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: 1269

1984 IMO Longlists, 51
Tags: algebra unsolved , algebra
Two cyclists leave simultaneously a point $P$ in a circular runway with constant velocities $v_1, v_2 (v_1 > v_2)$ and in the same sense. A pedestrian leaves $P$ at the same time, moving with velocity $v_3 = \frac{v_1+v_2}{12}$ . If the pedestrian and the cyclists move in opposite directions, the pedestrian meets the second cyclist $91$ seconds after he meets the first. If the pedestrian moves in the same direction as the cyclists, the first cyclist overtakes him $187$ seconds before the second does. Find the point where the first cyclist overtakes the second cyclist the first time.

1997 Polish MO Finals, 2
Tags: inequalities , algebra unsolved , algebra
Find all real solutions to: \begin{eqnarray*} 3(x^2 + y^2 + z^2) &=& 1 \\ x^2y^2 + y^2z^2 + z^2x^2 &=& xyz(x + y + z)^3. \end{eqnarray*}

2005 South East Mathematical Olympiad, 1
Tags: conics , parabola , function , calculus , derivative , algebra unsolved , algebra
Let $a \in \mathbb{R}$ be a parameter. (1) Prove that the curves of $y = x^2 + (a + 2)x - 2a + 1$ pass through a fixed point; also, the vertices of these parabolas all lie on the curve of a certain parabola. (2) If the function $x^2 + (a + 2)x - 2a + 1 = 0$ has two distinct real roots, find the value range of the larger root.

2003 China Team Selection Test, 3
Tags: induction , modular arithmetic , algebra , binomial theorem , algebra unsolved
Let $ \left(x_{n}\right)$ be a real sequence satisfying $ x_{0}=0$, $ x_{2}=\sqrt[3]{2}x_{1}$, and $ x_{n+1}=\frac{1}{\sqrt[3]{4}}x_{n}+\sqrt[3]{4}x_{n-1}+\frac{1}{2}x_{n-2}$ for every integer $ n\geq 2$, and such that $ x_{3}$ is a positive integer. Find the minimal number of integers belonging to this sequence.

2011 Uzbekistan National Olympiad, 2
Tags: floor function , limit , inequalities , algebra unsolved , algebra
Prove that $ \forall n\in\mathbb{N}$,$ \exists a,b,c\in$$\bigcup_{k\in\mathbb{N}}(k^{2},k^{2}+k+3\sqrt 3) $ such that $n=\frac{ab}{c}$.

1976 IMO Longlists, 30
Tags: algebra , polynomial , induction , integration , algebra unsolved
Prove that if $P(x) = (x-a)^kQ(x)$, where $k$ is a positive integer, $a$ is a nonzero real number, $Q(x)$ is a nonzero polynomial, then $P(x)$ has at least $k + 1$ nonzero coefficients.

2003 China Girls Math Olympiad, 5
Tags: inequalities , function , induction , algebra unsolved , algebra
Let $ \{a_n\}^{\infty}_1$ be a sequence of real numbers such that $ a_1 \equal{} 2,$ and \[ a_{n\plus{}1} \equal{} a^2_n \minus{} a_n \plus{} 1, \forall n \in \mathbb{N}.\] Prove that \[ 1 \minus{} \frac{1}{2003^{2003}} < \sum^{2003}_{i\equal{}1} \frac{1}{a_i} < 1.\]

2006 Nordic, 2
Tags: quadratics , algebra , algebra unsolved
Real numbers $x,y,z$ are not all equal and satisfy $x+\frac{1}{y} = y + \frac{1}{z} = z + \frac{1}{x}=k$. Find all possible values of $k$.

2005 China Team Selection Test, 2
Tags: logarithms , algebra unsolved , algebra
Determine whether $\sqrt{1001^2+1}+\sqrt{1002^2+1}+ \cdots + \sqrt{2000^2+1}$ be a rational number or not?

2007 Germany Team Selection Test, 2
Tags: absolute value , algebra unsolved , algebra
Determine the sum of absolute values for the complex roots of $ 20 x^8 \plus{} 7i x^7 \minus{}7ix \plus{} 20.$

2012 Kazakhstan National Olympiad, 1
Tags: algebra unsolved , algebra
Do there exist a infinite sequence of positive integers $(a_{n})$ ,such that for any $n\ge 1$ the relation $ a_{n+2}=\sqrt{a_{n+1}}+a_{n} $?

1977 IMO Longlists, 6
Tags: trigonometry , function , inequalities , induction , algebra unsolved , algebra
Let $x_1, x_2, \ldots , x_n \ (n \geq 1)$ be real numbers such that $0 \leq x_j \leq \pi, \ j = 1, 2,\ldots, n.$ Prove that if $\sum_{j=1}^n (\cos x_j +1) $ is an odd integer, then $\sum_{j=1}^n \sin x_j \geq 1.$

2004 Romania National Olympiad, 1
Tags: function , arithmetic sequence , algebra unsolved , algebra
Let $f : \mathbb{R} \to \mathbb{R}$ be a function such that $|f(x)-f(y)| \leq |x-y|$, for all $x,y \in \mathbb{R}$. Prove that if for any real $x$, the sequence $x,f(x),f(f(x)),\ldots$ is an arithmetic progression, then there is $a \in \mathbb{R}$ such that $f(x)=x+a$, for all $x \in \mathbb R$.

2009 Tuymaada Olympiad, 4
Tags: algebra unsolved , algebra
Is there a positive integer $ n$ such that among 200th digits after decimal point in the decimal representations of $ \sqrt{n}$, $ \sqrt{n\plus{}1}$, $ \sqrt{n\plus{}2}$, $ \ldots,$ $ \sqrt{n\plus{}999}$ every digit occurs 100 times? [i]Proposed by A. Golovanov[/i]

1991 IMTS, 3
Tags: modular arithmetic , algebra unsolved , algebra
Prove that a positive integer can be expressed in the form $3x^2+y^2$ iff it can also be expressed in form $u^2+uv+v^2$, where $x,y,u,v$ are all positive integers.

2003 Finnish National High School Mathematics Competition, 2
Tags: algebra unsolved , algebra
Find consecutive integers bounding the expression \[\frac{1}{x_1 + 1}+\frac{1}{x_2 + 1}+\frac{1}{x_3 + 1}+... +\frac{1}{x_{2001} + 1}+\frac{1}{x_{2002} + 1}\] where $x_1 = 1/3$ and $x_{n+1} = x_n^2 + x_n.$

2008 Middle European Mathematical Olympiad, 1
Tags: inequalities , algebra unsolved , algebra
Let $ (a_n)^{\infty}_{n\equal{}1}$ be a sequence of integers with $ a_{n} < a_{n\plus{}1}, \quad \forall n \geq 1.$ For all quadruple $ (i,j,k,l)$ of indices such that $ 1 \leq i < j \leq k < l$ and $ i \plus{} l \equal{} j \plus{} k$ we have the inequality $ a_{i} \plus{} a_{l} > a_{j} \plus{} a_{k}.$ Determine the least possible value of $ a_{2008}.$

1983 IMO Longlists, 30
Tags: algebra unsolved , algebra
Prove the existence of a unique sequence $\{u_n\} \ (n = 0, 1, 2 \ldots )$ of positive integers such that \[u_n^2 = \sum_{r=0}^n \binom{n+r}{r} u_{n-r} \qquad \text{for all } n \geq 0\]

2011 Postal Coaching, 3
Tags: function , algebra , functional equation , algebra unsolved
Suppose $f : \mathbb{R} \longrightarrow \mathbb{R}$ be a function such that \[2f (f (x)) = (x^2 - x)f (x) + 4 - 2x\] for all real $x$. Find $f (2)$ and all possible values of $f (1)$. For each value of $f (1)$, construct a function achieving it and satisfying the given equation.

2014 Saudi Arabia BMO TST, 1
Tags: function , induction , algebra unsolved , algebra
Find all functions $f:\mathbb{N}\rightarrow(0,\infty)$ such that $f(4)=4$ and \[\frac{1}{f(1)f(2)}+\frac{1}{f(2)f(3)}+\cdots+\frac{1}{f(n)f(n+1)}=\frac{f(n)}{f(n+1)},~\forall n\in\mathbb{N},\] where $\mathbb{N}=\{1,2,\dots\}$ is the set of positive integers.

2007 CHKMO, 2
Tags: algebra unsolved , algebra
For a positive integer k, let $f_{1}(k)$ be the square of the sum of the digits of k. (For example $f_{1}(123)=(1+2+3)^{2}=36$.) Let $f_{n+1}(k)=f_{1}(f_{n}(k))$. Determine the value of the $f_{2007}(2^{2006})$. Justify your claim.

1997 Federal Competition For Advanced Students, P2, 4
Tags: algebra unsolved , algebra
Determine all quadruples $ (a,b,c,d)$ of real numbers satisfying the equation: $ 256a^3 b^3 c^3 d^3\equal{}(a^6\plus{}b^2\plus{}c^2\plus{}d^2)(a^2\plus{}b^6\plus{}c^2\plus{}d^2)(a^2\plus{}b^2\plus{}c^6\plus{}d^2)(a^2\plus{}b^2\plus{}c^2\plus{}d^6).$

2005 Federal Competition For Advanced Students, Part 1, 2
Tags: algebra unsolved , algebra
For how many integers $a$ with $|a| \leq 2005$, does the system $x^2=y+a$ $y^2=x+a$ have integer solutions?

2002 India National Olympiad, 5
Tags: arithmetic sequence , algebra unsolved , algebra
Do there exist distinct positive integers $a$, $b$, $c$ such that $a$, $b$, $c$, $-a+b+c$, $a-b+c$, $a+b-c$, $a+b+c$ form an arithmetic progression (in some order).

2005 MOP Homework, 4
Tags: function , algebra unsolved , algebra
Find all functions $f:\mathbb{R} \rightarrow \mathbb{R}$ such that $f(x^3)-f(y^3)=(x^2+xy+y^2)(f(x)-f(y))$.