Olympiad problems

2016 JBMO Shortlist, 1

Tags: inequality , algebra

Let $a, b, c$ be positive real numbers such that $abc = 8$. Prove that $\frac{ab + 4}{a + 2}+\frac{bc + 4}{b + 2}+\frac{ca + 4}{c + 2}\ge 6$.

2016 Romanian Master of Mathematics, 4

Tags: algebra , inequalities

Let $x$ and $y$ be positive real numbers such that: $x+y^{2016}\geq 1$. Prove that $x^{2016}+y> 1-\frac{1}{100}$

1980 AMC 12/AHSME, 28

Tags: polynomial , algebra

The polynomial $x^{2n}+1+(x+1)^{2n}$ is not divisible by $x^2+x+1$ if $n$ equals $\text{(A)} \ 17 \qquad \text{(B)} \ 20 \qquad \text{(C)} \ 21 \qquad \text{(D)} \ 64 \qquad \text{(E)} \ 65$

1998 Spain Mathematical Olympiad, 1

Tags: algebra , search

Find the tangents of the angles of a triangle knowing that they are positive integers.

2025 Kosovo National Mathematical Olympiad`, P1

Tags: nice , system of equations , algebra

Find all real numbers $a$, $b$ and $c$ that satisfy the following system of equations: $$\begin{cases} ab-c = 3 \\ a+bc = 4 \\ a^2+c^2 = 5\end{cases}$$

1997 Moldova Team Selection Test, 5

Tags: polynomial , algebra

Let $P(x)\in\mathbb{Z}[x]$ with deg $P=2015$. Let $Q(x)=(P(x))^2-9$. Prove that: the number of distinct roots of $Q(x)$ can not bigger than $2015$

2022 Indonesia TST, A

Tags: function , inequalities , fe , algebra

Determine all functions $f : \mathbb{R} \to \mathbb{R}$ satisfying \[ f(a^2) - f(b^2) \leq (f(a)+b)(a-f(b)) \] for all $a,b \in \mathbb{R}$.

2005 Slovenia National Olympiad, Problem 1

Tags: system of equations , algebra

Find all real numbers $x,y$ such that $x^3-y^3=7(x-y)$ and $x^3+y^3=5(x+y)$.

2020 Dutch BxMO TST, 3

Tags: functional , algebra , functional equation

Find all functions $f: R \to R$ that satisfy $$f (x^2y) + 2f (y^2) =(x^2 + f (y)) \cdot f (y)$$ for all $x, y \in R$

1988 Swedish Mathematical Competition, 6

Tags: sequence , recurrence relation , radical , algebra , inequalities

The sequence $(a_n)$ is defined by $a_1 = 1$ and $a_{n+1} = \sqrt{a_n^2 +\frac{1}{a_n}}$ for $n \ge 1$. Prove that there exists $a$ such that $\frac{1}{2} \le \frac{a_n}{n^a} \le 2$ for $n \ge 1$.

1993 APMO, 3

Tags: inequalities , algebra , polynomial , ratio

Let \begin{eqnarray*} f(x) & = & a_n x^n + a_{n-1} x^{n-1} + \cdots + a_0 \ \ \mbox{and} \\ g(x) & = & c_{n+1} x^{n+1} + c_n x^n + \cdots + c_0 \end{eqnarray*} be non-zero polynomials with real coefficients such that $g(x) = (x+r)f(x)$ for some real number $r$. If $a = \max(|a_n|, \ldots, |a_0|)$ and $c = \max(|c_{n+1}|, \ldots, |c_0|)$, prove that $\frac{a}{c} \leq n+1$.

1976 Euclid, 4

Tags: polynomial division , remainder , algebra , polynomial

Source: 1976 Euclid Part B Problem 4 ----- The remainder when $f(x)=x^5-2x^4+ax^3-x^2+bx-2$ is divided by $x+1$ is $-7$. When $f(x)$ is divided by $x-2$ the remainder is $32$. Determine the remainder when $f(x)$ is divided by $x-1$.

1984 All Soviet Union Mathematical Olympiad, 372

Tags: radical , inequalities , algebra

Prove that every positive $a$ and $b$ satisfy inequality $$\frac{(a+b)^2}{2} + \frac{a+b}{4} \ge a\sqrt b + b\sqrt a$$

2018 Belarusian National Olympiad, 11.5

Tags: conic , hyperbola , algebra

The circle $S_1$ intersects the hyperbola $y=\frac1x$ at four points $A$, $B$, $C$, and $D$, and the other circle $S_2$ intersects the same hyperbola at four points $A$, $B$, $F$, and $G$. It's known that the radii of circles $S_1$ and $S_2$ are equal. Prove that the points $C$, $D$, $F$, and $G$ are the vertices of the parallelogram.

1998 Brazil Team Selection Test, Problem 3

Tags: polynomial , function , algebra

Find all functions $f: \mathbb N \to \mathbb N$ for which \[ f(n) + f(n+1) = f(n+2)f(n+3)-1996\] holds for all positive integers $n$.

2010 South East Mathematical Olympiad, 2

Tags: modular arithmetic , polynomial , number theory , algebra

For any set $A=\{a_1,a_2,\cdots,a_m\}$, let $P(A)=a_1a_2\cdots a_m$. Let $n={2010\choose99}$, and let $A_1, A_2,\cdots,A_n$ be all $99$-element subsets of $\{1,2,\cdots,2010\}$. Prove that $2010|\sum^{n}_{i=1}P(A_i)$.

2002 Croatia National Olympiad, Problem 3

Tags: number theory , algebra

Let $f(x)=x^{2002}-x^{2001}+1$. Prove that for every positive integer $m$, the numbers $m,f(m),f(f(m)),\ldots$ are pairwise coprime.

2021 China Team Selection Test, 5

Tags: function , algebra

Determine all $ f:R\rightarrow R $ such that $$ f(xf(y)+y^3)=yf(x)+f(y)^3 $$

2020 Czech and Slovak Olympiad III A, 3

Tags: parameter , parametric equation , algebra , system of equations

Consider the system of equations $\begin{cases} x^2 - 3y + p = z, \\ y^2 - 3z + p = x, \\ z^2 - 3x + p = y \end{cases}$ with real parameter $p$. a) For $p \ge 4$, solve the considered system in the field of real numbers. b) Prove that for $p \in (1, 4)$ every real solution of the system satisfies $x = y = z$. (Jaroslav Svrcek)

1987 IMO Longlists, 50

Tags: polynomial , algebra

Let $P,Q,R$ be polynomials with real coefficients, satisfying $P^4+Q^4 = R^2$. Prove that there exist real numbers $p, q, r$ and a polynomial $S$ such that $P = pS, Q = qS$ and $R = rS^2$. [hide="Variants"]Variants. (1) $P^4 + Q^4 = R^4$; (2) $\gcd(P,Q) = 1$ ; (3) $\pm P^4 + Q^4 = R^2$ or $R^4.$[/hide]

2003 Italy TST, 3

Tags: algebra , function

Determine all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ that satisfy \[f(f(x)+y)=2x+f(f(y)-x)\quad\text{for all real}\ x,y. \]

2018 Bulgaria National Olympiad, 3.

Tags: inequalities , algebra

Prove that \[ \left(\frac{6}{5}\right)^{\sqrt{3}}>\left(\frac{5}{4}\right)^{\sqrt{2}}. \]

2023 Portugal MO, 5

Tags: algebra

In the village of numbers the houses are numbered from $1$ to $n$. Meanwhile, one of the houses was demolished. Duarte calculated that the average number of houses that still exist is $\frac{202}{3}$ . How many houses were there in the village and what is the number of the demolished house?

2019 Romanian Master of Mathematics Shortlist, original P6

Tags: multivariate polynomial , algebra , polynomial

Let $P(x)$ be a nonconstant complex coefficient polynomial and let $Q(x,y)=P(x)-P(y).$ Suppose that polynomial $Q(x,y)$ has exactly $k$ linear factors unproportional two by tow (without counting repetitons). Let $R(x,y)$ be factor of $Q(x,y)$ of degree strictly smaller than $k$. Prove that $R(x,y)$ is a product of linear polynomials. [b]Note: [/b] The [i]degree[/i] of nontrivial polynomial $\sum_{m}\sum_{n}c_{m,n}x^{m}y^{n}$ is the maximum of $m+n$ along all nonzero coefficients $c_{m,n}.$ Two polynomials are [i]proportional[/i] if one of them is the other times a complex constant. [i]Proposed by Navid Safaie[/i]

2013 German National Olympiad, 2

Tags: upper bound on sequence , sequence , algebra

Let $\alpha$ be a real number with $\alpha>1$. Let the sequence $(a_n)$ be defined as $$a_n=1+\sqrt[\alpha]{2+\sqrt[\alpha]{3+\ldots+\sqrt[\alpha]{n+\sqrt[\alpha]{n+1}}}}$$ for all positive integers $n$. Show that there exists a positive real constant $C$ such that $a_n<C$ for all positive integers $n$.