Olympiad problems

2000 Finnish National High School Mathematics Competition, 3

Determine the positive integers $n$ such that the inequality \[n! > \sqrt{n^n}\] holds.

2007 Pan African, 1

Tags: algebra , system of equations , algebra unsolved

Solve the following system of equations for real $x,y$ and $z$: \begin{eqnarray*} x &=& \sqrt{2y+3}\\ y &=& \sqrt{2z+3}\\ z &=& \sqrt{2x+3}. \end{eqnarray*}

2000 All-Russian Olympiad, 1

Tags: quadratics , algebra unsolved , algebra

Let $a,b,c$ be distinct numbers such that the equations $x^2+ax+1=0$ and $x^2+bx+c=0$ have a common real root, and the equations $x^2+x+a=0$ and $x^2+cx+b$ also have a common real root. Compute the sum $a+b+c$.

2000 Irish Math Olympiad, 4

Tags: arithmetic sequence , algebra unsolved , algebra

The sequence $ a_1<a_2<...<a_M$ of real numbers is called a weak arithmetic progression of length $ M$ if there exists an arithmetic progression $ x_0,x_1,...,x_M$ such that: $ x_0 \le a_1<x_1 \le a_2<x_2 \le ... \le a_M<x_M.$ $ (a)$ Prove that if $ a_1<a_2<a_3$ then $ (a_1,a_2,a_3)$ is a weak arithmetic progression. $ (b)$ Prove that any subset of $ \{ 0,1,2,...,999 \}$ with at least $ 730$ elements contains a weak arithmetic progression of length $ 10$.

2005 Taiwan TST Round 2, 1

Tags: algebra unsolved , algebra

Prove that \[\displaystyle \sum_{\{i,j,k\}=\{1,2,3\}} \csc ^{13} \frac{2^i \pi}{7}\csc ^{14} \frac{2^j \pi}{7}\csc ^{2005} \frac{2^k\pi}{7}\] is rational. Here, $(i,j,k)$ is summed over all possible permutations of $(1,2,3)$.

1990 IMO Longlists, 86

Tags: function , trigonometry , algebra unsolved , algebra

Given function $f(x) = \sin x + \sin \pi x$ and positive number $d$. Prove that there exists real number $p$ such that $|f(x + p) - f(x)| < d$ holds for all real numbers $x$, and the value of $p$ can be arbitrarily large.

2008 Moldova National Olympiad, 9.2

Tags: algebra unsolved , algebra

Find $ f(x): (0,\plus{}\infty) \to \mathbb R$ such that \[ f(x)\cdot f(y) \plus{} f(\frac{2008}{x})\cdot f(\frac{2008}{y})\equal{}2f(x\cdot y)\] and $ f(2008)\equal{}1$ for $ \forall x \in (0,\plus{}\infty)$.

2010 Germany Team Selection Test, 3

Tags: function , trigonometry , algebra , functional equation , algebra unsolved

Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that \[f(x)f(y) = (x+y+1)^2 \cdot f \left( \frac{xy-1}{x+y+1} \right)\] $\forall x,y \in \mathbb{R}$ with $x+y+1 \neq 0$ and $f(x) > 1$ $\forall x > 0.$

2011 Irish Math Olympiad, 3

Tags: induction , modular arithmetic , number theory , greatest common divisor , algebra unsolved , algebra

The integers $a_0, a_1, a_2, a_3,\ldots$ are defined as follows: $a_0 = 1$, $a_1 = 3$, and $a_{n+1} = a_n + a_{n-1}$ for all $n \ge 1$. Find all integers $n \ge 1$ for which $na_{n+1} + a_n$ and $na_n + a_{n-1}$ share a common factor greater than $1$.

1979 IMO Longlists, 42

Tags: quadratics , algebra , polynomial , algebra unsolved

Let a quadratic polynomial $g(x) = ax^2 + bx + c$ be given and an integer $n \ge 1$. Prove that there exists at most one polynomial $f(x)$ of $n$th degree such that $f(g(x)) = g(f(x)).$

2006 Baltic Way, 3

Tags: algebra , polynomial , algebra unsolved

Prove that for every polynomial $P(x)$ with real coefficients there exist a positive integer $m$ and polynomials $P_{1}(x),\ldots , P_{m}(x)$ with real coefficients such that \[P(x) = (P_{1}(x))^{3}+\ldots +(P_{m}(x))^{3}\]

2014 Indonesia MO Shortlist, A6

Tags: algebra , polynomial , calculus , integration , algebra unsolved

Determine all polynomials with integral coefficients $P(x)$ such that if $a,b,c$ are the sides of a right-angled triangle, then $P(a), P(b), P(c)$ are also the sides of a right-angled triangle. (Sides of a triangle are necessarily positive. Note that it's not necessary for the order of sides to be preserved; if $c$ is the hypotenuse of the first triangle, it's not necessary that $P(c)$ is the hypotenuse of the second triangle, and similar with the others.)

2013 Moldova Team Selection Test, 4

Tags: limit , logarithms , algebra unsolved , algebra

Consider a positive real number $a$ and a positive integer $m$. The sequence $(x_k)_{k\in \mathbb{Z}^{+}}$ is defined as: $x_1=1$, $x_2=a$, $x_{n+2}=\sqrt[m+1]{x_{n+1}^mx_n}$. $a)$ Prove that the sequence is converging. $b)$ Find $\lim_{n\rightarrow \infty}{x_n}$.

2002 Austrian-Polish Competition, 10

Tags: induction , algebra unsolved , algebra

For all real number $x$ consider the family $F(x)$ of all sequences $(a_{n})_{n\geq 0}$ satisfying the equation \[a_{n+1}=x-\frac{1}{a_{n}}\quad (n\geq 0).\] A positive integer $p$ is called a [i]minimal period[/i] of the family $F(x)$ if (a) each sequence $\left(a_{n}\right)\in F(x)$ is periodic with the period $p$, (b) for each $0<q<p$ there exists $\left(a_{n}\right)\in F(x)$ such that $q$ is not a period of $\left(a_{n}\right)$. Prove or disprove that for each positive integer $P$ there exists a real number $x=x(P)$ such that the family $F(x)$ has the minimal period $p>P$.

1998 Vietnam Team Selection Test, 3

Tags: algebra unsolved , algebra

Let $p(1), p(2), \ldots, p(k)$ be all primes smaller than $m$, prove that \[\sum^{k}_{i=1} \frac{1}{p(i)} + \frac{1}{p(i)^2} > ln(ln(m)).\]

2003 Regional Competition For Advanced Students, 4

Tags: inequalities , algebra unsolved , algebra

For every real number $ b$ determine all real numbers $ x$ satisfying $ x\minus{}b\equal{} \sum_{k\equal{}0}^{\infty}x^k$.

1989 IMO Longlists, 56

Tags: algebra , polynomial , algebra unsolved

Let $ P_1(x), P_2(x), \ldots, P_n(x)$ be real polynomials, i.e. they have real coefficients. Show that there exist real polynomials $ A_r(x),B_r(x) \quad (r \equal{} 1, 2, 3)$ such that \[ \sum^n_{s\equal{}1} \left\{ P_s(x) \right \}^2 \equiv \left( A_1(x) \right)^2 \plus{} \left( B_1(x) \right)^2\] \[ \sum^n_{s\equal{}1} \left\{ P_s(x) \right \}^2 \equiv \left( A_2(x) \right)^2 \plus{} x \left( B_2(x) \right)^2\] \[ \sum^n_{s\equal{}1} \left\{ P_s(x) \right \}^2 \equiv \left( A_3(x) \right)^2 \minus{} x \left( B_3(x) \right)^2\]

2010 China National Olympiad, 1

Tags: inequalities , floor function , algebra unsolved , algebra

Let $m,n\ge 1$ and $a_1 < a_2 < \ldots < a_n$ be integers. Prove that there exists a subset $T$ of $\mathbb{N}$ such that \[|T| \leq 1+ \frac{a_n-a_1}{2n+1}\] and for every $i \in \{1,2,\ldots , m\}$, there exists $t \in T$ and $s \in [-n,n]$, such that $a_i=t+s$.

2001 Bulgaria National Olympiad, 2

Tags: algebra unsolved , algebra

Find all real values $t$ for which there exist real numbers $x$, $y$, $z$ satisfying : $3x^2 + 3xz + z^2 = 1$ , $3y^2 + 3yz + z^2 = 4$, $x^2 - xy + y^2 = t$.

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 China Western Mathematical Olympiad, 1

Tags: calculus , integration , induction , function , quadratics , algebra unsolved , algebra

The sequence $ \{a_n\}$ satisfies $ a_0 \equal{} 0, a_{n \plus{} 1} \equal{} ka_n \plus{} \sqrt {(k^2 \minus{} 1)a_n^2 \plus{} 1}, n \equal{} 0, 1, 2, \ldots$, where $ k$ is a fixed positive integer. Prove that all the terms of the sequence are integral and that $ 2k$ divides $ a_{2n}, n \equal{} 0, 1, 2, \ldots$.

2012 China Team Selection Test, 3

Tags: algebra , polynomial , inequalities , geometry , geometric transformation , algebra unsolved

Find the smallest possible value of a real number $c$ such that for any $2012$-degree monic polynomial \[P(x)=x^{2012}+a_{2011}x^{2011}+\ldots+a_1x+a_0\] with real coefficients, we can obtain a new polynomial $Q(x)$ by multiplying some of its coefficients by $-1$ such that every root $z$ of $Q(x)$ satisfies the inequality \[ \left\lvert \operatorname{Im} z \right\rvert \le c \left\lvert \operatorname{Re} z \right\rvert. \]

2006 Bulgaria Team Selection Test, 2

Tags: algebra , polynomial , algebra unsolved

Find all couples of polynomials $(P,Q)$ with real coefficients, such that for infinitely many $x\in\mathbb R$ the condition \[ \frac{P(x)}{Q(x)}-\frac{P(x+1)}{Q(x+1)}=\frac{1}{x(x+2)}\] Holds. [i] Nikolai Nikolov, Oleg Mushkarov[/i]

2004 Postal Coaching, 12

Tags: complex numbers , algebra unsolved , algebra

Suppose $z_1, z_2 , \cdots z_n$ are $n$ complex numbers such that $min_{j \not= k} | z_{j} - z_{k} | \geq max_{1 \leq j \leq n} |z_j|$. Find the maximum possible value of $n$. Further characterise all such maximal configurations.

1980 USAMO, 1

Tags: AMC , USA(J)MO , USAMO , algebra unsolved , algebra

A two-pan balance is innacurate since its balance arms are of different lengths and its pans are of different weights. Three objects of different weights $A$, $B$, and $C$ are each weighed separately. When placed on the left-hand pan, they are balanced by weights $A_1$, $B_1$, and $C_1$, respectively. When $A$ and $B$ are placed on the right-hand pan, they are balanced by $A_2$ and $B_2$, respectively. Determine the true weight of $C$ in terms of $A_1, B_1, C_1, A_2$, and $B_2$.