If $x, y, z, w$ are nonnegative real numbers satisfying \[\left\{ \begin{array}{l}y = x - 2003 \\ z = 2y - 2003 \\ w = 3z - 2003 \\ \end{array} \right. \] find the smallest possible value of $x$ and the values of $y, z, w$ corresponding to it.

Let $f$ be an injective function from ${1,2,3,\ldots}$ in itself. Prove that for any $n$ we have: $\sum_{k=1}^{n} f(k)k^{-2} \geq \sum_{k=1}^{n} k^{-1}.$

Solve in positive integers the equation $$m^{\frac{1}{n}}+n^{\frac{1}{m}}=2+\frac{2}{mn(m+n)^{\frac{1}{m}+\frac{1}{n}}}.$$

Find all functions $f : \mathbb{Z}\rightarrow \mathbb{Z}$ satisfying \[f^{a^{2} + b^{2}}(a+b) = af(a) +bf(b)\] for all integers $a$ and $b$

Let $P(X)$ be a polynomial with real coefficients such that $\{P(n)\}\leq\frac{1}{n}, \forall n\in\mathbb{N}$, where $\{a\}$ is the fractional part of the number $a$. Show that $P(n)\in\mathbb{Z}, \forall n\in\mathbb{N}$.

Two spheres in contact have a common tangent cone. These three surfaces divide the space into various parts, only one of which is bounded by all three surfaces, it is "ring-shaped." Being given the radii of the spheres, $r$ and $R$, find the volume of the "ring-shaped" part. (The desired expression is a rational function of $r$ and $R.$)

Given a polynomial \[P(x) = ab(a - c)x^3 + (a^3 - a^2c + 2ab^2 - b^2c + abc)x^2 +(2a^2b + b^2c + a^2c + b^3 - abc)x + ab(b + c),\] where $a, b, c \neq 0$, prove that $P(x)$ is divisible by \[Q(x) = abx^2 + (a^2 + b^2)x + ab\] and conclude that $P(x_0)$ is divisible by $(a + b)^3$ for $x_0 = (a + b + 1)^n, n \in \mathbb N$.

Let $p$ and $q$ be different primes, and $\alpha\in (0, 3)$ a real number. Prove that in sequence $$\left[ \alpha \right] , \left[ 2\alpha \right] , \left[ 3\alpha \right] \dots$$ exists number less than $2pq$, divisible by $p$ or $q$.

Let $k$ be a fixed real number. Find all functions $f: R \to R$ such that $f(x)+ (f(y))^2 = kf(x + y^2)$ for all real numbers $x$ and $y$.

Prove that among any $9$ distinct real numbers, there exist $4$ distinct numbers $a,b,c,d$ such that $$(ac+bd)^2\ge\frac{9}{10}(a^2+b^2)(c^2+d^2)$$

Consider the set $A_n=\{x_1,x_2,\ldots,x_n,y_1,y_2,\ldots,y_n\}$ of $2n$ variables. How many permutations of set $A_n$ are there for which it is possible to assign real values from the interval $(0,1)$ to the $2n$ variables so that: (i) $x_i+y_i=1$ for each $i$; (ii) $x_1<x_2<\ldots<x_n$; (iii) the $2n$ terms of the permutation form a strictly increasing sequence?

A fisherman, who was sailing in a rowing boat against the current of the river, had a hat falling from the bow of the boat into the water. After half an hour, the fisherman noticed the loss of his cap and immediately turned back. Find the speed of the river if the fisherman caught up with the cap at a distance of $a$ km from the place where it fell into the water (the speed of the river and the movement of the boat relative to the water is considered constant).

Let $a_1<a_2<a_3<a_4<\cdots$ be an infinite sequence of real numbers in the interval $(0,1)$. Show that there exists a number that occurs exactly once in the sequence \[ \frac{a_1}{1},\frac{a_2}{2},\frac{a_3}{3},\frac{a_4}{4},\ldots.\] [i]Merlijn Staps[/i]

The real numbers $x_1,x_2,..., x_n$ and $a_0,a_1,...,a_{n-1}$ with $x_i \ne 0$ for $i \in\{1,2,.., n\}$ are such that $$(x-x_1)(x-x_2)...(x-x_n)=x^n+a_{n-1}x^{n-1}+...+a_1x+a_0$$ Express $x_1^{-2}+x_2^{-2}+...+ x_n^{-2}$ in terms of $a_0,a_1,...,a_{n-1}$.

Show that we cannot find $171$ binary sequences (sequences of $0$’s and $1$’s), each of length $12$ such that any two of them differ in at least four positions.

Suppose that $a,b$ are real numbers, function $f(x) = ax+b$ satisfies $\mid f(x) \mid \leq 1$ for any $x \in [0,1]$. Find the range of values of $S= (a+1)(b+1).$

Find all pairs of real numbers $(a, b)$ such that the equality $$|ax+by|+ |bx + ay| = 2|x| + 2|y|$$ holds for all reals $x$ and $y$.

For which natural numbers $n$ does $2^n > 10n^2 -60n + 80$ hold?

Sequence {$x_n$} satisfies: $x_1=1$ , ${x_n=\sqrt{x_{n-1}^2+x_{n-1}}+x_{n-1}}$ ( ${n>=2}$ ) Find the general term of {$x_n$}

[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].

Find all continuous functions $f: \mathbb{R}\to \mathbb{R}$ such that \[ f(x)-f(y)\in \mathbb{Q}\quad \text{ for all }\quad x-y\in\mathbb{Q} \]

For each positive integer $n$, define $H_n(x)=(-1)^ne^{x^2}\frac{d^n}{dx^n}e^{-x^2}.$ (1) Find $H_1(x),\ H_2(x),\ H_3(x)$. (2) Express $\frac{d}{dx}H_n(x)$ interms of $H_n(x),\ H_{n+1}(x).$ Then prove that $H_n(x)$ is a polynpmial with degree $n$ by induction. (3) Let $a$ be real number. For $n\geq 3$, express $S_n(a)=\int_0^a xH_n(x)e^{-x^2}dx$ in terms of $H_{n-1}(a),\ H_{n-2}(a),\ H_{n-2}(0)$. (4) Find $\lim_{a\to\infty} S_6(a)$. If necessary, you may use $\lim_{x\to\infty}x^ke^{-x^2}=0$ for a positive integer $k$.

Let $n$ be a positive integer and let $x_1,\ldots,x_n,y_1,\ldots,y_n$ be integers satisfying the following condition: the numbers $x_1,\ldots,x_n$ are pairwise distinct and for every positive integer $m$ there exists a polynomial $P_m$ with integer coefficients such that $P_m(x_i) - y_i$, $i=1,\ldots,n$, are all divisible by $m$. Prove that there exists a polynomial $P$ with integer coefficients such that $P(x_i) = y_i$ for all $i=1,\ldots,n$.

Find all functions $f: [0, +\infty) \to [0, +\infty)$ satisfying the equation \[(y+1)f(x+y) = f\left(xf(y)\right)\] For all non-negative real numbers $x$ and $y.$

A function $f$ satisfies the conditions $f (x) \ne 0$ and $f (x+2) = f (x-1) f (x+5)$ for all real x. Show that $f (x+18) = f (x)$ for any real $x$.