Olympiad problems

1984 Vietnam National Olympiad, 1

Tags: calculus , integration , quadratics , number theory , algebra unsolved , algebra

$(a)$ Let $x, y$ be integers, not both zero. Find the minimum possible value of $|5x^2 + 11xy - 5y^2|$. $(b)$ Find all positive real numbers $t$ such that $\frac{9t}{10}=\frac{[t]}{t - [t]}$.

1987 IMO Longlists, 39

Tags: algebra , polynomial , algebra unsolved , Polynomials

Let $A$ be a set of polynomials with real coefficients and let them satisfy the following conditions: [b](i)[/b] if $f \in A$ and $\deg( f ) \leq 1$, then $f(x) = x - 1$; [b](ii)[/b] if $f \in A$ and $\deg( f ) \geq 2$, then either there exists $g \in A$ such that $f(x) = x^{2+\deg(g)} + xg(x) -1$ or there exist $g, h \in A$ such that $f(x) = x^{1+\deg(g)}g(x) + h(x)$; [b](iii)[/b] for every $g, h \in A$, both $x^{2+\deg(g)} + xg(x) -1$ and $x^{1+\deg(g)}g(x) + h(x)$ belong to $A.$ Let $R_n(f)$ be the remainder of the Euclidean division of the polynomial $f(x)$ by $x^n$. Prove that for all $f \in A$ and for all natural numbers $n \geq 1$ we have $R_n(f)(1) \leq 0$, and that if $R_n(f)(1) = 0$ then $R_n(f) \in A$.

1989 IMO Longlists, 9

Tags: algebra unsolved , algebra

Do there exist two sequences of real numbers $ \{a_i\}, \{b_i\},$ $ i \in \mathbb{N},$ satisfying the following conditions: \[ \frac{3 \cdot \pi}{2} \leq a_i \leq b_i\] and \[ \cos(a_i x) \minus{} \cos(b_i x) \geq \minus{} \frac{1}{i}\] $ \forall i \in \mathbb{N}$ and all $ x,$ with $ 0 < x < 1?$

2014 Cezar Ivănescu, 3

Tags: function , algebra unsolved , algebra

Let $f, g:\mathbb{N}\to\mathbb{N}$ be functions that satisfy the following equation: \[f(f(n))+g(f(n)) = n,\ \forall\ n\in\mathbb{N}\ .\] Prove that $g$ is the zero function on $\mathbb{N}$.

1992 China Team Selection Test, 2

Tags: inequalities , algebra unsolved , algebra

Let $n \geq 2, n \in \mathbb{N},$ find the least positive real number $\lambda$ such that for arbitrary $a_i \in \mathbb{R}$ with $i = 1, 2, \ldots, n$ and $b_i \in \left[0, \frac{1}{2}\right]$ with $i = 1, 2, \ldots, n$, the following holds: \[\sum^n_{i=1} a_i = \sum^n_{i=1} b_i = 1 \Rightarrow \prod^n_{i=1} a_i \leq \lambda \sum^n_{i=1} a_i b_i.\]

1979 IMO Longlists, 38

Tags: algebra , polynomial , algebra unsolved

Prove the following statement: If a polynomial $f(x)$ with real coefficients takes only nonnegative values, then there exists a positive integer $n$ and polynomials $g_1(x), g_2(x),\cdots, g_n(x)$ such that \[f(x) = g_1(x)^2 + g_2(x)^2 +\cdots+ g_n(x)^2\]

2012 France Team Selection Test, 1

Tags: function , algebra unsolved , algebra

Let $k>1$ be an integer. A function $f:\mathbb{N^*}\to\mathbb{N^*}$ is called $k$-[i]tastrophic[/i] when for every integer $n>0$, we have $f_k(n)=n^k$ where $f_k$ is the $k$-th iteration of $f$: \[f_k(n)=\underbrace{f\circ f\circ\cdots \circ f}_{k\text{ times}}(n)\] For which $k$ does there exist a $k$-tastrophic function?

1990 IMO Longlists, 35

Tags: calculus , derivative , geometric series , algebra unsolved , algebra

Prove that if $|x| < 1$, then \[ \frac{x}{(1-x)^2}+\frac{x^2}{(1+x^2)^2} + \frac{x^3}{(1-x^3)^2}+\cdots=\frac{x}{1-x}+\frac{2x^2}{1+x^2}+\frac{3x^3}{1-x^3}+\cdots\]

2006 Austrian-Polish Competition, 2

Tags: algebra , polynomial , algebra unsolved

Find all polynomials $P(x)$ with real coefficients satisfying the equation \[(x+1)^{3}P(x-1)-(x-1)^{3}P(x+1)=4(x^{2}-1) P(x)\] for all real numbers $x$.

2005 Tuymaada Olympiad, 2

Tags: algebra , polynomial , algebra unsolved

Six members of the team of Fatalia for the International Mathematical Olympiad are selected from $13$ candidates. At the TST the candidates got $a_1,a_2, \ldots, a_{13}$ points with $a_i \neq a_j$ if $i \neq j$. The team leader has already $6$ candidates and now wants to see them and nobody other in the team. With that end in view he constructs a polynomial $P(x)$ and finds the creative potential of each candidate by the formula $c_i = P(a_i)$. For what minimum $n$ can he always find a polynomial $P(x)$ of degree not exceeding $n$ such that the creative potential of all $6$ candidates is strictly more than that of the $7$ others? [i]Proposed by F. Petrov, K. Sukhov[/i]

2025 Romania EGMO TST, P1

Tags: algebra , polynomial , algebra unsolved

find all real coefficient polynomial $ P(x)$ such that $ P(x)P(x\plus{}1)\equal{}P(x^2\plus{}x\plus{}1)$ for all $ x$

1999 Federal Competition For Advanced Students, Part 2, 2

Tags: algebra , polynomial , algebra unsolved

Given a real number $A$ and an integer $n$ with $2 \leq n \leq 19$, find all polynomials $P(x)$ with real coefficients such that $P(P(P(x))) = Ax^n +19x+99$.

1988 IMO Longlists, 17

Tags: floor function , algebra unsolved , algebra

If $ n$ runs through all the positive integers, then $ f(n) \equal{} \left \lfloor n \plus{} \sqrt {3n} \plus{} \frac {1}{2} \right \rfloor$ runs through all positive integers skipping the terms of the sequence $ a_n \equal{} \left \lfloor \frac {n^2 \plus{} 2n}{3} \right \rfloor$.

2009 Tuymaada Olympiad, 1

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

Three real numbers are given. Fractional part of the product of every two of them is $ 1\over 2$. Prove that these numbers are irrational. [i]Proposed by A. Golovanov[/i]

2001 Czech-Polish-Slovak Match, 5

Tags: function , limit , algebra unsolved , algebra

Find all functions $f : \mathbb{R} \to \mathbb{R}$ that satisfy \[f(x^2 + y) + f(f(x) - y) = 2f(f(x)) + 2y^2\quad\text{ for all }x, y \in \mathbb{R}.\]

2010 Contests, 3

Tags: function , limit , algebra unsolved , algebra

Find all functions $f: \mathbb R \rightarrow \mathbb R$ such that \[f(x+xy+f(y)) = \left(f(x)+\frac{1}{2}\right) \left(f(y)+\frac{1}{2}\right)\] holds for all real numbers $x,y$.

1990 Romania Team Selection Test, 7

Tags: function , induction , algebra unsolved , algebra

The sequence $ (x_n)_{n \geq 1}$ is defined by: $ x_1\equal{}1$ $ x_{n\plus{}1}\equal{}\frac{x_n}{n}\plus{}\frac{n}{x_n}$ Prove that $ (x_n)$ increases and $ [x_n^2]\equal{}n$.

1994 China National Olympiad, 4

Tags: algebra , polynomial , algebra unsolved

Let $f(z)=c_0z^n+c_1z^{n-1}+ c_2z^{n-2}+\cdots +c_{n-1}z+c_n$ be a polynomial with complex coefficients. Prove that there exists a complex number $z_0$ such that $|f(z_0)|\ge |c_0|+|c_n|$, where $|z_0|\le 1$.

2005 Postal Coaching, 15

Tags: algebra unsolved , algebra

Let $X$ be a set with $|X| = n$ , and let $X_1 , X_2 ,... X_n$ be the $n$subsets eith $|X_j| \geq 2$, for $1 \leq j \leq n$. Suppose for each $2$ element subset $Y$ of $X$, there is a unique $j$ in the set $1,2,3....,n$ such that $Y \subset X_j$ . Prove that $X_j \cap X_k \not= \Phi$ for all $1 \leq j < k \leq n$

2007 Vietnam National Olympiad, 2

Tags: function , limit , algebra unsolved , algebra

Given a number $b>0$, find all functions $f: \mathbb{R}\rightarrow\mathbb{R}$ such that: $f(x+y)=f(x).3^{b^{y}+f(y)-1}+b^{x}.\left(3^{b^{y}+f(y)-1}-b^{y}\right) \forall x,y\in\mathbb{R}$

2011 Morocco National Olympiad, 3

Tags: function , algebra unsolved , algebra

Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that $\forall x\in \mathbb{R} \ \ f(x) = max(2xy-f(y))$ where $y\in \mathbb{R}$.

2000 Spain Mathematical Olympiad, 1

Tags: algebra , polynomial , parameterization , quadratics , algebra unsolved

Consider the polynomials \[P(x) = x^4 + ax^3 + bx^2 + cx + 1 \quad \text{and} \quad Q(x) = x^4 + cx^3 + bx^2 + ax + 1.\] Find the conditions on the parameters $a, b, $c with $a\neq c$ for which $P(x)$ and $Q(x)$ have two common roots and, in such cases, solve the equations $P(x) = 0$ and $Q(x) = 0.$

2003 Austrian-Polish Competition, 2

Tags: algebra unsolved , algebra , Sequence

The sequence $a_0, a_1, a_2, ..$ is defined by $a_0 = a, a_{n+1} = a_n + L(a_n)$, where $L(m)$ is the last digit of $m$ (eg $L(14) = 4$). Suppose that the sequence is strictly increasing. Show that infinitely many terms must be divisible by $d = 3$. For what other d is this true?

2010 Contests, 1

Tags: algebra , polynomial , algebra unsolved

Solve the system equations \[\left\{\begin{array}{cc}x^{4}-y^{4}=240\\x^{3}-2y^{3}=3(x^{2}-4y^{2})-4(x-8y)\end{array}\right.\]

2012 Puerto Rico Team Selection Test, 7

Tags: function , induction , strong induction , algebra unsolved , algebra

Let $f$ be a function with the following properties: 1) $f(n)$ is defined for every positive integer $n$; 2) $f(n)$ is an integer; 3) $f(2)=2$; 4) $f(mn)=f(m)f(n)$ for all $m$ and $n$; 5) $f(m)>f(n)$ whenever $m>n$. Prove that $f(n)=n$.