D\sharp}$ produces $11$. If the display shows $5$ and we press $\boxed{D\flat}$, we get $9$. If the display shows $5$ and we press the sequence $\boxed{D\sharp}$, $\boxed{D\flat}$, $\boxed{D\sharp}$, $\boxed{D\sharp}$, we get a display of $87$. [list=i] [*] Suppose the initial displayed number is $1$. Give a sequence of exactly eight button presses that will result in a display of $313$. [*] Suppose the initial displayed number is $1$, and we then perform exactly eight button presses. Describe all the numbers that can possibly result? Prove your answer by explaining how all these numbers can be produced and that no other numbers can be produced. [/list]