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: 1311

1995 Baltic Way, 8
Tags: inequalities , algebra proposed , algebra
The real numbers $a,b$ and $c$ satisfy the inequalities $|a|\ge |b+c|,|b|\ge |c+a|$ and $|c|\ge |a+b|$. Prove that $a+b+c=0$.

1985 Vietnam Team Selection Test, 3
Tags: function , algebra , functional equation , algebra proposed
Suppose a function $ f: \mathbb R\to \mathbb R$ satisfies $ f(f(x)) \equal{} \minus{} x$ for all $ x\in \mathbb R$. Prove that $ f$ has infinitely many points of discontinuity.

1993 Iran MO (3rd Round), 6
Tags: algebra proposed , algebra
Let $x_1, x_2, \ldots, x_{12}$ be twelve real numbers such that for each $1 \leq i \leq 12$, we have $|x_i| \geq 1$. Let $I=[a,b]$ be an interval such that $b-a \leq 2$. Prove that number of the numbers of the form $t= \sum_{i=1}^{12} r_ix_i$, where $r_i=\pm 1$, which lie inside the interval $I$, is less than $1000$.

2009 Indonesia TST, 3
Tags: algebra proposed , algebra
Find all triples $ (x,y,z)$ of positive real numbers which satisfy $ 2x^3 \equal{} 2y(x^2 \plus{} 1) \minus{} (z^2 \plus{} 1)$; $ 2y^4 \equal{} 3z(y^2 \plus{} 1) \minus{} 2(x^2 \plus{} 1)$; $ 2z^5 \equal{} 4x(z^2 \plus{} 1) \minus{} 3(y^2 \plus{} 1)$.

2008 ISI B.Math Entrance Exam, 5
Tags: algebra , polynomial , algebra proposed
If a polynomial $P$ with integer coefficients has three distinct integer zeroes , then show that $P(n)\neq 1$ for any integer $n$.

2024 Baltic Way, 2
Tags: algebra , functional equation , algebra proposed
Let $\mathbb{R}^+$ be the set of all positive real numbers. Find all functions $f: \mathbb{R}^+\to\mathbb{R}^+$ such that \[ \frac{f(a)}{1+a+ca}+\frac{f(b)}{1+b+ab}+\frac{f(c)}{1+c+bc} = 1 \] for all $a,b,c \in \mathbb{R}^+$ that satisfy $abc=1$.

2013 Tuymaada Olympiad, 5
Tags: algebra , polynomial , quadratics , linear algebra , matrix , parameterization , algebra proposed
Prove that every polynomial of fourth degree can be represented in the form $P(Q(x))+R(S(x))$, where $P,Q,R,S$ are quadratic trinomials. [i]A. Golovanov[/i] [b]EDIT.[/b] It is confirmed that assuming the coefficients to be [b]real[/b], while solving the problem, earned a maximum score.

2021 Baltic Way, 4
Tags: algebra , algebra proposed
Let $\Gamma$ be a circle in the plane and $S$ be a point on $\Gamma$. Mario and Luigi drive around the circle $\Gamma$ with their go-karts. They both start at $S$ at the same time. They both drive for exactly $6$ minutes at constant speed counterclockwise around the circle. During these $6$ minutes, Luigi makes exactly one lap around $\Gamma$ while Mario, who is three times as fast, makes three laps. While Mario and Luigi drive their go-karts, Princess Daisy positions herself such that she is always exactly in the middle of the chord between them. When she reaches a point she has already visited, she marks it with a banana. How many points in the plane, apart from $S$, are marked with a banana by the end of the $6$ minutes.

2009 Romania Team Selection Test, 1
Tags: algebra proposed , algebra
Given an integer $n\geq 2$, determine the maximum value the sum $x_1+\cdots+x_n$ may achieve, as the $x_i$ run through the positive integers, subject to $x_1\leq x_2\leq \cdots \leq x_n$ and $x_1+\cdots+x_n=x_1 x_2\cdots x_n$.

1994 Turkey Team Selection Test, 2
Tags: limit , logarithms , algebra proposed , algebra
Show that positive integers $n_i,m_i$ $(i=1,2,3, \cdots )$ can be found such that $ \mathop{\lim }\limits_{i \to \infty } \frac{2^{n_i}}{3^{m_i }} = 1$

1992 Baltic Way, 7
Tags: algebra proposed , algebra
Let $ a\equal{}\sqrt[1992]{1992}$. Which number is greater \[ \underbrace{a^{a^{a^{\ldots^{a}}}}}_{1992}\quad\text{or}\quad 1992? \]

2008 APMO, 4
Tags: function , induction , modular arithmetic , algebra proposed , algebra , binary representation
Consider the function $ f: \mathbb{N}_0\to\mathbb{N}_0$, where $ \mathbb{N}_0$ is the set of all non-negative integers, defined by the following conditions : $ (i)$ $ f(0) \equal{} 0$; $ (ii)$ $ f(2n) \equal{} 2f(n)$ and $ (iii)$ $ f(2n \plus{} 1) \equal{} n \plus{} 2f(n)$ for all $ n\geq 0$. $ (a)$ Determine the three sets $ L \equal{} \{ n | f(n) < f(n \plus{} 1) \}$, $ E \equal{} \{n | f(n) \equal{} f(n \plus{} 1) \}$, and $ G \equal{} \{n | f(n) > f(n \plus{} 1) \}$. $ (b)$ For each $ k \geq 0$, find a formula for $ a_k \equal{} \max\{f(n) : 0 \leq n \leq 2^k\}$ in terms of $ k$.

2009 CHKMO, 1
Tags: function , algebra , polynomial , induction , algebra proposed
Let $ f(x) \equal{} c_m x^m \plus{} c_{m\minus{}1} x^{m\minus{}1} \plus{}...\plus{} c_1 x \plus{} c_0$, where each $ c_i$ is a non-zero integer. Define a sequence $ \{ a_n \}$ by $ a_1 \equal{} 0$ and $ a_{n\plus{}1} \equal{} f(a_n)$ for all positive integers $ n$. (a) Let $ i$ and $ j$ be positive integers with $ i<j$. Show that $ a_{j\plus{}1} \minus{} a_j$ is a multiple of $ a_{i\plus{}1} \minus{} a_i$. (b) Show that $ a_{2008} \neq 0$

2009 Vietnam Team Selection Test, 2
Tags: algebra , polynomial , trigonometry , algebra proposed
Let a polynomial $ P(x) \equal{} rx^3 \plus{} qx^2 \plus{} px \plus{} 1$ $ (r > 0)$ such that the equation $ P(x) \equal{} 0$ has only one real root. A sequence $ (a_n)$ is defined by $ a_0 \equal{} 1, a_1 \equal{} \minus{} p, a_2 \equal{} p^2 \minus{} q, a_{n \plus{} 3} \equal{} \minus{} pa_{n \plus{} 2} \minus{} qa_{n \plus{} 1} \minus{} ra_n$. Prove that $ (a_n)$ contains an infinite number of nagetive real numbers.

2003 Silk Road, 3
Tags: algebra proposed , algebra
Let $0<a<b<1$ be reals numbers and \[g(x)=\left\{\begin{array}{cc}x+1-a,&\mbox{ if } 0<x<b\\b-a, & \mbox{ if } x=a \\x-a, & \mbox{ if } a<x<b\\1-a ,&\mbox{ if } x=b \\ x-a ,&\mbox{ if } b<x<1 \end{array}\right.\] Give that there exist $n+1$ reals numbers $0<x_0<x_1<...<x_n<1$, for which $g^{[n]}(x_i)=x_i \ (0 \leq i \leq n)$. Prove that there exists a positive integer $N$, such that $g^{[N]}(x)=x$ for all $0<x<1$. ($g^{[n]}(x)= \underbrace{g(g(....(g(x))....))}_{\text{n times}}$) Official solution [url=http://www.artofproblemsolving.com/Forum/viewtopic.php?f=125&t=365714&p=2011659#p2011659]here[/url]

2007 Bulgaria Team Selection Test, 2
Tags: inequalities , function , algebra proposed , algebra
Find all $a\in\mathbb{R}$ for which there exists a non-constant function $f: (0,1]\rightarrow\mathbb{R}$ such that \[a+f(x+y-xy)+f(x)f(y)\leq f(x)+f(y)\] for all $x,y\in(0,1].$

1976 Canada National Olympiad, 2
Tags: algebra proposed , algebra
Suppose \[ n(n\plus{}1)a_{n\plus{}1}\equal{}n(n\minus{}1)a_n\minus{}(n\minus{}2)a_{n\minus{}1} \] for every positive integer $ n\ge1$. Given that $ a_0\equal{}1,a_1\equal{}2$, find \[ \frac{a_0}{a_1}\plus{}\frac{a_1}{a_2}\plus{}\frac{a_2}{a_3}\plus{}\dots\plus{}\frac{a_{50}}{a_{51}}. \]

2010 Indonesia TST, 2
Tags: function , induction , algebra proposed , algebra , functional equation
Find all functions $ f: \mathbb{R} \rightarrow \mathbb{R}$ satisfying \[ f(x^3\plus{}y^3)\equal{}xf(x^2)\plus{}yf(y^2)\] for all real numbers $ x$ and $ y$. [i]Hery Susanto, Malang[/i]

2007 ISI B.Math Entrance Exam, 8
Tags: function , algebra proposed , algebra
Let $P:\mathbb{R} \to \mathbb{R}$ be a continuous function such that $P(X)=X$ has no real solution. Prove that $P(P(X))=X$ has no real solution.

2002 Balkan MO, 4
Tags: function , inequalities , algebra proposed , algebra
Determine all functions $f: \mathbb N\to \mathbb N$ such that for every positive integer $n$ we have: \[ 2n+2001\leq f(f(n))+f(n)\leq 2n+2002. \]

1997 Vietnam National Olympiad, 2
Tags: inequalities , number theory , relatively prime , algebra proposed , algebra
Let n be an integer which is greater than 1, not divisible by 1997. Let $ a_m\equal{}m\plus{}\frac{mn}{1997}$ for all m=1,2,..,1996 $ b_m\equal{}m\plus{}\frac{1997m}{n}$ for all m=1,2,..,n-1 We arrange the terms of two sequence $ (a_i), (b_j)$ in the ascending order to form a new sequence $ c_1\le c_2\le ...\le c_{1995\plus{}n}$ Prove that $ c_{k\plus{}1}\minus{}c_k<2$ for all k=1,2,...,1994+n

2009 Romania Team Selection Test, 3
Tags: function , complex analysis , trigonometry , inequalities , floor function , complex numbers , algebra proposed
Given an integer $n\geq 2$ and a closed unit disc, evaluate the maximum of the product of the lengths of all $\frac{n(n-1)}{2}$ segments determined by $n$ points in that disc.

2013 Federal Competition For Advanced Students, Part 2, 1
Tags: floor function , ceiling function , inequalities , algebra proposed , algebra
For each pair $(a,b)$ of positive integers, determine all non-negative integers $n$ such that \[b+\left\lfloor{\frac{n}{a}}\right\rfloor=\left\lceil{\frac{n+b}{a}}\right\rceil.\]

2021 Baltic Way, 2
Tags: inequalities , algebra , algebra proposed
Let $a$, $b$, $c$ be the side lengths of a triangle. Prove that $$ \sqrt[3]{(a^2+bc)(b^2+ca)(c^2+ab)} > \frac{a^2+b^2+c^2}{2}. $$

2006 Costa Rica - Final Round, 1
Tags: algebra proposed , algebra
Consider the set $S=\{1,2,...,n\}$. For every $k\in S$, define $S_{k}=\{X \subseteq S, \ k \notin X, X\neq \emptyset\}$. Determine the value of the sum \[S_{k}^{*}=\sum_{\{i_{1},i_{2},...,i_{r}\}\in S_{k}}\frac{1}{i_{1}\cdot i_{2}\cdot...\cdot i_{r}}\] [hide]in fact, this problem was taken from an austrian-polish[/hide]