Find all functions $f:\mathbb{Z}\to \mathbb{Z}$ satisfying the condition $f(n)-f(n+f(m))=m$ for all $m,n\in \mathbb{Z}$

Twenty children, ten boys and ten girls, are standing in a line. Each boy counted the number of children standing to the right of him. Each girl counted the number of children standing to the left of her. Prove that the sums of numbers counted by the boys and the girls are the same.

Find $P(x)\in Z[x]$ st : $P(n)|2557^{n}+213.2014$ with any $n\in N^{*}$

Find how many integer values $3\le n \le 99$ satisfy that the polynomial $x^2 + x + 1$ divides $x^{2^n} + x + 1$.

Let $p(x)$ be a nonzero polynomial of degree less than $1992$ having no nonconstant factor in common with $x^3 -x$. Let $$ \frac{d^{1992}}{dx^{1992}} \left( \frac{p(x)}{x^3 -x } \right) =\frac{f(x)}{g(x)}$$ for polynomials $f(x)$ and $g(x).$ Find the smallest possible degree of $f(x)$.

Given the Fibonacci sequence with $f_0=f_1=1$and for $n\geq 1, f_{n+1}=f_n+f_{n-1}$, find all real solutions to the equation: $$x^{2024}=f_{2023}x+f_{2022}.$$

For all positive integer $n$, define $f_n(x)$ such that $f_n(x) = \sum_{k=1}^n{|x - k|}$. Determine all solution from the inequality $f_n(x) < 41$ for all positive $2$-digit integers $n$ (in decimal notation).

Done with her new problems, Wendy takes a break from math. Still without any fresh reading material, she feels a bit antsy. She starts to feel annoyed that Michael's loose papers clutter the family van. Several of them are ripped, and bits of paper litter the floor. Tired of trying to get Michael to clean up after himself, Wendy spends a couple of minutes putting Michael's loose papers in the trash. "That seems fair to me," confirms Hannah encouragingly. While collecting Michael's scraps, Wendy comes across a corner of a piece of paper with part of a math problem written on it. There is a monic polynomial of degree $n$, with real coefficients. The first two terms after $x^n$ are $a_{n-1}x^{n-1}$ and $a_{n-2}x^{n-2}$, but the rest of the polynomial is cut off where Michael's page is ripped. Wendy barely makes out a little of Michael's scribbling, showing that $a_{n-1}=-a_{n-2}$. Wendy deciphers the goal of the problem, which is to find the sum of the squares of the roots of the polynomial. Wendy knows neither the value of $n$, nor the value of $a_{n-1}$, but still she finds a [greatest] lower bound for the answer to the problem. Find the absolute value of that lower bound.

Determine all positive integers $k$ for which there exist positive integers $r$ and $s$ that satisfy the equation $$(k^2-6k+11)^{r-1}=(2k-7)^{s}.$$

Define $\text{sgn}(x)$ to be $1$ when $x$ is positive, $-1$ when $x$ is negative, and $0$ when $x$ is $0.$ Compute $$\sum_{n=1}^{\infty} \frac{\text{sgn}(\sin(2^n))}{2^n}.$$ (The arguments to $\sin$ are in radians.)

Let $ d$ be the sum of the lengths of all the diagonals of a plane convex polygon with $ n$ vertices (where $ n>3$). Let $ p$ be its perimeter. Prove that: \[ n\minus{}3<{2d\over p}<\Bigl[{n\over2}\Bigr]\cdot\Bigl[{n\plus{}1\over 2}\Bigr]\minus{}2,\] where $ [x]$ denotes the greatest integer not exceeding $ x$.

Prove that the fraction $0.123456789101112...$ is not periodic.

Let $a$, $b$, $c$ be real numbers, not all of them are equal. Prove that $a+b+c=0$ if and only if $a^2+ab+b^2=b^2+bc+c^2=c^2+ca+a^2$.

let $x,y$ be non-zero reals such that : $x^3+y^3+3x^2y^2=x^3y^3$ find all values of $\frac{1}{x}+\frac{1}{y}$

Let $a$ and $b$ be positive real numbers with $a^2 + b^2 =\frac12$. Prove that $$\frac{1}{1 - a}+\frac{1}{1-b}\ge 4.$$ When does equality hold? [i](Walther Janous)[/i]

Calculate the sum $1+\frac{\binom{2}{1}}{8}+\frac{\binom{4}{2}}{8^2}+\frac{\binom{6}{3}}{8^3}+...+\frac{\binom{2n}{n}}{8^n}+...$

$f(x),g(x)$ - two square trinomials and $a,b,c,d$ - some reals. $f(a)=2,f(b)=3,f(c)=7,f(d)=10$ and $g(a)=16,g(b)=15,g(c)=11$ Find $g(d)$

The negative real numbers $x, y, z, t$ satisfy simultaneously equalities, $$x + y + z = t, \,\,\,\,\frac{1}{x}+ \frac{1}{y}+\frac{1}{z}= \frac{1}{t}, \\,\,\,\, x^3 + y^3 + z^3 = 1000^3$$ Compute $x + y + z + t$.

Let $ \pi_n(x)$ be a polynomial of degree not exceeding $ n$ with real coefficients such that \[ |\pi_n(x)| \leq \sqrt{1\minus{}x^2} \;\textrm{for}\ \;\minus{}1\leq x \leq 1 \ .\] Then \[ |\pi'_n(x)| \leq 2(n\minus{}1).\] [i]P. Turan[/i]

Let $\langle x_n\rangle$ be a sequence of real numbers defined as \[x_1=1; x_n=\dfrac{2n}{(n-1)^2}\sum_{i=1}^{n-1}x_i\] Show that the sequence $y_n=x_{n+1}-x_n$ has finite limits as $n\to \infty.$

An infinite sequence of numbers $a, b, c, d,...$ is obtained by term-by-term addition of two geometric progressions. Can this sequence begin with the following numbers:. a) $1,1,3,5$ ? b) $1,2,3,5$ ? c) $1,2,3, 4$ ?

In a year that has $53$ Saturdays, what day of the week is May $12$? Give all chances.

Let $N$ be a positive integer. Let $R$ denote the smallest positive number that is the sum of $m$ terms $\sum^m_{i=1}{\pm \sqrt{a_i}}$, where each $a_i, i=1,\cdots, m$ is an integer not larger than $N$. Prove that \[R\le C\cdot N^{-m+\frac{3}{2}}\] for some positive real number $C$. [i]Proposed by Navid[/i] [i](Clarification: note that the constant is allowed to depend on $m$ but should be independent of $N$, i.e. the equation $R(m,N)\le C(m)\cdot N^{-m+\frac{3}{2}}$ should hold for all positive integers $N$)[/i]

Natalia and Marcela count $1$ by $1$ starting together at $1$, but Marcela's speed is triple that of Natalia (when Natalia says her second number, Marcela says the fourth number). When the difference of the numbers that they say in unison is any of the multiples of $ 29$, between $500$ and $600$, Natalia continues counting normally and Marcela begins to count downwards in such a way that, at one point, the two say in unison the same number. What is said number?

Given real positive numbers $a, b, c$ and $d$, for which the equalities $a^2 + ab + b^2 = 3c^2$ and $a^3 + a^2b + ab^2 + b^3 = 4d^3$ are fulfilled. Prove that $$a + b + d \le 3c.$$