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

1999 India National Olympiad, 3
Tags: algebra , polynomial , calculus , integration , quadratics , number theory , algebra unsolved
Show that there do not exist polynomials $p(x)$ and $q(x)$ each having integer coefficients and of degree greater than or equal to 1 such that \[ p(x)q(x) = x^5 +2x +1 . \]

1996 China Team Selection Test, 3
Tags: complex numbers , algebra unsolved , algebra
Does there exist non-zero complex numbers $a, b, c$ and natural number $h$ such that if integers $k, l, m$ satisfy $|k| + |l| + |m| \geq 1996$, then $|ka + lb + mc| > \frac {1}{h}$ is true?

2004 China Team Selection Test, 1
Tags: algebra unsolved , algebra
Given sequence $ \{ c_n \}$ satisfying the conditions that $ c_0\equal{}1$, $ c_1\equal{}0$, $ c_2\equal{}2005$, and $ c_{n\plus{}2}\equal{}\minus{}3c_n \minus{} 4c_{n\minus{}1} \plus{}2008$, ($ n\equal{}1,2,3, \cdots$). Let $ \{ a_n \}$ be another sequence such that $ a_n\equal{}5(c_{n\plus{}1} \minus{} c_n) \cdot (502 \minus{} c_{n\minus{}1} \minus{} c_{n\minus{}2}) \plus{} 4^n \times 2004 \times 501$, ($ n\equal{}2,3, \cdots$). Is $ a_n$ a perfect square for every $ n > 2$?

1995 China Team Selection Test, 3
Tags: quadratics , algebra , polynomial , algebra unsolved
Prove that the interval $\lbrack 0,1 \rbrack$ can be split into black and white intervals for any quadratic polynomial $P(x)$, such that the sum of weights of the black intervals is equal to the sum of weights of the white intervals. (Define the weight of the interval $\lbrack a,b \rbrack$ as $P(b) - P(a)$.) Does the same result hold with a degree 3 or degree 5 polynomial?

2003 China Team Selection Test, 2
Tags: algebra , polynomial , algebra unsolved
Can we find positive reals $a_1, a_2, \dots, a_{2002}$ such that for any positive integer $k$, with $1 \leq k \leq 2002$, every complex root $z$ of the following polynomial $f(x)$ satisfies the condition $|\text{Im } z| \leq |\text{Re } z|$, \[f(x)=a_{k+2001}x^{2001}+a_{k+2000}x^{2000}+ \cdots + a_{k+1}x+a_k,\] where $a_{2002+i}=a_i$, for $i=1,2, \dots, 2001$.

2007 Federal Competition For Advanced Students, Part 2, 2
Tags: algebra , system of equations , algebra unsolved
Find all tuples $ (x_1,x_2,x_3,x_4,x_5,x_6)$ of non-negative integers, such that the following system of equations holds: $ x_1x_2(1\minus{}x_3)\equal{}x_4x_5 \\ x_2x_3(1\minus{}x_4)\equal{}x_5x_6 \\ x_3x_4(1\minus{}x_5)\equal{}x_6x_1 \\ x_4x_5(1\minus{}x_6)\equal{}x_1x_2 \\ x_5x_6(1\minus{}x_1)\equal{}x_2x_3 \\ x_6x_1(1\minus{}x_2)\equal{}x_3x_4$

2012 Baltic Way, 4
Tags: quadratics , algebra , quadratic formula , algebra unsolved
Prove that for infinitely many pairs $(a,b)$ of integers the equation \[x^{2012} = ax + b\] has among its solutions two distinct real numbers whose product is 1.

2000 Brazil National Olympiad, 5
Tags: function , algebra unsolved , algebra
Let $ X$ the set of all sequences $ \{a_1, a_2,\ldots , a_{2000}\}$, such that each of the first 1000 terms is 0, 1 or 2, and each of the remaining terms is 0 or 1. The [i]distance[/i] between two members $ a$ and $ b$ of $ X$ is defined as the number of $ i$ for which $ a_i$ and $ b_i$ are different. Find the number of functions $ f : X \to X$ which preserve the distance.

2008 Tuymaada Olympiad, 5
Tags: algebra unsolved , algebra
A loader has a waggon and a little cart. The waggon can carry up to 1000 kg, and the cart can carry only up to 1 kg. A finite number of sacks with sand lie in a storehouse. It is known that their total weight is more than 1001 kg, while each sack weighs not more than 1 kg. What maximum weight of sand can the loader carry in the waggon and the cart, regardless of particular weights of sacks? [i]Author: M.Ivanov, D.Rostovsky, V.Frank[/i]

1997 Pre-Preparation Course Examination, 3
Tags: function , algebra unsolved , algebra
Suppose that $f : \mathbb R^+ \to \mathbb R^+$ is a decreasing function such that \[f(x+y)+f(f(x)+f(y))=f(f(x+f(y))+f(y+f(x)), \quad \forall x,y \in \mathbb R^+.\] Prove that $f(x) = f^{-1}(x).$

2010 Brazil National Olympiad, 1
Tags: function , Vieta , algebra unsolved , algebra , functional equation
Find all functions $f$ from the reals into the reals such that \[ f(ab) = f(a+b) \] for all irrational $a, b$.

2000 All-Russian Olympiad, 1
Tags: floor function , algebra unsolved , algebra
Evaluate the sum \[ \left\lfloor \frac{2^0}{3} \right\rfloor + \left\lfloor \frac{2^1}{3} \right\rfloor + \left\lfloor \frac{2^2}{3} \right\rfloor + \cdots + \left\lfloor \frac{2^{1000}}{3} \right\rfloor. \]

1995 IberoAmerican, 3
Tags: function , floor function , induction , calculus , integration , algebra unsolved , algebra
A function $f: \N\rightarrow\N$ is circular if for every $p\in\N$ there exists $n\in\N,\ n\leq{p}$ such that $f^n(p)=p$ ($f$ composed with itself $n$ times) The function $f$ has repulsion degree $k>0$ if for every $p\in\N$ $f^i(p)\neq{p}$ for every $i=1,2,\dots,\lfloor{kp}\rfloor$. Determine the maximum repulsion degree can have a circular function. [b]Note:[/b] Here $\lfloor{x}\rfloor$ is the integer part of $x$.

2011 Moldova Team Selection Test, 4
Tags: algebra unsolved , algebra
Let $n$ be an integer satisfying $n\geq2$. Find the greatest integer not exceeding the expression: $E=1+\sqrt{1+\frac{2^2}{3!}}+\sqrt[3]{1+\frac{3^2}{4!}}+\dots+\+\sqrt[n]{1+\frac{n^2}{(n+1)!}}$

2004 Vietnam Team Selection Test, 2
Tags: function , calculus , integration , search , algebra unsolved , algebra
Find all real values of $\alpha$, for which there exists one and only one function $f: \mathbb{R} \mapsto \mathbb{R}$ and satisfying the equation \[ f(x^2 + y + f(y)) = (f(x))^2 + \alpha \cdot y \] for all $x, y \in \mathbb{R}$.

1999 Polish MO Finals, 1
Tags: algebra unsolved , algebra
For which $n$ do the equations have a solution in integers: \begin{eqnarray*}x_1 ^2 + x_2 ^2 + 50 &=& 16x_1 + 12x_2 \\ x_2 ^2 + x_3 ^2 + 50 &=& 16x_2 + 12x_3 \\ \cdots \quad \cdots \quad \cdots & \cdots & \cdots \quad \cdots \\ x_{n-1} ^2 + x_n ^2 + 50 &=& 16x_{n-1} + 12x_n \\ x_n ^2 + x_1 ^2 + 50 &=& 16x_n + 12x_1 \end{eqnarray*}

2018-IMOC, N4
Tags: algebra unsolved , algebra , number theory , pen
Let a sequence $\{a_n\}$, $n \in \mathbb{N}^{*}$ given, satisfying the condition \[0 < a_{n+1} - a_n \leq 2001\] for all $n \in \mathbb{N}^{*}$ Show that there are infinitely many pairs of positive integers $(p, q)$ such that $p < q$ and $a_p$ is divisor of $a_q$.

2004 Regional Competition For Advanced Students, 4
Tags: induction , number theory , prime numbers , algebra unsolved , algebra
The sequence $ < x_n >$ is defined through: $ x_{n \plus{} 1} \equal{} \left(\frac {n}{2004} \plus{} \frac {1}{n}\right)x_n^2 \minus{} \frac {n^3}{2004} \plus{} 1$ for $ n > 0$ Let $ x_1$ be a non-negative integer smaller than $ 204$ so that all members of the sequence are non-negative integers. Show that there exist infinitely many prime numbers in this sequence.

2011 Morocco National Olympiad, 2
Tags: quadratics , algebra unsolved , algebra
Prove that the equation $x^{2}+p|x| = qx - 1 $ has 4 distinct real solutions if and only if $p+|q|+2<0$ ($p$ and $q$ are two real parameters).

1989 IMO Longlists, 78
Tags: algebra , polynomial , algebra unsolved
Let $ P(x)$ be a polynomial with integer coefficients such that \[ P(m_1) \equal{} P(m_2) \equal{} P(m_3) \equal{} P(m_4) \equal{} 7\] for given distinct integers $ m_1,m_2,m_3,$ and $ m_4.$ Show that there is no integer m such that $ P(m) \equal{} 14.$

1986 Vietnam National Olympiad, 3
Tags: algebra , polynomial , logarithms , algebra unsolved
Suppose $ M(y)$ is a polynomial of degree $ n$ such that $ M(y) \equal{} 2^y$ for $ y \equal{} 1, 2, \ldots, n \plus{} 1$. Compute $ M(n \plus{} 2)$.

2007 Korea National Olympiad, 4
Tags: induction , strong induction , algebra unsolved , algebra
Two real sequence $ \{x_{n}\}$ and $ \{y_{n}\}$ satisfies following recurrence formula; $ x_{0}\equal{} 1$, $ y_{0}\equal{} 2007$ $ x_{n\plus{}1}\equal{} x_{n}\minus{}(x_{n}y_{n}\plus{}x_{n\plus{}1}y_{n\plus{}1}\minus{}2)(y_{n}\plus{}y_{n\plus{}1})$, $ y_{n\plus{}1}\equal{} y_{n}\minus{}(x_{n}y_{n}\plus{}x_{n\plus{}1}y_{n\plus{}1}\minus{}2)(x_{n}\plus{}x_{n\plus{}1})$ Then show that for all nonnegative integer $ n$, $ {x_{n}}^{2}\leq 2007$.

1992 Vietnam Team Selection Test, 2
Tags: algebra , polynomial , algebra unsolved
Let a polynomial $f(x)$ be given with real coefficients and has degree greater or equal than 1. Show that for every real number $c > 0$, there exists a positive integer $n_0$ satisfying the following condition: if polynomial $P(x)$ of degree greater or equal than $n_0$ with real coefficients and has leading coefficient equal to 1 then the number of integers $x$ for which $|f(P(x))| \leq c$ is not greater than degree of $P(x)$.

1990 Vietnam Team Selection Test, 2
Tags: limit , algebra unsolved , algebra
Let be given four positive real numbers $ a$, $ b$, $ A$, $ B$. Consider a sequence of real numbers $ x_1$, $ x_2$, $ x_3$, $ \ldots$ is given by $ x_1 \equal{} a$, $ x_2 \equal{} b$ and $ x_{n \plus{} 1} \equal{} A\sqrt [3]{x_n^2} \plus{} B\sqrt [3]{x_{n \minus{} 1}^2}$ ($ n \equal{} 2, 3, 4, \ldots$). Prove that there exist limit $ \lim_{n\to \plus{} \propto}x_n$ and find this limit.

2010 Vietnam National Olympiad, 2
Tags: algebra unsolved , algebra
Let $\{a_{n}\}$ be a sequence which satisfy $a_{1}=5$ and $a_{n=}\sqrt[n]{a_{n-1}^{n-1}+2^{n-1}+2.3^{n-1}} \qquad \forall n\geq2$ [b](a)[/b] Find the general fomular for $a_{n}$ [b](b)[/b] Prove that $\{a_{n}\}$ is decreasing sequences