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

2018 India PRMO, 23
Tags: inequalities , algebra
What is the largest positive integer $n$ such that $$\frac{a^2}{\frac{b}{29} + \frac{c}{31}}+\frac{b^2}{\frac{c}{29} + \frac{a}{31}}+\frac{c^2}{\frac{a}{29} + \frac{b}{31}} \ge n(a+b+c)$$holds for all positive real numbers $a,b,c$.

1987 All Soviet Union Mathematical Olympiad, 441
Tags: algebra , combinatorics , sum of squares , tournament
Ten sportsmen have taken part in a table-tennis tournament (each pair has met once only, no draws). Let $xi$ be the number of $i$-th player victories, $yi$ -- losses. Prove that $$x_1^2 + ... + x_{10}^2 = y_1^2 + ... + y_{10}^2$$

2019 HMIC, 3
Tags: algebra
Do there exist four points $P_i = (x_i, y_i) \in \mathbb{R}^2\ (1\leq i \leq 4)$ on the plane such that: [list] [*] for all $i = 1,2,3,4$, the inequality $x_i^4 + y_i^4 \le x_i^3+ y_i^3$ holds, and [*] for all $i \neq j$, the distance between $P_i$ and $P_j$ is greater than $1$? [/list] [i]Pakawut Jiradilok[/i]

1974 IMO Longlists, 25
Tags: algebra , trigonometry , limit
Let $f : \mathbb R \to \mathbb R$ be of the form $f(x) = x + \epsilon \sin x,$ where $0 < |\epsilon| \leq 1.$ Define for any $x \in \mathbb R,$ \[x_n=\underbrace{f \ o \ \ldots \ o \ f}_{n \text{ times}} (x).\] Show that for every $x \in \mathbb R$ there exists an integer $k$ such that $\lim_{n\to \infty } x_n = k\pi.$

2022 Turkey Team Selection Test, 2
Tags: functional equation , algebra , functional equation in r
Find all functions $f: \mathbb{Q^+} \rightarrow \mathbb{Q}$ satisfying $f(x)+f(y)= \left(f(x+y)+\frac{1}{x+y} \right) (1-xy+f(xy))$ for all $x, y \in \mathbb{Q^+}$.

V Soros Olympiad 1998 - 99 (Russia), 11.1
Tags: inequalities , algebra , trigonometry
Find all $x$ for which the inequality holds $$9 \sin x +40 \cos x \ge 41.$$

2016 Thailand Mathematical Olympiad, 9
Tags: functional equation , functional , algebra
A real number $a \ne 0$ is given. Determine all functions $f : R \to R$ satisfying $f(x)f(y) + f(x + y) = axy$ for all real numbers $x, y$.

2021 Iran Team Selection Test, 4
Tags: algebra , number theory , polynomial , prime number , combi , function
Assume $\Omega(n),\omega(n)$ be the biggest and smallest prime factors of $n$ respectively . Alireza and Amin decided to play a game. First Alireza chooses $1400$ polynomials with integer coefficients. Now Amin chooses $700$ of them, the set of polynomials of Alireza and Amin are $B,A$ respectively . Amin wins if for all $n$ we have : $$\max_{P \in A}(\Omega(P(n))) \ge \min_{P \in B}(\omega(P(n)))$$ Who has the winning strategy. Proposed by [i]Alireza Haghi[/i]

2015 Switzerland Team Selection Test, 7
Tags: functional equation , algebra , function
Find all finite and non-empty sets $A$ of functions $f: \mathbb{R} \mapsto \mathbb{R}$ such that for all $f_1, f_2 \in A$, there exists $g \in A$ such that for all $x, y \in \mathbb{R}$ $$f_1 \left(f_2 (y)-x\right)+2x=g(x+y)$$

2022 Belarusian National Olympiad, 10.7
Tags: algebra , polynomial
Find all positive integers $a$ for which there exists a polynomial $p(x)$ with integer coefficients such that $p(\sqrt{2}+1)=2-\sqrt{2}$ and $p(\sqrt{2}+2)=a$

1994 All-Russian Olympiad Regional Round, 10.1
Tags: combinatorics , algebra
We have seven equal pails with water, filled to one half, one third, one quarter, one fifth, one eighth, one ninth, and one tenth, respectively. We are allowed to pour water from one pail into another until the first pail empties or the second one fills to the brim. Can we obtain a pail that is filled to (a) one twelfth, (b) one sixth after several such steps?

2001 Balkan MO, 3
Tags: algebra , inequalities
Let $a$, $b$, $c$ be positive real numbers with $abc \leq a+b+c$. Show that \[ a^2 + b^2 + c^2 \geq \sqrt 3 abc. \] [i]Cristinel Mortici, Romania[/i]

2010 Iran Team Selection Test, 12
Tags: function , algebra , number theory , polynomial
Prove that for each natural number $m$, there is a natural number $N$ such that for each $b$ that $2\leq b\leq1389$ sum of digits of $N$ in base $b$ is larger than $m$.

2024 Mozambique National Olympiad, P5
Tags: algebra , number theory , diophantine equation
Find all pairs of positive integers $x,y$ such that $\frac{4}{x}+\frac{2}{y}=1$

2009 Albania Team Selection Test, 2
Tags: function , polynomial , induction , calculus , derivative , limit , algebra
Find all the functions $ f :\mathbb{R}\mapsto\mathbb{R} $ with the following property: $ \forall x$ $f(x)= f(x/2) + (x/2)f'(x)$

1994 Vietnam National Olympiad, 1
Tags: algebra , system of equations , logarithm
Find all real solutions to \[x^{3}+3x-3+\ln{(x^{2}-x+1)}=y,\] \[y^{3}+3y-3+\ln{(y^{2}-y+1)}=z,\] \[z^{3}+3z-3+\ln{(z^{2}-z+1)}=x.\]

2007 IMO Shortlist, 1
Tags: sequence , algebra
Real numbers $ a_{1}$, $ a_{2}$, $ \ldots$, $ a_{n}$ are given. For each $ i$, $ (1 \leq i \leq n )$, define \[ d_{i} \equal{} \max \{ a_{j}\mid 1 \leq j \leq i \} \minus{} \min \{ a_{j}\mid i \leq j \leq n \} \] and let $ d \equal{} \max \{d_{i}\mid 1 \leq i \leq n \}$. (a) Prove that, for any real numbers $ x_{1}\leq x_{2}\leq \cdots \leq x_{n}$, \[ \max \{ |x_{i} \minus{} a_{i}| \mid 1 \leq i \leq n \}\geq \frac {d}{2}. \quad \quad (*) \] (b) Show that there are real numbers $ x_{1}\leq x_{2}\leq \cdots \leq x_{n}$ such that the equality holds in (*). [i]Author: Michael Albert, New Zealand[/i]

2005 VJIMC, Problem 4
Tags: algebra , limit , sequence , recurrence relation
Let $(x_n)_{n\ge2}$ be a sequence of real numbers such that $x_2>0$ and $x_{n+1}=-1+\sqrt[n]{1+nx_n}$ for $n\ge2$. Find (a) $\lim_{n\to\infty}x_n$, (b) $\lim_{n\to\infty}nx_n$.

1911 Eotvos Mathematical Competition, 1
Tags: inequalities , algebra
Show that, if the real numbers $a, b, c, A, B, C$ satisfy $$aC -2bB + cA = 0 \ \ and \ \ ac - b^2 > 0,$$ then $$AC - B^2 \le 0.$$

2001 Junior Balkan Team Selection Tests - Moldova, 6
Tags: algebra , inequalities
Let the nonnegative numbers $a_1, a_2,... a_9$, where $a_1 = a_9 = 0$ and let at least one of the numbers is nonzero. Denote the sentence $(P)$: '' For $2 \le i \le 8$ there is a number $a_i$, such that $a_{i - 1} + a_{i + 1} <ka_i $”. a) Show that the sentence $(P)$ is true for $k = 2$. b) Determine whether is the sentence $(P)$ true for $k = \frac{19}{10}$

2018 Singapore Senior Math Olympiad, 3
Tags: algebra , polynomial
Determine the largest positive integer $n$ such that the following statement is true: There exists $n$ real polynomials, $P_1(x),\ldots,P_n(x)$ such that the sum of any two of them have no real roots but the sum of any three does.

2009 QEDMO 6th, 4
Tags: algebra , sum , combination
Let $a$ and $b$ be two real numbers and let $n$ be a nonnegative integer. Then prove that $$\sum_{k=0}^{n} {n \choose k} (a + k)^k (b - k)^{n-k} = \sum_{k=0}^{n} \frac{n!}{t!} (a + b)^t $$

1986 Traian Lălescu, 2.1
Tags: function , functional equation , algebra , real analysis , primitive , calculus , integration
Let be a nonnegative integer $ n. $ Find all continuous functions $ f:\mathbb{R}_{\ge 0}\longrightarrow\mathbb{R} $ for which the following equation holds: $$ (1+n)\int_0^x f(t) dt =nxf(x) ,\quad\forall x>0. $$

2005 iTest, 35
Tags: algebra
How many values of $x$ satisfy the equation $$(x^2 - 9x + 19)^{x^2 + 16x + 60 }= 1?$$

2022 Kyiv City MO Round 2, Problem 3
Tags: algebra
Nonzero real numbers $x_1, x_2, \ldots, x_n$ satisfy the following condition: $$x_1 - \frac{1}{x_2} = x_2 - \frac{1}{x_3} = \ldots = x_{n-1} - \frac{1}{x_n} = x_n - \frac{1}{x_1}$$ Determine all $n$ for which $x_1, x_2, \ldots, x_n$ have to be equal. [i](Proposed by Oleksii Masalitin, Anton Trygub)[/i]