Find the largest real constant $k$ for which the inequality \[(a^2 + 3)(b^2 + 3)(c^2 + 3)(d^2 + 3) + k(a - 1)(b - 1)(c - 1)(d - 1) \ge 0\] holds for all real numbers $a$, $b$, $c$, and $d$.

Given an integer $k$. $f(n)$ is defined on negative integer set and its values are integers. $f(n)$ satisfies \[ f(n)f(n+1)=(f(n)+n-k)^2, \] for $n=-2,-3,\cdots$. Find an expression of $f(n)$.

Juanita wrote the numbers from $1$ to $13$ , calculated the sum of all the digits he had written and obtained $$1+2+3+4+5+6+7+8+9+(1+0)+(1+1)+(1+2)+(1+3)=55.$$ His brother Ariel wrote the numbers from $1$ to $100$ and calculated the sum of all the digits written. Find the value of Ariel's sum.

$f_{1},f_{2},\dots,f_{n}$ are polynomials with integer coefficients. Prove there exist a reducible $g(x)$ with integer coefficients that $f_{1}+g,f_{2}+g,\dots,f_{n}+g$ are irreducible.

Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $$f(f(x)-f(y))+2f(xy)=x^2f(x)+f(y^2)$$ for all real numbers $x,y$.

Determine all functions $ f: \mathbb{R} \mapsto \mathbb{R}$ with $ x,y \in \mathbb{R}$ such that \[ f(x \minus{} f(y)) \equal{} f(x\plus{}y) \plus{} f(y)\]

Prove that $$x\cos x \le \frac{\pi^2}{16}$$ for $0 \le x \le \frac{\pi}{2}$

$$\frac{p}{q} = \sum_{n = 1}^\infty \frac{1}{2^{n + 6}} \frac{(10 - 4\cos^2(\frac{\pi n}{24})) (1 - (-1)^n) - 3\cos(\frac{\pi n}{24}) (1 + (-1)^n)}{25 - 16\cos^2(\frac{\pi n}{24})}$$ where $p$ and $q$ are relatively prime positive integers. Find $p + q$.

Solve the system of equations: $\begin{cases} x_1 + x_2 + x_3 = 6 \\ x_2 + x_3 + x_4 = 9 \\ x_3 + x_4 + x_5 = 3 \\ x_4 + x_5 + x_6 = -3 \\ x_5 + x_6 + x_7 = -9 \\ x_6 + x_7 + x_8 = -6 \\ x_7 + x_8 + x_1 = -2 \\ x_8 + x_1 + x_2 = 2 \end{cases}$

Let $\{a_1,a_2,...,a_{100}\}$ be a sequence of $100$ distinct real numbers. Show that there exists either an increasing subsequence $a_{i_1}<a_{i_2}<...<a_{i_{10}}$ $(i_1<i_2<...<i_{10})$ of $10$ numbers, or a decreasing subsequence $ a_{j_1}>a_{j_2}>...>a_{j_{12}}$ $(j_1<j_2<...<j_{12})$ of $12$ numbers, or both.

[b]p1.[/b] A bus leaves San Mateo with $n$ fairies on board. When it stops in San Francisco, each fairy gets off, but for each fairy that gets off, $n$ fairies get on. Next it stops in Oakland where $6$ times as many fairies get off as there were in San Mateo. Finally the bus arrives at Berkeley, where the remaining $391$ fairies get off. How many fairies were on the bus in San Mateo? [b]p2.[/b] Let $a$ and $b$ be two real solutions to the equation $x^2 + 8x - 209 = 0$. Find $\frac{ab}{a+b}$ . Express your answer as a decimal or a fraction in lowest terms. [b]p3.[/b] Let $a$, $b$, and $c$ be positive integers such that the least common multiple of $a$ and $b$ is $25$ and the least common multiple of $b$ and $c$ is $27$. Find $abc$. [b]p4.[/b] It takes Justin $15$ minutes to finish the Speed Test alone, and it takes James $30$ minutes to finish the Speed Test alone. If Justin works alone on the Speed Test for $3$ minutes, then how many minutes will it take Justin and James to finish the rest of the test working together? Assume each problem on the Speed Test takes the same amount of time. [b]p5.[/b] Angela has $128$ coins. $127$ of them have the same weight, but the one remaining coin is heavier than the others. Angela has a balance that she can use to compare the weight of two collections of coins against each other (that is, the balance will not tell Angela the weight of a collection of coins, but it will say which of two collections is heavier). What is the minumum number of weighings Angela must perform to guarantee she can determine which coin is heavier? PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find all real numbers $c$ for which there exists a function $f\colon\mathbb R\rightarrow \mathbb R$ such that for each $x, y\in\mathbb R$ it's true that $$f(f(x)+f(y))+cxy=f(x+y).$$

Professor Oak is feeding his $100$ Pokémon. Each Pokémon has a bowl whose capacity is a positive real number of kilograms. These capacities are known to Professor Oak. The total capacity of all the bowls is $100$ kilograms. Professor Oak distributes $100$ kilograms of food in such a way that each Pokémon receives a non-negative integer number of kilograms of food (which may be larger than the capacity of the bowl). The [i]dissatisfaction level[/i] of a Pokémon who received $N$ kilograms of food and whose bowl has a capacity of $C$ kilograms is equal to $\lvert N-C\rvert$. Find the smallest real number $D$ such that, regardless of the capacities of the bowls, Professor Oak can distribute food in a way that the sum of the dissatisfaction levels over all the $100$ Pokémon is at most $D$. [i]Oleksii Masalitin, Ukraine[/i]

Find all triplets $ (x,y,z)$, that satisfy: $ \{\begin{array}{c}\ \ x^2 - 2x - 4z = 3\ y^2 - 2y - 2x = - 14 \ z^2 - 4y - 4z = - 18 \end{array}$

Solve for integers $x,y,z$: \[ \{ \begin{array}{ccc} x + y &=& 1 - z \\ x^3 + y^3 &=& 1 - z^2 . \end{array} \]

Positive numbers $x,y$ satisfy $x^2+y^3 \ge x^3+y^4$. Prove that $x^3+y^3 \le 2$.

A positive interger $n$ is called [i][u]rising[/u][/i] if its decimal representation $a_ka_{k-1}\cdots a_0$ satisfies the condition $a_k\le a_{k-1}\le\cdots \le a_0$. Polynomial $P$ with real coefficents is called [i][u]interger-valued[/u][/i] if for all integer numbers $n$, $P(n)$ takes interger values. $P(n)$ is called [i][u]rising-valued[/u][/i] if for all [i]rising[/i] numbers $n$, $P(n)$ takes integer values. Does it necessarily mean that, "every [i]rising-valued[/i] $P$ is also [i]interger-valued[/i] $P$"?

Prove the inequality $$\frac{a + b}{a^2 -ab + b^2} \le \frac{4}{ |a + b|}$$ for all real numbers $a$ and $b$ with $a + b\ne 0$. When does equality hold?

It is known about the function $f : R \to R$ that $\bullet$ $f(x) > f(y)$ for all real $x > y$ $\bullet$ $f(x) > x$ for all real $x$ $\bullet$ $f(2x - f (x)) = x$ for all real $x$. Prove that $f(x) = x + f(0)$ for all real numbers $x$.

Distinct real numbers \( a, b, c \) satisfy the following condition: \[ \frac{a - b}{a^3b^3} + \frac{b - c}{b^3c^3} + \frac{c - a}{c^3a^3} = 0. \] What are the possible values of the expression \[ \frac{a^4 + b^4 + c^4}{a^2b^2 + b^2c^2 + c^2a^2}? \] [i]Proposed by Vadym Solomka[/i]

A game is played in three moves. The first player picks any real number, then the second player makes it the coefficient of a cubic, except that the coefficient of $x^3$ is already fixed at $1$. Can the first player make his choices so that the final cubic has three distinct integer roots?

Find maximal value of the function $f(x)=8-3\sin^2 (3x)+6 \sin (6x)$

Equation $$x^n + a_1x^{n-1} + a_2x^{n-2} +...+ a_{n-1}x + a_n = 0$$ with integer non-zero coefficients $a_1$, $a_2$, $...$ , $a_n$ has $n$ different integer roots. Prove that if any two roots are relatively prime, then the numbers $a_{n-1}$ and $a_n$ are coprime.

Let $a,b>1$ be two real numbers. Prove that $a>b$ if and only if there exists a function $f: (0,\infty)\to\mathbb{R}$ such that i) the function $g:\mathbb{R}\to\mathbb{R}$, $g(x)=f(a^x)-x$ is increasing; ii) the function $h:\mathbb{R}\to\mathbb{R}$, $h(x)=f(b^x)-x$ is decreasing.

Find all positive integer solutions $ x, y, z$ of the equation $ 3^x \plus{} 4^y \equal{} 5^z.$