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

2010 Postal Coaching, 3
Tags: function , algebra unsolved , algebra
Find all functions $f: \mathbb{Z} \rightarrow \mathbb{Z}$ such that $\boxed{1} \ f(1) = 1$ $\boxed{2} \ f(m+n)(f(m)-f(n)) = f(m-n)(f(m)+f(n)) \ \forall \ m,n \in \mathbb{Z}$

2004 Federal Competition For Advanced Students, Part 1, 4
Tags: calculus , integration , algebra unsolved , algebra
Each of the $2N = 2004$ real numbers $x_1, x_2, \ldots , x_{2004}$ equals either $\sqrt 2 -1 $ or $\sqrt 2 +1$. Can the sum $\sum_{k=1}^N x_{2k-1}x_2k$ take the value $2004$? Which integral values can this sum take?

2002 Moldova Team Selection Test, 1
Tags: algebra , polynomial , algebra unsolved
Prove that for every positive integer n, there exists a polynomial p(x) with integer coefficients such that p(1), p(2),..., p(n-1), p(n) are distinct powers of 2.

2006 China Team Selection Test, 3
Tags: algebra , polynomial , pigeonhole principle , algebra unsolved
$k$ and $n$ are positive integers that are greater than $1$. $N$ is the set of positive integers. $A_1, A_2, \cdots A_k$ are pairwise not-intersecting subsets of $N$ and $A_1 \cup A_2 \cup \cdots \cup A_k = N$. Prove that for some $i \in \{ 1,2,\cdots,k \}$, there exsits infinity many non-factorable n-th degree polynomials so that coefficients of one polynomial are pairwise distinct and all the coeficients are in $A_i$.

2007 Moldova National Olympiad, 11.4
Tags: function , algebra unsolved , algebra
The function $f: \mathbb{R}\rightarrow\mathbb{R}$ satisfies $f(\textrm{cot}x)=\sin2x+\cos2x$, for any $x\in(0,\pi)$. Find the minimum and maximum value of $g: [-1;1]\rightarrow\mathbb{R}$, $g(x)=f(x)\cdot f(1-x)$.

2002 Italy TST, 3
Tags: function , algebra unsolved , algebra
Find all functions $f:\mathbb{R}^+\rightarrow\mathbb{R}^+$ which satisfy the following conditions: $(\text{i})$ $f(x+f(y))=f(x)f(y)$ for all $x,y>0;$ $(\text{ii})$ there are at most finitely many $x$ with $f(x)=1$.

2011 Albania National Olympiad, 4
Tags: induction , algebra unsolved , algebra
The sequence $(a_{n})$ is defined by $a_1=1$ and $a_n=n(a_1+a_2+\cdots+a_{n-1})$ , $\forall n>1$. [b](a)[/b] Prove that for every even $n$, $a_{n}$ is divisible by $n!$. [b](b)[/b] Find all odd numbers $n$ for the which $a_{n}$ is divisible by $n!$.

2002 All-Russian Olympiad Regional Round, 10.1
Tags: modular arithmetic , arithmetic sequence , algebra unsolved , algebra
What is the largest possible length of an arithmetic progression of positive integers $ a_{1}, a_{2},\cdots , a_{n}$ with difference $ 2$, such that $ {a_{k}}^{2}\plus{}1$ is prime for $ k \equal{} 1, 2, . . . , n$?

2004 Austrian-Polish Competition, 3
Tags: linear algebra , matrix , trigonometry , algebra , system of equations , algebra unsolved
Solve the following system of equations in $\mathbb{R}$ where all square roots are non-negative: $ \begin{matrix} a - \sqrt{1-b^2} + \sqrt{1-c^2} = d \\ b - \sqrt{1-c^2} + \sqrt{1-d^2} = a \\ c - \sqrt{1-d^2} + \sqrt{1-a^2} = b \\ d - \sqrt{1-a^2} + \sqrt{1-b^2} = c \\ \end{matrix} $

2003 All-Russian Olympiad, 1
Tags: algebra unsolved , algebra
Suppose that $M$ is a set of $2003$ numbers such that, for any distinct $a, b \in M$, the number $a^2 +b\sqrt 2$ is rational. Prove that $a\sqrt 2$ is rational for all $a \in M.$

2013 Romania Team Selection Test, 3
Tags: function , induction , inequalities , algebra unsolved , algebra
Determine all injective functions defined on the set of positive integers into itself satisfying the following condition: If $S$ is a finite set of positive integers such that $\sum\limits_{s\in S}\frac{1}{s}$ is an integer, then $\sum\limits_{s\in S}\frac{1}{f\left( s\right) }$ is also an integer.

2005 Polish MO Finals, 1
Tags: algebra unsolved , algebra
Given real $c > -2$. Prove that for positive reals $x_1,...,x_n$satisfying:$\sum\limits_{i=1}^n \sqrt{x_i ^2+cx_ix_{i+1}+x_{i+1}^2}=\sqrt{c+2}\left( \sum\limits_{i=1}^n x_i \right)$ holds $c=2$ or $x_1=...=x_n$

1991 Irish Math Olympiad, 2
Tags: algebra , polynomial , algebra unsolved
Problem: Find all polynomials satisfying the equation $ f(x^2) = (f(x))^2 $ for all real numbers x. I'm not exactly sure where to start though it doesn't look too difficult. Thanks!

2011 Mediterranean Mathematics Olympiad, 1
Tags: algebra , polynomial , Vieta , inequalities , algebra unsolved
A Mediterranean polynomial has only real roots and it is of the form \[ P(x) = x^{10}-20x^9+135x^8+a_7x^7+a_6x^6+a_5x^5+a_4x^4+a_3x^3+a_2x^2+a_1x+a_0 \] with real coefficients $a_0\ldots,a_7$. Determine the largest real number that occurs as a root of some Mediterranean polynomial. [i](Proposed by Gerhard Woeginger, Austria)[/i]

2006 Federal Competition For Advanced Students, Part 1, 2
Tags: algebra unsolved , algebra
Show that the sequence $ a_n \equal{} \frac {(n \plus{} 1)^nn^{2 \minus{} n}}{7n^2 \plus{} 1}$ is strictly monotonically increasing, where $ n \equal{} 0,1,2, \dots$.

1978 IMO Longlists, 47
Tags: algebra , polynomial , induction , algebra unsolved
Given the expression \[P_n(x) =\frac{1}{2^n}\left[(x +\sqrt{x^2 - 1})^n+(x-\sqrt{x^2 - 1})^n\right],\] prove: $(a) P_n(x)$ satisfies the identity \[P_n(x) - xP_{n-1}(x) + \frac{1}{4}P_{n-2}(x) \equiv 0.\] $(b) P_n(x)$ is a polynomial in $x$ of degree $n.$

1989 IMO Longlists, 45
Tags: logarithms , number theory , algebra unsolved , algebra
Let $ (\log_2(x))^2 \minus{} 4 \cdot \log_2(x) \minus{} m^2 \minus{} 2m \minus{} 13 \equal{} 0$ be an equation in $ x.$ Prove: [b](a)[/b] For any real value of $ m$ the equation has two distinct solutions. [b](b)[/b] The product of the solutions of the equation does not depend on $ m.$ [b](c)[/b] One of the solutions of the equation is less than 1, while the other solution is greater than 1. Find the minimum value of the larger solution and the maximum value of the smaller solution.

1988 China Team Selection Test, 2
Tags: function , algebra , functional equation , algebra unsolved
Find all functions $f: \mathbb{Q} \mapsto \mathbb{C}$ satisfying (i) For any $x_1, x_2, \ldots, x_{1988} \in \mathbb{Q}$, $f(x_{1} + x_{2} + \ldots + x_{1988}) = f(x_1)f(x_2) \ldots f(x_{1988})$. (ii) $\overline{f(1988)}f(x) = f(1988)\overline{f(x)}$ for all $x \in \mathbb{Q}$.

1994 Hong Kong TST, 2
Tags: function , modular arithmetic , algebra unsolved , algebra
Given that, a function $f(n)$, defined on the natural numbers, satisfies the following conditions: (i)$f(n)=n-12$ if $n>2000$; (ii)$f(n)=f(f(n+16))$ if $n \leq 2000$. (a) Find $f(n)$. (b) Find all solutions to $f(n)=n$.

2002 Bundeswettbewerb Mathematik, 2
Tags: algebra unsolved , algebra
We consider the sequences strictely increasing $(a_0,a_1,...)$ of naturals which have the following property : For every natural $n$, there is exactly one representation of $n$ as $a_i+2a_j+4a_k$, where $i,j,k$ can be equal. Prove that there is exactly a such sequence and find $a_{2002}$

2002 Italy TST, 3
Tags: function , algebra unsolved , algebra
Find all functions $f:\mathbb{R}^+\rightarrow\mathbb{R}^+$ which satisfy the following conditions: $(\text{i})$ $f(x+f(y))=f(x)f(y)$ for all $x,y>0;$ $(\text{ii})$ there are at most finitely many $x$ with $f(x)=1$.

2014 India IMO Training Camp, 3
Tags: geometry , invariant , algebra unsolved , algebra
Starting with the triple $(1007\sqrt{2},2014\sqrt{2},1007\sqrt{14})$, define a sequence of triples $(x_{n},y_{n},z_{n})$ by $x_{n+1}=\sqrt{x_{n}(y_{n}+z_{n}-x_{n})}$ $y_{n+1}=\sqrt{y_{n}(z_{n}+x_{n}-y_{n})}$ $ z_{n+1}=\sqrt{z_{n}(x_{n}+y_{n}-z_{n})}$ for $n\geq 0$.Show that each of the sequences $\langle x_n\rangle _{n\geq 0},\langle y_n\rangle_{n\geq 0},\langle z_n\rangle_{n\geq 0}$ converges to a limit and find these limits.

1993 Polish MO Finals, 2
Tags: function , continued fraction , algebra unsolved , algebra
Find all real-valued functions $f$ on the reals such that $f(-x) = -f(x)$, $f(x+1) = f(x) + 1$ for all $x$, and $f\left(\dfrac{1}{x}\right) = \dfrac{f(x)}{x^2}$ for $x \not = 0$.

1998 Vietnam National Olympiad, 3
Tags: modular arithmetic , algebra unsolved , algebra
The sequence $\{a_{n}\}_{n\geq 0}$ is defined by $a_{0}=20,a_{1}=100,a_{n+2}=4a_{n+1}+5a_{n}+20(n=0,1,2,...)$. Find the smallest positive integer $h$ satisfying $1998|a_{n+h}-a_{n}\forall n=0,1,2,...$

1999 Moldova Team Selection Test, 13
Tags: function , algebra , floor function , algebra unsolved
Let $N$ be a natural number. Find (with prove) the number of solutions in the segment $[1,N]$ of the equation $x^2-[x^2]=(x-[x])^2$, where $[x]$ means the floor function of $x$.