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$.

For how many integers $n$ between 1 and 100 does $x^2+x-n$ factor into the product of two linear factors with integer coefficients? $\text{(A)} \ 0 \qquad \text{(B)} \ 1 \qquad \text{(C)} \ 2 \qquad \text{(D)} \ 9 \qquad \text{(E)} \ 10$

The real-valued function $f(x)$ is defined on the reals. It satisfies $|f(x)| \le A$, $|f''(x)| \le B$ for some positive $A, B$ (and all $x$). Show that $|f'(x)| \le C$, for some fixed$ C$, which depends only on $A$ and $B$. What is the smallest possible value of $C$?

Let $p$ be a prime satisfying $p^2\mid 2^{p-1}-1$, and let $n$ be a positive integer. Define \[ f(x) = \frac{(x-1)^{p^n}-(x^{p^n}-1)}{p(x-1)}. \] Find the largest positive integer $N$ such that there exist polynomials $g(x)$, $h(x)$ with integer coefficients and an integer $r$ satisfying $f(x) = (x-r)^N g(x) + p \cdot h(x)$. [i]Proposed by Victor Wang[/i]

Let $p,q,r \in \mathbb{R}^+$ and for every $n \in \mathbb{N}$ where $pqr=1$ , denote $$ \frac{1}{p^n+q^n+1} + \frac{1}{q^n+r^n+1} + \frac{1}{r^n+p^n+ 1} \leq 1$$ [i](Art-Ninja)[/i]

Let $\alpha$ and $\beta$ be the roots of $x^{2} - qx + 1$, where $q$ is a rational number larger than $2$. Let $s_1 = \alpha + \beta$, $t_1 = 1$, and for all integers $n \geq 2$: $s_n = \alpha^n + \beta^n$ $t_n = s_{n-1} + 2s_{n-2} + \cdot \cdot \cdot + (n - 1)s_{1} + n$ Prove that, for all odd integers $n$, $t_n$ is the square of a rational number.

Find all polynomials with real coefficients such that one can find an integer valued series $a_0, a_1, \dots$ satisfying $\lfloor P(x) \rfloor = a_{ \lfloor x^2 \rfloor}$ for all $x$ real numbers.

Simplify the fraction: $\frac{(1^4+4)\cdot (5^4+4)\cdot (9^4+4)\cdot ... (69^4+4)\cdot(73^4+4)}{(3^4+4)\cdot (7^4+4)\cdot (11^4+4)\cdot ... (71^4+4)\cdot(75^4+4)}$.

A polynomial $P$ has integer coefficients and $P(3)=4$ and $P(4)=3$. For how many $x$ we might have $P(x)=x$?

Let $\{a_n\}_{n=0}^{\infty}$ be a sequence of integers numbers. Let $\Delta^1a_n=a_{n+1}-a_n$ for a non-negative integer $n$. Define $\Delta^Ma_n= \Delta^{M-1}a_{n+1}- \Delta^{M-1}a_n$. A sequence is [i]patriota[/i] if there are positive integers $k,l$ such that $a_{n+k}=\Delta^Ma_{n+l}$ for all non-negative integers $n$. Determine, with proof, whether exists a sequence that the last value of $M$ for which the sequence is [i]patriota[/i] is $2022$.

Assume $c$ is a real number. If there exists $x\in[1,2]$ such that $\max\left\{\left |x+\frac cx\right |, \left |x+\frac cx + 2\right |\right\}\geq 5$, please find the value range of $c$.

Find all functions $f : R -\{0\} \to R$ that satisfy $\frac{1}{x}f(-x)+ f\left(\frac{1}{x}\right)= x$ for all $x \ne 0$.

Let $a, b, c$ be real numbers with the property as $ab + bc + ca = 1$. Show that: $$\frac {(a + b) ^ 2 + 1} {c ^ 2 + 2} + \frac {(b + c) ^ 2 + 1} {a ^ 2 + 2} + \frac {(c + a) ^ 2 + 1} {b ^ 2 + 2} \ge 3 $$.

Let $P(x) = x^4-x^3+x$. a) Prove that for all positive real numbers $a$, the polynomial $P(x) - a$ has a unique positive zero. b) A sequence $(a_n)$ is defined by $a_1 = \dfrac{1}{3}$ and for all $n \geq 1$, $a_{n+1}$ is the positive zero of the polynomial $P(x) - a_n$. Prove that the sequence $(a_n)$ converges, and find the limit of the sequence.

Let $f$ be a quadratic function which satisfies the following condition. Find the value of $\frac{f(8)-f(2)}{f(2)-f(1)}$. For two distinct real numbers $a,b$, if $f(a)=f(b)$, then $f(a^2-6b-1)=f(b^2+8)$.

For two given positive integers $ m,n > 1$, let $ a_{ij} (i = 1,2,\cdots,n, \; j = 1,2,\cdots,m)$ be nonnegative real numbers, not all zero, find the maximum and the minimum values of $ f$, where \[ f = \frac {n\sum_{i = 1}^{n}(\sum_{j = 1}^{m}a_{ij})^2 + m\sum_{j = 1}^{m}(\sum_{i= 1}^{n}a_{ij})^2}{(\sum_{i = 1}^{n}\sum_{j = 1}^{m}a_{ij})^2 + mn\sum_{i = 1}^{n}\sum_{j=1}^{m}a_{ij}^2}. \]

If $p = (1- \cos x)(1+ \sin x)$ and $q = (1+ \cos x)(1- \sin x)$, write the expression $$\cos^2 x - \cos^4 x - \sin2x + 2$$ in terms of $p$ and $q$.

A positive integer $n$ is [i]magical[/i] if $\lfloor \sqrt{\lceil \sqrt{n} \rceil} \rfloor=\lceil \sqrt{\lfloor \sqrt{n} \rfloor} \rceil$. Find the number of magical integers between $1$ and $10,000$ inclusive.

Four distinct points are marked in a line. For each point, the sum of the distances from said point to the other three is calculated; getting in total 4 numbers. Decide whether these 4 numbers can be, in some order: a) $29,29,35,37$ b) $28,29,35,37$ c) $28,34,34,37$

Find all functions $R-->R$ such that: $f(x^2) - f(y^2) + 2x + 1 = f(x + y)f(x - y)$

Let $n\geqslant 2$ be a positive integer and $a_1,a_2, \ldots ,a_n$ be real numbers such that \[a_1+a_2+\dots+a_n=0.\] Define the set $A$ by \[A=\left\{(i, j)\,|\,1 \leqslant i<j \leqslant n,\left|a_{i}-a_{j}\right| \geqslant 1\right\}\] Prove that, if $A$ is not empty, then \[\sum_{(i, j) \in A} a_{i} a_{j}<0.\]

Let $a_1, a_2, \ldots , a_n, \ldots $ be a sequence of real numbers such that $0 \leq a_n \leq 1$ and $a_n - 2a_{n+1} + a_{n+2} \geq 0$ for $n = 1, 2, 3, \ldots$. Prove that \[0 \leq (n + 1)(a_n - a_{n+1}) \leq 2 \qquad \text{ for } n = 1, 2, 3, \ldots\]

Let $p$ and $a$ be positive integer numbers having no common divisors except of $1$. Prove that $p$ is prime if and only if all the coefficients of the polynomial \[ F(x) = (x-a)^p - (x^p - a) \] are divisible by $p$.

[b]p4.[/b] What is gcd $(2^6 - 1, 2^9 - 1)$? [b]p5.[/b] Sarah is walking along a sidewalk at a leisurely speed of $\frac12$ m/s. Annie is some distance behind her, walking in the same direction at a faster speed of $s$ m/s. What is the minimum value of $s$ such that Sarah and Annie spend no more than one second within one meter of each other? [b]p6.[/b] You have a choice to play one of two games. In both games, a coin is flipped four times. In game $1$, if (at least) two flips land heads, you win. In game $2$, if (at least) two consecutive flips land heads, you win. Let $N$ be the number of the game that gives you a better chance of winning, and let $p$ be the absolute difference in the probabilities of winning each game. Find $N + p$. PS. You should use hide for answers.

Let $k$ be a positive number and $P$ a Polynomio with integer coeficients. Prove that exists a $n$ positive integer such that: $P(1)+P(2)+\dots+P(N)$ is divisible by $k$.