Consider the polynomial $ f(x) \equal{} ax^2 \plus{} bx \plus{} c$, with degree less than or equal to 2. When $ f$ varies with subject to the constrain $ f(0) \equal{} 0,\ f(2) \equal{} 2$, find the minimum value of $ S\equal{}\int_0^2 |f'(x)|\ dx$.

Let $f$ be a function defined on the positive integers with $f(n) \ge 0$ and $f(n) \le f(n+1)$ for all $n$. Prove that if \[\sum_{n = 1}^{\infty} \frac{f(n)}{n^2}\] diverges, there exists a sequence $a_1, a_2, \dots$ such that the sequence $\tfrac{a_n}{n}$ hits every natural number, while \[a_{n+m} \le a_n + a_m + f(n+m)\] holds for every pair $n$, $m$.

Let $\mathbb{N} = \{1, 2, 3, \ldots\}$ be the set of positive integers. Find all functions $f$, defined on $\mathbb{N}$ and taking values in $\mathbb{N}$, such that $(n-1)^2< f(n)f(f(n)) < n^2+n$ for every positive integer $n$.

Let $a_1, a_2, \ldots, a_{2023}$ be nonnegative real numbers such that $a_1 + a_2 + \ldots + a_{2023} = 100$. Let $A = \left \{ (i,j) \mid 1 \leqslant i \leqslant j \leqslant 2023, \, a_ia_j \geqslant 1 \right\}$. Prove that $|A| \leqslant 5050$ and determine when the equality holds. [i]Proposed by Yunhao Fu[/i]

Give all possible representations of $2022$ as a sum of at least two consecutive positive integers and prove that these are the only representations.

Determine all $n \in \mathbb{N}$ for which $n^{10} + n^5 + 1$ is prime.

Alice has five real numbers $a < b < c < d < e$. She takes the sum of each pair of numbers and writes down the ten sums. The three smallest sums are $32$, $36$ and $37$, while the two largest sums are $48$ and $51$. Determine $e$.

There are $100$ people in a room. Some are [i]wise[/i] and some are [i]optimists[/i]. $\quad \bullet~$ A [i]wise[/i] person can look at someone and know if they are wise or if they are an optimist. $\quad \bullet~$ An [i]optimist[/i] thinks everyone is wise (including themselves). Everyone in the room writes down what they think is the number of wise people in the room. What is the smallest possible value for the average? $$ \mathrm a. ~ 10\qquad \mathrm b.~25\qquad \mathrm c. ~50 \qquad \mathrm d. ~75 \qquad \mathrm e. ~100 $$

Show that the quadratic equation $x^2 + 7x - 14 (q^2 +1) =0$ , where $q$ is an integer, has no integer root.

For any real $ a$ define $ f_a : \mathbb{R} \rightarrow \mathbb{R}^2$ by the law $ f_a(t) \equal{} \left( \sin(t), \cos(at) \right)$. a) Prove that $ f_{\pi}$ is not periodic. b) Determine the values of the parameter $ a$ for which $ f_a$ is periodic. [b]Remark[/b]. L. Euler proved in $ 1737$ that $ \pi$ is irrational.

[b]1.[/b] For each real number $r$ between $0$ and $1$ we can represent $r$ as an infinite decimal $r = 0.r_1r_2r_3\dots$ with $0 \leq r_i \leq 9$. For example, $\frac{1}{4} = 0.25000\dots$, $\frac{1}{3} = 0.333\dots$ and $\frac{1}{\sqrt{2}} = 0.707106\dots$. a) Show that we can choose two rational numbers $p$ and $q$ between $0$ and $1$ such that, from their decimal representations $p = 0.p_1p_2p_3\dots$ and $q = 0.q_1q_2q_3\dots$, it's possible to construct an irrational number $\alpha = 0.a_1a_2a_3\dots$ such that, for each $i = 1, 2, 3, \dots$, we have $a_i = p_1$ or $a_1 = q_i$. b) Show that there's a rational number $s = 0.s_1s_2s_3\dots$ and an irrational number $\beta = 0.b_1b_2b_3\dots$ such that, for all $N \geq 2017$, the number of indexes $1 \leq i \leq N$ satisfying $s_i \neq b_i$ is less than or equal to $\frac{N}{2017}$.

Brenda and Sally run in opposite directions on a circular track, starting at diametrically opposite points. They first meet after Brenda has run $ 100$ meters. They next meet after Sally has run $ 150$ meters past their first meeting point. Each girl runs at a constant speed. What is the length of the track in meters? $ \textbf{(A)}\ 250 \qquad \textbf{(B)}\ 300 \qquad \textbf{(C)}\ 350 \qquad \textbf{(D)}\ 400\qquad \textbf{(E)}\ 500$

Positive integers $a$, $b$, $c$ are given. It is known that $\frac{c}{b}=\frac{b}{a}$, and the number $b^2-a-c+1$ is a prime. Prove that $a$ and $c$ are double of a squares of positive integers.

If $a$, $b$ and $c$ are nonnegative real numbers such that $a^2+b^2+c^2=1$, prove that $$\frac{1}{2} \leq \frac{a}{1+a^4}+\frac{b}{1+b^4}+\frac{c}{1+c^4} \leq \frac{9\sqrt{3}}{10}$$

Let $a_n$ be a sequence given by $a_1 = 18$, and $a_n = a_{n-1}^2+6a_{n-1}$, for $n>1$. Prove that this sequence contains no perfect powers.

Let $ n$ be a positive integer, and let $ x$ and $ y$ be a positive real number such that $ x^n \plus{} y^n \equal{} 1.$ Prove that \[ \left(\sum^n_{k \equal{} 1} \frac {1 \plus{} x^{2k}}{1 \plus{} x^{4k}} \right) \cdot \left( \sum^n_{k \equal{} 1} \frac {1 \plus{} y^{2k}}{1 \plus{} y^{4k}} \right) < \frac {1}{(1 \minus{} x) \cdot (1 \minus{} y)}. \] [i]Author: Juhan Aru, Estonia[/i]

If three nonnegative reals $a$, $b$, $c$ satisfy $a+b+c=1$, prove that $2 \leq \left(1-a^{2}\right)^{2}+\left(1-b^{2}\right)^{2}+\left(1-c^{2}\right)^{2}\leq \left(1+a\right)\left(1+b\right)\left(1+c\right)$.

Find all real triples $(a,b,c)$, for which $$a(b^2+c)=c(c+ab)$$ $$b(c^2+a)=a(a+bc)$$ $$c(a^2+b)=b(b+ca).$$

Find all pairs of real numbers $(x, y)$ that satisfy the equation $\frac{x + 6}{y}+\frac{13}{xy}=\frac{4-y}{x}$

(a) If $c(a^3+b^3) = a(b^3+c^3) = b(c^3+a^3)$ with $a, b, c$ positive real numbers, does $a = b = c$ necessarily hold? (b) If $a(a^3+b^3) = b(b^3+c^3) = c(c^3+a^3)$ with $a, b, c$ positive real numbers, does $a = b = c$ necessarily hold?

[b]p1.[/b] An Egyptian fraction has the form $1/n$, where $n$ is a positive integer. In ancient Egypt, these were the only fractions allowed. Other fractions between zero and one were always expressed as a sum of distinct Egyptian fractions. For example, $3/5$ was seen as $1/2 + 1/10$, or $1/3 + 1/4 + 1/60$. The preferred method of representing a fraction in Egypt used the "greedy" algorithm, which at each stage, uses the Egyptian fraction that eats up as much as possible of what is left of the original fraction. Thus the greedy fraction for $3/5$ would be $1/2 + 1/10$. a) Find the greedy Egyptian fraction representations for $2/13$. b) Find the greedy Egyptian fraction representations for $9/10$. c) Find the greedy Egyptian fraction representations for $2/(2k+1)$, where $k$ is a positive integer. d) Find the greedy Egyptian fraction representations for $3/(6k+1)$, where $k$ is a positive integer. [b]p2.[/b] a) The smaller of two concentric circles has radius one unit. The area of the larger circle is twice the area of the smaller circle. Find the difference in their radii. [img]https://cdn.artofproblemsolving.com/attachments/8/1/7c4d81ebfbd4445dc31fa038d9dc68baddb424.png[/img] b) The smaller of two identically oriented equilateral triangles has each side one unit long. The smaller triangle is centered within the larger triangle so that the perpendicular distance between parallel sides is always the same number $d$. The area of the larger triangle is twice the area of the smaller triangle. Find $d$. [img]https://cdn.artofproblemsolving.com/attachments/8/7/1f0d56d8e9e42574053c831fa129eb40c093d9.png[/img] [b]p3.[/b] Suppose that the domain of a function $f$ is the set of real numbers and that $f$ takes values in the set of real numbers. A real number $x_0$ is a fixed point of f if $f(x_0) = x_0$. a) Let $f(x) = m x + b$. For which $m$ does $f$ have a fixed point? b) Find the fixed point of f$(x) = m x + b$ in terms of m and b, when it exists. c) Consider the functions $f_c(x) = x^2 - c$. i. For which values of $c$ are there two different fixed points? ii. For which values of $c$ are there no fixed points? iii. In terms of $c$, find the value(s) of the fixed point(s). d) Find an example of a function that has exactly three fixed points. [b]p4.[/b] A square based pyramid is made out of rubber balls. There are $100$ balls on the bottom level, 81 on the next level, etc., up to $1$ ball on the top level. a) How many balls are there in the pyramid? b) If each ball has a radius of $1$ meter, how tall is the pyramid? c) What is the volume of the solid that you create if you place a plane against each of the four sides and the base of the balls? [b]p5.[/b] We wish to consider a general deck of cards specified by a number of suits, a sequence of denominations, and a number (possibly $0$) of jokers. The deck will consist of exactly one card of each denomination from each suit, plus the jokers, which are "wild" and can be counted as any possible card of any suit. For example, a standard deck of cards consists of $4$ suits, $13$ denominations, and $0$ jokers. a) For a deck with $3$ suits $\{a, b, c\}$ and $7$ denominations $\{1, 2, 3, 4, 5, 6, 7\}$, and $0$ jokers, find the probability that a 3-card hand will be a straight. (A straight consists of $3$ cards in sequence, e.g., $1 \heartsuit$ ,$2 \spadesuit$ , $3\clubsuit$ , $2\diamondsuit$ but not $6 \heartsuit$ ,$7 \spadesuit$ , $1\diamondsuit$). b) For a deck with $3$ suits, $7$ denominations, and $0$ jokers, find the probability that a $3$-card hand will consist of $3$ cards of the same suit (i.e., a flush). c) For a deck with $3$ suits, $7$ denominations, and $1$ joker, find the probability that a $3$-card hand dealt at random will be a straight and also the probability that a $3$-card hand will be a flush. d) Find a number of suits and the length of the denomination sequence that would be required if a deck is to contain $1$ joker and is to have identical probabilities for a straight and a flush when a $3$-card hand is dealt. The answer that you find must be an answer such that a flush and a straight are possible but not certain to occur. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Given a sequence $a_n$: \[ 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, \dots \] (one '1', two '2' and so on) and another sequence $b_n$ such that $a_{b_n}=b_{a_n}$ for all positive integers $n$. It is known that $b_k=1$ for some $k>100$. Prove that $b_m=1$ for all $m>k$.

Greater or less than one is the number $0.99999^{1.00001} \cdot 1.00001^{0.99999}$?

Let $n$ be a positive integer number and let $a_1, a_2, \ldots, a_n$ be $n$ positive real numbers. Prove that $f : [0, \infty) \rightarrow \mathbb{R}$, defined by \[f(x) = \dfrac{a_1 + x}{a_2 + x} + \dfrac{a_2 + x}{a_3 + x} + \cdots + \dfrac{a_{n-1} + x}{a_n + x} + \dfrac{a_n + x}{a_1 + x}, \] is a decreasing function. [i]Dan Marinescu et al.[/i]

Let $a$ and $b$ be real, positive and distinct numbers. We consider the set: \[M=\{ ax+by\mid x,y\in\mathbb{R},\ x>0,\ y>0,\ x+y=1\} \] Prove that: (i) $\frac{2ab}{a+b}\in M;$ (ii) $\sqrt{ab}\in M.$