Compute $1 \times 4 - 2 \times 3 + 2 \times 5 - 3 \times 4 + 3 \times 6 - 4 \times 5 + 4 \times 7 - 5 \times 6 + 5 \times 8 - 6 \times 7.$

You have a list of $2023$ numbers, where each one can be $-1$, $0$, $1$ or $2$. The sum of all numbers is $19$ and the sum of their squares is $99$. What are the minimum and maximum values of the sum of the cubes of those $2023$ numbers?

For each subset $T$ of $S = \{1, 2, ... , 7\}$, the result $r(T)$ of T is computed as follows: the elements of $T$ are written, largest to smallest, and alternating signs $(+, -)$ starting with $+$ are put in front of each number. The value of the resulting expression is$ r(T)$. (For example, for $T =\{2, 4, 7\}$, we have $r(T) = +7 - 4 + 2 = 5$.) Compute the sum of $r(T)$ as $T$ ranges over all subsets of $S$.

Consider those functions $ f: \mathbb{N} \mapsto \mathbb{N}$ which satisfy the condition \[ f(m \plus{} n) \geq f(m) \plus{} f(f(n)) \minus{} 1 \] for all $ m,n \in \mathbb{N}.$ Find all possible values of $ f(2007).$ [i]Author: Nikolai Nikolov, Bulgaria[/i]

Determine all functions $f: \mathbb R \to \mathbb R$ with the property \[f(f(x)+y)=2x+f(f(y)-x), \quad \forall x,y \in \mathbb R.\]

Let $a,b,c$ be positive numbers. Prove that a triangle with sides $a,b,c$ exists if and only if the system of equations $$\begin{cases}\dfrac{y}{z}+\dfrac{z}{y}=\dfrac{a}{x} \\ \\ \dfrac{z}{x}+\dfrac{x}{z}=\dfrac{b}{y} \\ \\ \dfrac{x}{y}+\dfrac{y}{x}=\dfrac{c}{z}\end{cases}$$ has a real solution.

Let $f(x) = ax^7+bx^3+cx-5$, where $a,b$ and $c$ are constants. If $f(-7) = 7$, the $f(7)$ equals $\textbf {(A) } -17 \qquad \textbf {(B) } -7 \qquad \textbf {(C) } 14 \qquad \textbf {(D) } 21\qquad \textbf {(E) } \text{not uniquely determined}$

From the sequence $1,\frac12, \frac13, ...$ can one choose (a) a subsequence of $100$ different numbers, (b) an infinite subsequence such that each number (beginning from the third) is equal to the difference between the two preceding numbers ($a_k=a_{k-2}-a_{k-1}$)? (SI Tokarev)

The driver starts driving every morning at the same time from office to the house of his boss, picks up the boss and then drives back to the office. He always drives with the same speed on the same road. Because the time of arrival of the car to the boss's house is predetermined, the boss always leaves the house on time, and thus the driver does not spend any time waiting for his boss. Once the driver started driving from the office $42$ minutes later, than usual. The boss saw that the car didn't come and started walking in the direction of office. When he met the car on the road, the driver picked him up and started driving back to the office. The speed of the boss is 20 times lower than the speed of the car, and the time usually spent on the route from office to the house is at least an hour. Determine did the car come earlier or later to the office and by how many minutes.

Solve equation $x^4 = 2x^2 + \lfloor x \rfloor$, where $ \lfloor x \rfloor$ is an integral part of $x$.

Let $x,y,z{}$ be positive real numbers. Prove that \[(xy+yz+zx)\left(\frac{1}{x^2+y^2}+\frac{1}{y^2+z^2}+\frac{1}{z^2+x^2}\right)>\frac{5}{2}.\]

Let $P(x)$ and $Q(x)$ be polynomials with integer coefficients. Let $a_n = n! +n$. Show that if $\frac{P(a_n)}{Q(a_n)}$ is an integer for every $n$, then $\frac{P(n)}{Q(n)}$ is an integer for every integer $n$ such that $Q(n)\neq 0$.

Let $x, y \in \mathbb{R}^+$. Prove that: \[ \left( 1 + \frac{1}{x} \right) \left( 1 + \frac{1}{y} \right) \geq \left( 1 + \frac{2}{x + y} \right)^2. \]

Show that \[ (2a-1) \sin x + (1-a) \sin(1-a)x \geq 0 \] for $0 \leq a \leq 1$ and $0 \leq x \leq \pi$.

Let $ f(x)$ be a function with the two properties: [list=a] [*] for any two real numbers $ x$ and $ y$, $ f(x \plus{} y) \equal{} x \plus{} f(y)$, and [*] $ f(0) \equal{} 2$ [/list] What is the value of $ f(1998)$? $ \textbf{(A)}\ 0\qquad \textbf{(B)}\ 2\qquad \textbf{(C)}\ 1996\qquad \textbf{(D)}\ 1998\qquad \textbf{(E)}\ 2000$

Find all functions $f: R \to R$ and $g:R \to R$ such that $f(x-f(y))=xf(y)-yf(x)+g(x)$ for all real numbers $x,y$. I.Voronovich

Let $x$ and $y$ be positive real numbers that satisfy $(\log x)^2+(\log y)^2=\log(x^2)+\log(y^2)$. Compute the maximum possible value of $(\log(xy))^2$.

Find the least possible value of $(x + y)(y + z)$ for positive reals satisfying $(x + y + z) xyz = 1$.

[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].

The sequences $a_n$, $b_n$ and $c_n$ are defined recursively in the following way: $a_0 = 1/6$, $b_0 = 1/2$, $c_0 = 1/3,$ $$a_{n+1}= \frac{(a_n + b_n)(a_n + c_n)}{(a_n - b_n)(a_n - c_n)},\,\, b_{n+1}= \frac{(b_n + a_n)(b_n + c_n)}{(b_n - a_n)(b_n - c_n)},\,\, c_{n+1}= \frac{(c_n + a_n)(c_n + b_n)}{(c_n - a_n)(c_n - b_n)}$$ For each natural number $N$, the following polynomials are defined: $A_n(x) =a_o+a_1 x+ ...+ a_{2N}x^{2N}$ $B_n(x) =b_o+a_1 x+ ...+ a_{2N}x^{2N}$ $C_n(x) =a_o+a_1 x+ ...+ a_{2N}x^{2N}$ Assume the sequences are well defined. Show that there is no real $c$ such that $A_N(c) = B_N(c) = C_N(c) = 0$.

To obtain the value of a polynomial of degree $n$, whose coefficients are $$a_0, a_1, . . . ,a_n$$ (starting with the term of highest degree), when the variable $x$ is given the value $b$, the process indicated in the attached flowchart can be applied, which develops the actions required to apply Ruffini's rule. It is requested to build another flowchart analogous that allows to express the calculation of the value of the derivative of the given polynomial, also for $x = b$. [img]https://cdn.artofproblemsolving.com/attachments/a/a/27563a0e97e74553a270fcd743f22176aed83b.png[/img]

The numbers in the sequence 101, 104, 109, 116, $\dots$ are of the form $a_n = 100 + n^2$, where $n = 1$, 2, 3, $\dots$. For each $n$, let $d_n$ be the greatest common divisor of $a_n$ and $a_{n + 1}$. Find the maximum value of $d_n$ as $n$ ranges through the positive integers.

Let $a>b>0$. Prove that $\sqrt2 a^3+ \frac{3}{ab-b^2}\ge 10$ When does equality hold?

For $a,b,c$ positive real numbers such that $ab+bc+ca=3$, prove: $ \frac{a}{\sqrt{a^3+5}}+\frac{b}{\sqrt{b^3+5}}+\frac{c}{\sqrt{c^3+5}} \leq \frac{\sqrt{6}}{2}$ [i]Proposed by Dorlir Ahmeti, Albania[/i]

[b]p1.[/b] In the picture below, the two parallel cuts divide the square into three pieces of equal area. The distance between the two parallel cuts is $d$. The square has length $s$. Find and prove a formula that expresses $s$ as a function of $d$. [img]https://cdn.artofproblemsolving.com/attachments/c/b/666074d28de50cdbf338a2c667f88feba6b20c.png[/img] [b]p2.[/b] Let $S$ be a subset of $\{1, 2, 3, . . . 10, 11\}$. We say that $S$ is lucky if no two elements of $S$ differ by $4$ or $7$. (a) Give an example of a lucky set with five elements. (b) Is it possible to find a lucky set with six elements? Explain why or why not.[/quote] [b]p3.[/b] Find polynomials $p(x)$ and $q(x)$ with real coefficients such that (a) $p(x) - q(x) = x^3 + x^2 - x - 1$ for all real $x$, (b) $p(x) > 0$ for all real $x$, (c) $q(x) > 0$ for all real $x$. [b]p4.[/b] A permutation on $\{1, 2, 3, …, n\}$ is a rearrangement of the symbols. For example $32154$ is a permutation on $\{1, 2, 3, 4, 5\}$. Given a permutation $a_1a_2a_3…a_n$, an inversion is a pair of $a_i$ and $a_j$ such that $a_i > a_j$ but $i < j$. For example, $32154$ has $4$ inversions. Suppose you are only allowed to exchange adjacent symbols. For any permutation, show that the minimum number of exchanges required to put all the symbols in their natural positions (that is, $123 …n$) is the number of inversions. [b]p5.[/b] We say a number $N$ is a nontrivial sum of consecutive positive integers if it can be written as the sum of $2$ or more consecutive positive integers. What is the set of numbers from $1000$ to $2000$ that are NOT nontrivial sums of consecutive positive integers? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].