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.

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Found problems: 15925

2013 Vietnam National Olympiad, 1
Tags: algebra , matrix , linear algebra , trigonometry
Solve with full solution: \[\left\{\begin{matrix}\sqrt{(\sin x)^2+\frac{1}{(\sin x)^2}}+\sqrt{(\cos y)^2+\frac{1}{(\cos y)^2}}=\sqrt\frac{20y}{x+y} \\\sqrt{(\sin y)^2+\frac{1}{(\sin y)^2}}+\sqrt{(\cos x)^2+\frac{1}{(\cos x)^2}}=\sqrt\frac{20x}{x+y}\end{matrix}\right. \]

2013 Iran MO (3rd Round), 3
Tags: complex number , algebra
For every positive integer $n \geq 2$, Prove that there is no $n-$tuple of distinct complex numbers $(x_1,x_2,\dots,x_n)$ such that for each $1 \leq k \leq n$ following equality holds. $\prod_{\underset{i \neq k}{1 \leq i \leq n}}^{ } (x_k - x_i) = \prod_{\underset{i \neq k}{1 \leq i \leq n}}^{ } (x_k + x_i) $ (20 points)

2023 Israel National Olympiad, P6
Tags: set , algebra
Determine if there exists a set $S$ of $5783$ different real numbers with the following property: For every $a,b\in S$ (not necessarily distinct) there are $c\neq d$ in $S$ so that $a\cdot b=c+d$.

STEMS 2021 Math Cat C, Q1
Tags: combinatorics , algebra , function
Let $M>1$ be a natural number. Tom and Jerry play a game. Jerry wins if he can produce a function $f: \mathbb{N} \rightarrow \mathbb{N}$ satisfying [list] [*]$f(M) \ne M$ [/*] [*] $f(k)<2k$ for all $k \in \mathbb{N}$[/*] [*] $f^{f(n)}(n)=n$ for all $n \in \mathbb{N}$. For each $\ell>0$ we define $f^{\ell}(n)=f\left(f^{\ell-1}(n)\right)$ and $f^0(n)=n$[/*] [/list] Tom wins otherwise. Prove that for infinitely many $M$, Tom wins, and for infinitely many $M$, Jerry wins. [i]Proposed by Anant Mudgal[/i]

2013 CHMMC (Fall), 2
Tags: algebra , polynomial
Suppose the roots of $$x^4 - 3x^2 + 6x - 12 = 1$$ are $\alpha$, $\beta$, $\gamma$ , and $\delta$. What is the value of $$\frac{\alpha+ \beta+ \gamma }{\delta^2}+\frac{\alpha+ \delta+ \gamma}{\beta^2}+\frac{\alpha+ \beta+ \delta}{\gamma^2}+\frac{\delta+ \beta+ \gamma }{\alpha^2}?$$

2016 Belarus Team Selection Test, 1
Tags: algebra , inequality , inequalities
Prove for positive $a,b,c$ that $$ (a^2+\frac{b^2}{c^2})(b^2+\frac{c^2}{a^2})(c^2+\frac{a^2}{b^2}) \geq abc (a+\frac{1}{a})(b+\frac{1}{b})(c+\frac{1}{c})$$

2009 IMS, 3
Tags: domain , topology , functional analysis , function , integration , algebra , calculus
Let $ A\subset \mathbb C$ be a closed and countable set. Prove that if the analytic function $ f: \mathbb C\backslash A\longrightarrow \mathbb C$ is bounded, then $ f$ is constant.

2022 Philippine MO, 1
Tags: functional equation , algebra
Find all functions $f:\mathbb{R} \rightarrow \mathbb{R}$ such that \[ f(a-b)f(c-d) + f(a-d)f(b-c) \leq (a-c)f(b-d) \] for all real numbers $a, b, c,$ and $d$.

2000 Slovenia National Olympiad, Problem 2
Tags: polynomial , algebra , inequalities
Consider the polynomial $p(x)=a_nx^n+\ldots+a_1x+a_0$ with real coefficients such that $0\le a_i\le a_0$ for each $i=1,2,\ldots,n$. If $a$ is the coefficient of $x^{n+1}$ in the polynomial $q(x)=p(x)^2$, prove that $2a\le p(1)^2$.

1986 Tournament Of Towns, (122) 4
Tags: sum , reciprocal , product , algebra
Consider subsets of the set $1 , 2,..., N$. For each such subset we can compute the product of the reciprocals of each member. Find the sum of all such products.

1992 Romania Team Selection Test, 7
Tags: algebra
Let $(a_{n})_{n\geq 1}$ and $(b_{n})_{n\geq 1}$ be the sequence of positive integers defined by $a_{n+1}=na_{n}+1$ and $b_{n+1}=nb_{n}-1$ for $n\geq 1$. Show that the two sequence cannot have infinitely many common terms. [i]Laurentiu Panaitopol[/i]

2007 Estonia Team Selection Test, 5
Tags: continuous , algebra , functional , functional equation
Find all continuous functions $f: R \to R$ such that for all reals $x$ and $y$, $f(x+f(y)) = y+f(x+1)$.

2022 Flanders Math Olympiad, 4
Tags: algebra , polynomial
Determine all real polynomials $P$ of degree at most $22$ for which $$kP (k + 1) - (k + 1)P (k) = k^2 + k + 1$$ for all $k \in \{1, 2, 3, . . . , 21, 22\}$.

2021 Romania National Olympiad, 1
Tags: complex number , algebra
Find the complex numbers $x,y,z$,with $\mid x\mid=\mid y\mid=\mid z\mid$,knowing that $x+y+z$ and $x^{3}+y^{3}+z^{3}$ are be real numbers.

2001 District Olympiad, 1
Tags: arithmetic sequence , algebra , induction
Let $(a_n)_{n\ge 1}$ be a sequence of real numbers such that \[a_1\binom{n}{1}+a_2\binom{n}{2}+\ldots+a_n\binom{n}{n}=2^{n-1}a_n,\ (\forall)n\in \mathbb{N}^*\] Prove that $(a_n)_{n\ge 1}$ is an arithmetical progression. [i]Lucian Dragomir[/i]

1941 Moscow Mathematical Olympiad, 077
Tags: integer root , integer polynomial , polynomial , algebra
A polynomial $P(x)$ with integer coefficients takes odd values at $x = 0$ and $x = 1$. Prove that $P(x)$ has no integer roots.

2005 iTest, 14
Tags: algebra
A bottle contains $5$ gallons of a $10\%$ solution of oil. How many gallons of pure oil must be added to make a $30\%$ oil solution? (round to the nearest hundredth)

2005 Gheorghe Vranceanu, 1
Tags: equation , algebra
Solve in the real numbers the equation $ 3^{x+1}=(x-1)(x-3). $

2007 Tournament Of Towns, 2
Tags: polynomial , algebra
The polynomial $x^3 + px^2 + qx + r$ has three roots in the interval $(0,2)$. Prove that $-2 <p + q + r < 0$.

1994 BMO TST – Romania, 3:
Tags: algebra , combinatorics
Let $M_1, M_2, . . ., M_{11}$ be $5-$element sets such that $M_i \cap M_j \neq {\O}$ for all $i, j \in \{1, . . ., 11\}$. Determine the minimum possible value of the greatest number of the given sets that have nonempty intersection.

2007 QEDMO 5th, 3
Tags: algebra
Let $a,$ $b,$ $c,$ $d$ be four positive reals such that $d=a+b+c+2\sqrt{ab+bc+ca}.$ Prove that $a=b+c+d-2\sqrt{bc+cd+db}.$ Darij Grinberg

2007 IMAC Arhimede, 1
Tags: algebra , fibonacci , fibonacci sequence , inequalities , number theory
Let $(f_n) _{n\ge 0}$ be the sequence defined by$ f_0 = 0, f_1 = 1, f_{n + 2 }= f_{n + 1} + f_n$ for $n> 0$ (Fibonacci string) and let $t_n =$ ${n+1}\choose{2}$ for $n \ge 1$ . Prove that: a) $f_1^2+f_2^2+...+f_n^2 = f_n \cdot f_{n+1}$ for $n \ge 1$ b) $\frac{1}{n^2} \cdot \Sigma_{k=1}^{n}\left( \frac{t_k}{f_k}\right)^2 \ge \frac{t_{n+1}^2}{9 f_n \cdot f_{n+1}}$

1993 IMO Shortlist, 4
Tags: matrix , linear algebra , system of equations , algebra
Solve the following system of equations, in which $a$ is a given number satisfying $|a| > 1$: $\begin{matrix} x_{1}^2 = ax_2 + 1 \\ x_{2}^2 = ax_3 + 1 \\ \ldots \\ x_{999}^2 = ax_{1000} + 1 \\ x_{1000}^2 = ax_1 + 1 \\ \end{matrix}$

2010 Contests, 1
Tags: number theory , quadratic , algebra
Let $a,b,c\in\{0,1,2,\cdots,9\}$.The quadratic equation $ax^2+bx+c=0$ has a rational root. Prove that the three-digit number $abc$ is not a prime number.

2004 Alexandru Myller, 1
Tags: cyclic equality , number theory , algebra
Let be a nonnegative integer $ n $ and three real numbers $ a,b,c $ satisfying $$ a^n+c=b^n+a=c^n+b=a+b+c. $$ Show that $ a=b=c. $ [i]Gheorghe Iurea[/i]