This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

AND:
OR:
NO:

Found problems: 15925

1982 AMC 12/AHSME, 1
Tags: algebra , polynomial
When the polynomial $x^3-2$ is divided by the polynomial $x^2-2$, the remainder is $\textbf{(A)} \ 2 \qquad \textbf{(B)} \ -2 \qquad \textbf{(C)} \ -2x-2 \qquad \textbf{(D)} \ 2x+2 \qquad \textbf{(E)} \ 2x-2$

2012 Vietnam National Olympiad, 3
Tags: algebra , limit , function
Find all $f:\mathbb{R} \to \mathbb{R}$ such that: (a) For every real number $a$ there exist real number $b$:$f(b)=a$ (b) If $x>y$ then $f(x)>f(y)$ (c) $f(f(x))=f(x)+12x.$

2000 Moldova Team Selection Test, 3
Tags: function , polynomial , calculus , probability , trigonometry , integration , algebra
For each positive integer $ n$, evaluate the sum \[ \sum_{k\equal{}0}^{2n}(\minus{}1)^{k}\frac{\binom{4n}{2k}}{\binom{2n}{k}}\]

2020 IMO Shortlist, A1
Tags: inequality , algebra , inequalities
[i]Version 1[/i]. Let $n$ be a positive integer, and set $N=2^{n}$. Determine the smallest real number $a_{n}$ such that, for all real $x$, \[ \sqrt[N]{\frac{x^{2 N}+1}{2}} \leqslant a_{n}(x-1)^{2}+x . \] [i]Version 2[/i]. For every positive integer $N$, determine the smallest real number $b_{N}$ such that, for all real $x$, \[ \sqrt[N]{\frac{x^{2 N}+1}{2}} \leqslant b_{N}(x-1)^{2}+x . \]

2019 Middle European Mathematical Olympiad, 1
Tags: algebra , functional equation , injective function
Find all functions $f:\mathbb{R} \to \mathbb{R}$ such that for any two real numbers $x,y$ holds $$f(xf(y)+2y)=f(xy)+xf(y)+f(f(y)).$$ [i]Proposed by Patrik Bak, Slovakia[/i]

2021 Romania Team Selection Test, 3
Tags: algebra , sequence
Let $\alpha$ be a real number in the interval $(0,1).$ Prove that there exists a sequence $(\varepsilon_n)_{n\geq 1}$ where each term is either $0$ or $1$ such that the sequence $(s_n)_{n\geq 1}$ \[s_n=\frac{\varepsilon_1}{n(n+1)}+\frac{\varepsilon_2}{(n+1)(n+2)}+...+\frac{\varepsilon_n}{(2n-1)2n}\]verifies the inequality \[0\leq \alpha-2ns_n\leq\frac{2}{n+1}\] for any $n\geq 2.$

2019 South East Mathematical Olympiad, 1
Tags: algebra
Let $[a]$ represent the largest integer less than or equal to $a$, for any real number $a$. Let $\{a\} = a - [a]$. Are there positive integers $m,n$ and $n+1$ real numbers $x_0,x_1,\hdots,x_n$ such that $x_0=428$, $x_n=1928$, $\frac{x_{k+1}}{10} = \left[\frac{x_k}{10}\right] + m + \left\{\frac{x_k}{5}\right\}$ holds? Justify your answer.

2014 Korea National Olympiad, 2
Tags: function , functional equation , algebra
Determine all the functions $f : \mathbb{R}\rightarrow\mathbb{R}$ that satisfies the following. $f(xf(x)+f(x)f(y)+y-1)=f(xf(x)+xy)+y-1$

2014 Poland - Second Round, 3.
Tags: algebra , polynomial
For each positive integer $n$, determine the smallest possible value of the polynomial $$ W_n(x)=x^{2n}+2x^{2n-1}+3x^{2n-2}+\ldots + (2n-1)x^2+2nx. $$

2012 India Regional Mathematical Olympiad, 3
Tags: number theory , fractional part , solve- integer part , integer part , solve- fractional part , algebra
Solve for real $x$ : $2^{2x} \cdot 2^{3\{x\}} = 11 \cdot 2^{5\{x\}} + 5 \cdot 2^{2[x]}$ (For a real number $x, [x]$ denotes the greatest integer less than or equal to x. For instance, $[2.5] = 2$, $[-3.1] = -4$, $[\pi ] = 3$. For a real number $x, \{x\}$ is defined as $x - [x]$.)

2009 BMO TST, 4
Tags: algebra , polynomial
Find all the polynomials $P(x)$ of a degree $\leq n$ with real non-negative coefficients such that $P(x) \cdot P(\frac{1}{x}) \leq [P(1)]^2$ , $ \forall x>0$.

1949-56 Chisinau City MO, 10
Tags: root , rational , algebra
Get rid of irrationality in the denominator of a fraction $$\frac{1}{\sqrt[3]{4}+\sqrt[3]{2}+2}$$.

2019 Saint Petersburg Mathematical Olympiad, 5
Tags: algebra , combinatorics , number theory , sum
Call the [i]improvement [/i] of a positive number its replacement by a power of two. (i.e. one of the numbers $1, 2, 4, 8, ...$), for which it increases, but not more than than $3$ times. Given $2^{100}$ positive numbers with a sum of $2^{100}$. Prove that you can erase some of them, and [i]improve [/i] each of the other numbers so that the sum the resulting numbers were again $2^{100}$.

2008 Germany Team Selection Test, 3
Tags: function , algebra
Determine all functions $ f: \mathbb{R} \mapsto \mathbb{R}$ with $ x,y \in \mathbb{R}$ such that \[ f(x \minus{} f(y)) \equal{} f(x\plus{}y) \plus{} f(y)\]

2019 Azerbaijan IMO TST, 3
Tags: sequence , maximum value , algebra
Let $a_0,a_1,a_2,\dots $ be a sequence of real numbers such that $a_0=0, a_1=1,$ and for every $n\geq 2$ there exists $1 \leq k \leq n$ satisfying \[ a_n=\frac{a_{n-1}+\dots + a_{n-k}}{k}. \]Find the maximum possible value of $a_{2018}-a_{2017}$.

2022 VN Math Olympiad For High School Students, Problem 8
Tags: number theory , algebra
Given [i]Fibonacci[/i] sequence $(F_n),$ and a positive integer $m$, denote $k(m)$ by the smallest positive integer satisfying $F_{n+k(m)}\equiv F_n(\bmod m),$ for all natural numbers $n$. Prove that: $k(m)$ is even for all $m>2.$

2018-IMOC, A4
Tags: functional equation , fe , algebra
Find all functions $f:\mathbb R\to\mathbb R$ such that $$f\left(x^2+f(y)\right)-y=(f(x+y)-y)^2$$holds for all $x,y\in\mathbb R$.

2025 Romania National Olympiad, 4
Tags: sequence , upper bound on sequence , algebra
Let $m \geq 2$ be a fixed positive integer, and $(a_n)_{n\geq 1}$ be a sequence of nonnegative real numbers such that, for all $n\geq 1$, we have that $a_{n+1} \leq a_n - a_{mn}$. a) Prove that the sequence $b_n = \sum_{k=1}^{n} a_k$ is bounded above. b) Prove that the sequence $c_n = \sum_{k=1}^{n} k^2 a_k$ is bounded above.

2003 AIME Problems, 10
Tags: symmetry , quadratic , algebra
Two positive integers differ by $60.$ The sum of their square roots is the square root of an integer that is not a perfect square. What is the maximum possible sum of the two integers?

2019 Switzerland - Final Round, 6
Tags: functional , functional equation , algebra
Show that there exists no function $f : Z \to Z$ such that for all $m, n \in Z$ $$f(m + f(n)) = f(m) - n.$$

1959 AMC 12/AHSME, 20
Tags: algebra , direct proportion , inversely proportional
It is given that $x$ varies directly as $y$ and inversely as the square of $z$, and that $x=10$ when $y=4$ and $z=14$. Then, when $y=16$ and $z=7$, $x$ equals: $ \textbf{(A)}\ 180\qquad\textbf{(B)}\ 160\qquad\textbf{(C)}\ 154\qquad\textbf{(D)}\ 140\qquad\textbf{(E)}\ 120 $

1976 Chisinau City MO, 126
Tags: polynomial , algebra , divisible
Let $P (x)$ be a polynomial with integer coefficients and $P (n) =m$ for some integers $n, m$ ($m \ne 10$). Prove that $P (n + km)$ is divisible by $m$ for any integer $k$.

2019 IFYM, Sozopol, 5
Tags: algebra , functional equation
The non-decreasing functions $f,g: \mathbb{R}\rightarrow \mathbb{R}$ are such that $f(r)\leq g(r)$ for $\forall$ rational numbers $r$. Is it true that $f(x)\leq g(x)$ for $\forall$ real numbers $x$?

2021 Indonesia MO, 4
Tags: algebra , inequalities
Let $x,y$ and $z$ be positive reals such that $x + y + z = 3$. Prove that \[ 2 \sqrt{x + \sqrt{y}} + 2 \sqrt{y + \sqrt{z}} + 2 \sqrt{z + \sqrt{x}} \le \sqrt{8 + x - y} + \sqrt{8 + y - z} + \sqrt{8 + z - x} \]

2006 Hong kong National Olympiad, 2
Tags: logarithm , modular arithmetic , algebra , function
For a positive integer $k$, let $f_1(k)$ be the square of the sum of the digits of $k$. Define $f_{n+1}$ = $f_1 \circ f_n$ . Evaluate $f_{2007}(2^{2006} )$.