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

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Found problems: 15925

2013 Ukraine Team Selection Test, 9
Tags: function , algebra
Determine all functions $f:\Bbb{R}\to\Bbb{R}$ such that \[ f^2(x+y)=f^2(x)+2f(xy)+f^2(y), \] for all $x,y\in \Bbb{R}.$

2012 IFYM, Sozopol, 6
Tags: algebra
Find all triples $(x,y,z)$ of real numbers satisfying the system of equations $\left\{\begin{matrix} 3(x+\frac{1}{x})=4(y+\frac{1}{y})=5(z+\frac{1}{z}),\\ xy+yz+zx=1.\end{matrix}\right.$

1984 AIME Problems, 7
Tags: domain , function , functional equation , algebra
The function $f$ is defined on the set of integers and satisfies \[ f(n)=\begin{cases} n-3 & \text{if } n\ge 1000 \\ f(f(n+5)) & \text{if } n<1000\end{cases} \] Find $f(84)$.

2021 Iran Team Selection Test, 3
Tags: number theory , polynomial , inequalities , algebra , relatively prime
Prove there exist two relatively prime polynomials $P(x),Q(x)$ having integer coefficients and a real number $u>0$ such that if for positive integers $a,b,c,d$ we have: $$|\frac{a}{c}-1|^{2021} \le \frac{u}{|d||c|^{1010}}$$ $$| (\frac{a}{c})^{2020}-\frac{b}{d}| \le \frac{u}{|d||c|^{1010}}$$ Then we have : $$bP(\frac{a}{c})=dQ(\frac{a}{c})$$ (Two polynomials are relatively prime if they don't have a common root) Proposed by [i]Navid Safaii[/i] and [i]Alireza Haghi[/i]

2024 Euler Olympiad, Round 1, 1
Tags: euler , algebra
Using each of the ten digits exactly once, construct two five-digit numbers such that their difference is minimized. Determine this minimal difference. [i]Proposed by Giorgi Arabidze, Georgia [/i]

2008 Germany Team Selection Test, 1
Tags: functional inequality , algebra
Consider those functions $ f: \mathbb{N} \mapsto \mathbb{N}$ which satisfy the condition \[ f(m \plus{} n) \geq f(m) \plus{} f(f(n)) \minus{} 1 \] for all $ m,n \in \mathbb{N}.$ Find all possible values of $ f(2007).$ [i]Author: Nikolai Nikolov, Bulgaria[/i]

1991 Spain Mathematical Olympiad, 2
Tags: integer , subset , linear algebra , algebra , matrix
Given two distinct elements $a,b \in \{-1,0,1\}$, consider the matrix $A$ . Find a subset $S$ of the set of the rows of $A$, of minimum size, such that every other row of $A$ is a linear combination of the rows in $S$ with integer coefficients.

2023 AMC 10, 9
Tags: special factorizations , difference of squares , algebra
The numbers $16$ and $25$ are a pair of consecutive perfect squares whose difference is $9$. How many pairs of consecutive positive perfect squares have a difference of less than or equal to $2023$? $\textbf{(A) } 674 \qquad \textbf{(B) } 1011 \qquad \textbf{(C) } 1010 \qquad \textbf{(D) } 2019 \qquad \textbf{(E) } 2017$

2014 Swedish Mathematical Competition, 3
Tags: algebra , functional equation , functional
Determine all functions $f: \mathbb R \to \mathbb R$, such that $$ f (f (x + y) - f (x - y)) = xy$$ for all real $x$ and $y$.

2022 JBMO TST - Turkey, 2
Tags: floor function , algebra
For a real number $a$, $[a]$ denotes the largest integer not exceeding $a$. Find all positive real numbers $x$ satisfying the equation $$x\cdot [x]+2022=[x^2]$$

2011 Saudi Arabia Pre-TST, 4.3
Tags: inequalities , algebra
Let $x_1,x_2,...,x_n$ be positive real numbers for which $$\frac{1}{1+x_1}+\frac{1}{1+x_2}+...+\frac{1}{1+x_n}=1$$ Prove that $x_1x_2...x_n \ge (n -1)^n$.

2010 Gheorghe Vranceanu, 2
Tags: imo-like , polynomial , algebra
Find all polynomials $ P $ with integer coefficients that have the property that for any natural number $ n $ the polynomial $ P-n $ has at least a root whose square is integer.

1964 All Russian Mathematical Olympiad, 046
Tags: radical , algebra , number theory
Find integer solutions $(x,y)$ of the equation ($1964$ times "$\sqrt{}$"): $$\sqrt{x+\sqrt{x+\sqrt{....\sqrt{x+\sqrt{x}}}}}=y$$

1990 China Team Selection Test, 4
Tags: function , algebra
Number $a$ is such that $\forall a_1, a_2, a_3, a_4 \in \mathbb{R}$, there are integers $k_1, k_2, k_3, k_4$ such that $\sum_{1 \leq i < j \leq 4} ((a_i - k_i) - (a_j - k_j))^2 \leq a$. Find the minimum of $a$.

2013 NZMOC Camp Selection Problems, 10
Tags: inequalities , algebra
Find the largest possible real number $C$ such that for all pairs $(x, y)$ of real numbers with $x \ne y$ and $xy = 2$, $$\frac{((x + y)^2- 6))(x-y)^2 + 8))}{(x-y)^2} \ge C.$$ Also determine for which pairs $(x, y)$ equality holds.

2020 Purple Comet Problems, 8
Tags: algebra
Camilla drove $20$ miles in the city at a constant speed and $40$ miles in the country at a constant speed that was $20$ miles per hour greater than her speed in the city. Her entire trip took one hour. Find the number of minutes that Camilla drove in the country rounded to the nearest minute.

2019 LIMIT Category A, Problem 5
Tags: trigonometry , summation , algebra
If $\sum_{i=1}^n\cos^{-1}(\alpha_i)=0$, then find $\sum_{i=1}^n\alpha_i$. $\textbf{(A)}~\frac n2$ $\textbf{(B)}~n$ $\textbf{(C)}~n\pi$ $\textbf{(D)}~\frac{n\pi}2$

2001 Tournament Of Towns, 1
Tags: algebra , polynomial
Find at least one polynomial $P(x)$ of degree 2001 such that $P(x)+P(1- x)=1$ holds for all real numbers $x$.

2012 Indonesia TST, 1
Tags: function , algebra
Find all functions $f : \mathbb{R} \rightarrow \mathbb{R}$ such that \[f(x+y) + f(x)f(y) = f(xy) + (y+1)f(x) + (x+1)f(y)\] for all $x,y \in \mathbb{R}$.

2000 Turkey Junior National Olympiad, 3
Tags: polynomial , algebra
$f:\mathbb{R}\rightarrow \mathbb{R}$ satisfies the equation \[f(x)f(y)-af(xy)=x+y\] , for every real numbers $x,y$. Find all possible real values of $a$.

2009 Junior Balkan Team Selection Tests - Romania, 4
Tags: inequalities , algebra
Let $a,b,c > 0$ be real numbers with the sum equal to $3$. Show that: $$\frac{a+3}{3a+bc}+\frac{b+3}{3b+ca}+\frac{c+3}{3c+ab} \ge 3$$

2009 Purple Comet Problems, 25
Tags: algebra , function , rational function , number theory , relatively prime , polynomial
The polynomial $P(x)=a_0+a_1x+a_2x^2+...+a_8x^8+2009x^9$ has the property that $P(\tfrac{1}{k})=\tfrac{1}{k}$ for $k=1,2,3,4,5,6,7,8,9$. There are relatively prime positive integers $m$ and $n$ such that $P(\tfrac{1}{10})=\tfrac{m}{n}$. Find $n-10m$.

2012 Ukraine Team Selection Test, 1
Tags: algebra , inequalities
Let $a, b, c$ be positive reals. Prove that $\sqrt{2a^2+bc}+\sqrt{2b^2+ac}+\sqrt{2c^2+ab}\ge 3 \sqrt{ab+bc+ca}$

2017 BMT Spring, 2
Tags: algebra
Find all solutions to $3^x-9^{x-1} = 2.$

2024 Pan-American Girls’ Mathematical Olympiad, 5
Tags: algebra , real , constant , functional equation
Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that $f(f(x+y) - f(x)) + f(x)f(y) = f(x^2) - f(x+y),$ for all real numbers $x, y$.