Olympiad problems

2003 Bosnia and Herzegovina Team Selection Test, 3

Prove that for every positive integer $n$ holds: $(n-1)^n+2n^n \leq (n+1)^{n} \leq 2(n-1)^n+2n^{n}$

2001 China Western Mathematical Olympiad, 1

Tags: induction , integration , 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$.

II Soros Olympiad 1995 - 96 (Russia), 9.9

Tags: algebra

There are $5$ ingots weighing $1$, $2$, $3$, $4$ and $5$ kg with an unknown copper content that varies in different ingots. Each ingot must be divided into $5$ parts and $5$ new ingots of the same mass of $1$, $2$, $3$, $4$ and $5$ kg must be made. This requires that the percentage of copper in all pieces be the same, regardless of what it was in the original pieces. What parts should each piece be divided into?

1996 South africa National Olympiad, 2

Tags: function , trigonometry , algebra , inequalities

Find all real numbers for which $3^x+4^x=5^x$.

1999 IMO Shortlist, 6

Tags: algebra , modular arithmetic , arithmetic sequence , divisibility , sum of digits

Prove that for every real number $M$ there exists an infinite arithmetic progression such that: - each term is a positive integer and the common difference is not divisible by 10 - the sum of the digits of each term (in decimal representation) exceeds $M$.

2018 Benelux, 1

Tags: inequalities , algebra

(a) Determine the minimal value of $\displaystyle\left(x+\dfrac{1}{y}\right)\left(x+\dfrac{1}{y}-2018\right)+\left(y+\dfrac{1}{x}\right)\left(y+\dfrac{1}{x}-2018\right), $ where $x$ and $y$ vary over the positive reals. (b) Determine the minimal value of $\displaystyle\left(x+\dfrac{1}{y}\right)\left(x+\dfrac{1}{y}+2018\right)+\left(y+\dfrac{1}{x}\right)\left(y+\dfrac{1}{x}+2018\right), $ where $x$ and $y$ vary over the positive reals.

2013 Online Math Open Problems, 30

Tags: algebra , polynomial , number theory , relatively prime , area of a triangle , geometry

Let $P(t) = t^3+27t^2+199t+432$. Suppose $a$, $b$, $c$, and $x$ are distinct positive reals such that $P(-a)=P(-b)=P(-c)=0$, and \[ \sqrt{\frac{a+b+c}{x}} = \sqrt{\frac{b+c+x}{a}} + \sqrt{\frac{c+a+x}{b}} + \sqrt{\frac{a+b+x}{c}}. \] If $x=\frac{m}{n}$ for relatively prime positive integers $m$ and $n$, compute $m+n$. [i]Proposed by Evan Chen[/i]

2020 European Mathematical Cup, 4

Tags: algebra , functional equation , function

Let $\mathbb{R^+}$ denote the set of all positive real numbers. Find all functions $f: \mathbb{R^+}\rightarrow \mathbb{R^+}$ such that $$xf(x + y) + f(xf(y) + 1) = f(xf(x))$$ for all $x, y \in\mathbb{R^+}.$ [i]Proposed by Amadej Kristjan Kocbek, Jakob Jurij Snoj[/i]

1992 IMO Longlists, 79

Tags: floor function , algebra , sequence , fibonacci , fibonacci sequence , inequality

Let $ \lfloor x \rfloor$ denote the greatest integer less than or equal to $ x.$ Pick any $ x_1$ in $ [0, 1)$ and define the sequence $ x_1, x_2, x_3, \ldots$ by $ x_{n\plus{}1} \equal{} 0$ if $ x_n \equal{} 0$ and $ x_{n\plus{}1} \equal{} \frac{1}{x_n} \minus{} \left \lfloor \frac{1}{x_n} \right \rfloor$ otherwise. Prove that \[ x_1 \plus{} x_2 \plus{} \ldots \plus{} x_n < \frac{F_1}{F_2} \plus{} \frac{F_2}{F_3} \plus{} \ldots \plus{} \frac{F_n}{F_{n\plus{}1}},\] where $ F_1 \equal{} F_2 \equal{} 1$ and $ F_{n\plus{}2} \equal{} F_{n\plus{}1} \plus{} F_n$ for $ n \geq 1.$

2017 Auckland Mathematical Olympiad, 2

Tags: algebra , inequalities

The sum of the three nonnegative real numbers $ x_1, x_2, x_3$ is not greater than $\frac12$. Prove that $(1 - x_1)(1 - x_2)(1 - x_3) \ge \frac12$

2000 Junior Balkan Team Selection Tests - Romania, 3

Tags: ceiling function , cardinal , algebra

Let be a real number $ a. $ For any real number $ p $ and natural number $ k, $ let be the set $$ A_k(p)=\{ px\in\mathbb{Z}\mid k=\lceil x \rceil \} . $$ Find all real numbers $ b $ such that $ \# A_n(a)=\# A_n(b) , $ for any natural number $ n. $ $ \# $ [i]denotes the cardinal.[/i] [i]Eugen Păltânea[/i]

2019 Brazil Undergrad MO, 3

Tags: calculus , system of equations , algebra

Let $a,b,c$ be constants and $a,b,c$ are positive real numbers. Prove that the equations $2x+y+z=\sqrt{c^2+z^2}+\sqrt{c^2+y^2}$ $x+2y+z=\sqrt{b^2+x^2}+\sqrt{b^2+z^2}$ $x+y+2z=\sqrt{a^2+x^2}+\sqrt{a^2+y^2}$ have exactly one real solution $(x,y,z)$ with $x,y,z \geq 0$.

2023 Princeton University Math Competition, 11

Tags: algebra , complex number

11. Let $f(z)=\frac{a z+b}{c z+d}$ for $a, b, c, d \in \mathbb{C}$. Suppose that $f(1)=i, f(2)=i^{2}$, and $f(3)=i^{3}$. If the real part of $f(4)$ can be written as $\frac{m}{n}$ for relatively prime positive integers $m, n$, find $m^{2}+n^{2}$.

2006 Turkey Team Selection Test, 3

Tags: algebra , inequalities

If $x,y,z$ are positive real numbers and $xy+yz+zx=1$ prove that \[ \frac{27}{4} (x+y)(y+z)(z+x) \geq ( \sqrt{x+y} +\sqrt{ y+z} + \sqrt{z+x} )^2 \geq 6 \sqrt 3. \]

1986 Traian Lălescu, 1.3

Tags: algebra , polynom , supremum , polynomial function , real analysis , extremal

Let be four real numbers. Find the polynom of least degree such that two of these numbers are some locally extreme values, and the other two are the respective points of local extrema.

2014 Contests, 3

Tags: quadratic , function , algebra , quadratic formula , inequalities

Let $ x,y,z $ be three non-negative real numbers such that \[x^2+y^2+z^2=2(xy+yz+zx). \] Prove that \[\dfrac{x+y+z}{3} \ge \sqrt[3]{2xyz}.\]

2019 Tuymaada Olympiad, 1

Tags: algebra , sequence , inequalities

In a sequence $a_1, a_2, ..$ of real numbers the product $a_1a_2$ is negative, and to define $a_n$ for $n > 2$ one pair $(i, j)$ is chosen among all the pairs $(i, j), 1 \le i < j < n$, not chosen before, so that $a_i +a_j$ has minimum absolute value, and then $a_n$ is set equal to $a_i + a_j$ . Prove that $|a_i| < 1$ for some $i$.

2013 Hanoi Open Mathematics Competitions, 2

Tags: function , minimum , algebra

The smallest value of the function $f(x) =|x| +\left|\frac{1 - 2013x}{2013 - x}\right|$ where $x \in [-1, 1] $ is: (A): $\frac{1}{2012}$, (B): $\frac{1}{2013}$, (C): $\frac{1}{2014}$, (D): $\frac{1}{2015}$, (E): None of the above.

2011 Swedish Mathematical Competition, 6

Tags: algebra , functional , functional equation

How many functions $f:\mathbb N \to \mathbb N$ are there such that $f(0)=2011$, $f(1) = 111$, and $$f\left(\max \{x + y + 2, xy\}\right) = \min \{f (x + y), f (xy + 2)\}$$ for all non-negative integers $x$, $y$?

2014 Iran MO (3rd Round), 5

Tags: ratio , algebra , polynomial , number theory

Can an infinite set of natural numbers be found, such that for all triplets $(a,b,c)$ of it we have $abc + 1 $ perfect square? (20 points )

1983 IMO Longlists, 38

Tags: polynomial , algebra

Let $\{u_n \}$ be the sequence defined by its first two terms $u_0, u_1$ and the recursion formula \[u_{n+2 }= u_n - u_{n+1}.\] [b](a)[/b] Show that $u_n$ can be written in the form $u_n = \alpha a^n + \beta b^n$, where $a, b, \alpha, \beta$ are constants independent of $n$ that have to be determined. [b](b)[/b] If $S_n = u_0 + u_1 + \cdots + u_n$, prove that $S_n + u_{n-1}$ is a constant independent of $n.$ Determine this constant.

2013 HMNT, 8

Tags: algebra

Define the sequence $\{x_i\}_{i \ge 0}$ by $x_0 = x_1 = x_2 = 1$ and $x_k = \frac{x_{k-1}+x_{k-2}+1}{x_{k-3}}$ for $k > 2$. Find $x_{2013}$.

1993 Swedish Mathematical Competition, 6

Tags: function , algebra

For real numbers $a$ and $b$ define $f(x) = \frac{1}{ax+b}$. For which $a$ and $b$ are there three distinct real numbers $x_1,x_2,x_3$ such that $f(x_1) = x_2$, $f(x_2) = x_3$ and $f(x_3) = x_1$?

2009 Belarus Team Selection Test, 2

Tags: algebra , sequence

a) Prove that there is not an infinte sequence $(x_n)$, $n=1,2,...$ of positive real numbers satisfying the relation $x_{n+2}=\sqrt{x_{n+1}}-\sqrt{x_{n}}$, $\forall n \in N$ (*) b) Do there exist sequences satisfying (*) and containing arbitrary many terms? I.Voronovich

2010 Slovenia National Olympiad, 1

Tags: algebra

Let $a,b$ be real numbers such that $|a| \neq |b|$ and $\frac{a+b}{a-b}+\frac{a-b}{a+b}=6.$ Find the value of the expression $\frac{a^3+b^3}{a^3-b^3}+\frac{a^3-b^3}{a^3+b^3}.$