Olympiad problems

2009 Croatia Team Selection Test, 1

Determine the lowest positive integer n such that following statement is true: If polynomial with integer coefficients gets value 2 for n different integers, then it can't take value 4 for any integer.

1997 China Team Selection Test, 3

Tags: calculus , integration , algebra unsolved , algebra

Prove that there exists $m \in \mathbb{N}$ such that there exists an integral sequence $\lbrace a_n \rbrace$ which satisfies: [b]I.[/b] $a_0 = 1, a_1 = 337$; [b]II.[/b] $(a_{n + 1} a_{n - 1} - a_n^2) + \frac{3}{4}(a_{n + 1} + a_{n - 1} - 2a_n) = m, \forall$ $n \geq 1$; [b]III. [/b]$\frac{1}{6}(a_n + 1)(2a_n + 1)$ is a perfect square $\forall$ $n \geq 1$.

2013 India Regional Mathematical Olympiad, 3

Tags: algebra unsolved , algebra

Consider the expression \[2013^2+2014^2+2015^2+ \cdots+n^2\] Prove that there exists a natural number $n > 2013$ for which one can change a suitable number of plus signs to minus signs in the above expression to make the resulting expression equal $9999$

2018 BMT Spring, 10

Tags: algebra unsolved

Let $a$,$b$,$c$ be the roots of the equation $x^{3} - 2018x +2018 = 0$. Let $q$ be the smallest positive integer for which there exists an integer $p, \, 0 < p \leq q$, such that $$\frac {a^{p+q} + b^{p+q} + c^{p+q}} {p+q} = \left(\frac {a^{p} + b^{p} + c^{p}} {p}\right)\left(\frac {a^{q} + b^{q} + c^{q}} {q}\right).$$ Find $p^{2} + q^{2}$.

1998 German National Olympiad, 4

Tags: algebra , polynomial , quadratics , algebra unsolved

Let $a$ be a positive real number. Then prove that the polynomial \[ p(x)=a^3x^3+a^2x^2+ax+a \] has integer roots if and only if $a=1$ and determine those roots.

1994 India National Olympiad, 6

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 $x \not = 0$.

2000 Croatia National Olympiad, Problem 1

Tags: algebra unsolved , algebra

Find all positive integer solutions $x,y,z$ such that $1/x +2/y - 3/z=1$

2004 Brazil National Olympiad, 6

Tags: algebra unsolved , algebra

Let $a$ and $b$ be real numbers. Define $f_{a,b}\colon R^2\to R^2$ by $f_{a,b}(x;y)=(a-by-x^2;x)$. If $P=(x;y)\in R^2$, define $f^0_{a,b}(P) = P$ and $f^{k+1}_{a,b}(P)=f_{a,b}(f_{a,b}^k(P))$ for all nonnegative integers $k$. The set $per(a;b)$ of the [i]periodic points[/i] of $f_{a,b}$ is the set of points $P\in R^2$ such that $f_{a,b}^n(P) = P$ for some positive integer $n$. Fix $b$. Prove that the set $A_b=\{a\in R \mid per(a;b)\neq \emptyset\}$ admits a minimum. Find this minimum.

2010 Germany Team Selection Test, 2

Tags: algebra unsolved , algebra

We are given $m,n \in \mathbb{Z}^+.$ Show the number of solution $4-$tuples $(a,b,c,d)$ of the system \begin{align*} ab + bc + cd - (ca + ad + db) &= m\\ 2 \left(a^2 + b^2 + c^2 + d^2 \right) - (ab + ac + ad + bc + bd + cd) &= n \end{align*} is divisible by 10.

2003 Greece National Olympiad, 2

Tags: algebra unsolved , algebra

Find all real solutions of the system \[\begin{cases}x^2 + y^2 - z(x + y) = 2, \\ y^2 + z^2 - x(y + z) = 4, \\ z^2 + x^2 - y(z + x) = 8.\end{cases}\]

1988 IMO Longlists, 24

Tags: algebra unsolved , algebra

Find the positive integers $x_1, x_2, \ldots, x_{29}$ at least one of which is greater that 1988 so that \[ x^2_1 + x^2_2 + \ldots x^2_{29} = 29 \cdot x_1 \cdot x_2 \ldots x_{29}. \]

2011 Iran Team Selection Test, 5

Tags: function , algebra unsolved , algebra

Find all surjective functions $f: \mathbb R \to \mathbb R$ such that for every $x,y\in \mathbb R,$ we have \[f(x+f(x)+2f(y))=f(2x)+f(2y).\]

1988 IMO Longlists, 87

Tags: algebra unsolved , algebra

In a row written in increasing order all the irreducible positive rational numbers, such that the product of the numerator and the denominator is less than 1988. Prove that any two adjacent fractions $\frac{a}{b}$ and $\frac{c}{d},$ $\frac{a}{b} < \frac{c}{d},$ satisfy the equation $b \cdot c - a \cdot d = 1.$

1992 IMTS, 3

Tags: algebra unsolved , algebra

For $n$ a positive integer, denote by $P(n)$ the product of all positive integers divisors of $n$. Find the smallest $n$ for which \[ P(P(P(n))) > 10^{12} \]

1992 Cono Sur Olympiad, 3

Tags: algebra unsolved , algebra

Consider the set $S$ of $100$ numbers: $1; \frac{1}{2}; \frac{1}{3}; ... ; \frac{1}{100}$. Any two numbers, $a$ and $b$, are eliminated in $S$, and the number $a+b+ab$ is added. Now, there are $99$ numbers on $S$. After doing this operation $99$ times, there's only $1$ number on $S$. What values can this number take?

2012 Finnish National High School Mathematics Competition, 2

Tags: algebra unsolved , algebra , system of equations

Let $x\ne 1,y\ne 1$ and $x\ne y.$ Show that if \[\frac{yz-x^2}{1-x}=\frac{zx-y^2}{1-y},\] then \[\frac{yz-x^2}{1-x}=\frac{zx-y^2}{1-y}=x+y+z.\]

1989 IMO Longlists, 10

Tags: algebra unsolved , algebra

Find the maximum number $ c$ such that for all $n \in \mathbb{N}$ to have \[ \{n \cdot \sqrt{2}\} \geq \frac{c}{n}\] where $ \{n \cdot \sqrt{2}\} \equal{} n \cdot \sqrt{2} \minus{} [n \cdot \sqrt{2}]$ and $ [x]$ is the integer part of $ x.$ Determine for this number $ c,$ all $ n \in \mathbb{N}$ for which $ \{n \cdot \sqrt{2}\} \equal{} \frac{c}{n}.$

1972 IMO Longlists, 2

Tags: parameterization , algebra , system of equations , algebra unsolved

Find all real values of the parameter $a$ for which the system of equations \[x^4 = yz - x^2 + a,\] \[y^4 = zx - y^2 + a,\] \[z^4 = xy - z^2 + a,\] has at most one real solution.

2013 Silk Road, 3

Tags: function , inequalities , algebra unsolved , algebra

Find all non-decreasing functions $ f\,:\,\mathbb{N}\to\mathbb{N} $, such that $f(f(m)f(n)+m)=f(mf(n))+f(m)$

1995 India National Olympiad, 2

Tags: quadratics , number theory , relatively prime , Diophantine equation , algebra unsolved , algebra

Show that there are infintely many pairs $(a,b)$ of relatively prime integers (not necessarily positive) such that both the equations \begin{eqnarray*} x^2 +ax +b &=& 0 \\ x^2 + 2ax + b &=& 0 \\ \end{eqnarray*} have integer roots.

2002 China Second Round Olympiad, 2

Tags: algebra unsolved , algebra

There are real numbers $a,b$ and $c$ and a positive number $\lambda$ such that $f(x)=x^3+ax^2+bx+c$ has three real roots $x_1, x_2$ and $x_3$ satisfying $(1) x_2-x_1=\lambda$ $(2) x_3>\frac{1}{2}(x_1+x_2)$. Find the maximum value of $\frac{2a^3+27c-9ab}{\lambda^3}$

2001 China Western Mathematical Olympiad, 1

Tags: induction , integration , algebra unsolved , algebra

The sequence $ \{x_n\}$ satisfies $ x_1 \equal{} \frac {1}{2}, x_{n \plus{} 1} \equal{} x_n \plus{} \frac {x_n^2}{n^2}$. Prove that $ x_{2001} < 1001$.

2008 Germany Team Selection Test, 1

Tags: inequalities , algebra unsolved , algebra

Determine $ Q \in \mathbb{R}$ which is so big that a sequence with non-negative reals elements $ a_1 ,a_2, \ldots$ which satisfies the following two conditions: [b](i)[/b] $ \forall m,n \geq 1$ we have $ a_{m \plus{} n} \leq 2 \left(a_m \plus{} a_n \right)$ [b](ii)[/b] $ \forall k \geq 0$ we have $ a_{2^k} \leq \frac {1}{(k \plus{} 1)^{2008}}$ such that for each sequence element we have the inequality $ a_n \leq Q.$

2008 Argentina National Olympiad, 2

Tags: abstract algebra , absolute value , algebra unsolved , algebra

In every cell of a $ 60 \times 60$ board is written a real number, whose absolute value is less or equal than $ 1$. The sum of all numbers on the board equals $ 600$. Prove that there is a $ 12 \times 12$ square in the board such that the absolute value of the sum of all numbers on it is less or equal than $ 24$.

2006 Regional Competition For Advanced Students, 2

Tags: algebra , system of equations , algebra unsolved

Let $ n>1$ be a positive integer an $ a$ a real number. Determine all real solutions $ (x_1,x_2,\dots,x_n)$ to following system of equations: $ x_1\plus{}ax_2\equal{}0$ $ x_2\plus{}a^2x_3\equal{}0$ … $ x_k\plus{}a^kx_{k\plus{}1}\equal{}0$ … $ x_n\plus{}a^nx_1\equal{}0$