Olympiad problems

2025 Kyiv City MO Round 1, Problem 1

Find all triples of positive integers $ a, b, c $ that satisfy the equation: \[ a + \frac{1}{b + \frac{1}{c}} = 20.25. \]

2012 Iran MO (3rd Round), 2

Tags: continued fraction , inequalities , algebra

Suppose $N\in \mathbb N$ is not a perfect square, hence we know that the continued fraction of $\sqrt{N}$ is of the form $\sqrt{N}=[a_0,\overline{a_1,a_2,...,a_n}]$. If $a_1\neq 1$ prove that $a_i\le 2a_0$.

2009 China Girls Math Olympiad, 8

Tags: pell equation , diophantine equation , continued fraction , modular arithmetic , induction , floor function , number theory

For a positive integer $ n,$ $ a_{n}\equal{}n\sqrt{5}\minus{} \lfloor n\sqrt{5}\rfloor$. Compute the maximum value and the minimum value of $ a_{1},a_{2},\ldots ,a_{2009}.$

1994 India National Olympiad, 6

Tags: continued fraction , algebra , function

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 $x \not = 0$.

2010 Stanford Mathematics Tournament, 15

Tags: graphing lines , continued fraction , analytic geometry , slope

Find the best approximation of $\sqrt{3}$ by a rational number with denominator less than or equal to $15$

2011 Kyrgyzstan National Olympiad, 7

Tags: function , continued fraction , induction , algebra

Given that $g(n) = \frac{1}{{2 + \frac{1}{{3 + \frac{1}{{... + \frac{1}{{n - 1}}}}}}}}$ and $k(n) = \frac{1}{{2 + \frac{1}{{3 + \frac{1}{{... + \frac{1}{{n - 1 + \frac{1}{n}}}}}}}}}$, for natural $n$. Prove that $\left| {g(n) - k(n)} \right| \le \frac{1}{{(n - 1)!n!}}$.

2004 IMO Shortlist, 3

Tags: continued fraction , function , functional equation , number theory , algebra

Does there exist a function $s\colon \mathbb{Q} \rightarrow \{-1,1\}$ such that if $x$ and $y$ are distinct rational numbers satisfying ${xy=1}$ or ${x+y\in \{0,1\}}$, then ${s(x)s(y)=-1}$? Justify your answer. [i]Proposed by Dan Brown, Canada[/i]

1992 Baltic Way, 11

Tags: continued fraction , function , algebra

Let $ Q^\plus{}$ denote the set of positive rational numbers. Show that there exists one and only one function $f: Q^\plus{}\to Q^\plus{}$ satisfying the following conditions: (i) If $ 0<q<1/2$ then $ f(q)\equal{}1\plus{}f(q/(1\minus{}2q))$, (ii) If $ 1<q\le2$ then $ f(q)\equal{}1\plus{}f(q\minus{}1)$, (iii) $ f(q)\cdot f(1/q)\equal{}1$ for all $ q\in Q^\plus{}$.

2015 CCA Math Bonanza, T4

Tags: continued fraction

Evaluate the continued fraction $$1+\frac{2}{2+\frac{2}{2+\ldots}}$$ [i]2015 CCA Math Bonanza Team Round #4[/i]

2003 Baltic Way, 1

Tags: continued fraction , function , algebra

Find all functions $f:\mathbb{Q}^{+}\rightarrow \mathbb{Q}^{+}$ which for all $x \in \mathbb{Q}^{+}$ fulfil \[f\left(\frac{1}{x}\right)=f(x) \ \ \text{and} \ \ \left(1+\frac{1}{x}\right)f(x)=f(x+1). \]

2024 USA TSTST, 2

Tags: continued fraction , polynomial , algebra

Let $p$ be an odd prime number. Suppose $P$ and $Q$ are polynomials with integer coefficients such that $P(0)=Q(0)=1$, there is no nonconstant polynomial dividing both $P$ and $Q$, and \[ 1 + \cfrac{x}{1 + \cfrac{2x}{1 + \cfrac{\ddots}{1 + (p-1)x}}}=\frac{P(x)}{Q(x)}. \] Show that all coefficients of $P$ except for the constant coefficient are divisible by $p$, and all coefficients of $Q$ are [i]not[/i] divisible by $p$. [i]Andrew Gu[/i]

1995 Putnam, 4

Tags: induction , continued fraction

Evaluate : \[ \sqrt[8]{2207-\frac{1}{2207-\frac{1}{2207-\cdots}}} \] Express your expression in the form $\frac{a+b\sqrt{c}}{d}$ where $a,b,c,d\in \Bbb{Z}$.

1966 Putnam, A6

Tags: continued fraction

Justify the statement that $$3=\sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{1+5\sqrt{1+\dots}}}}}.$$

KoMaL A Problems 2024/2025, A. 903

Tags: floor function , irrational number , continued fraction , number theory

Let the irrational number \[\alpha =1-\cfrac{1}{2a_1-\cfrac{1}{2a_2-\cfrac{1}{2a_3-\cdots}}}\] where coefficients $a_1, a_2, \ldots$ are positive integers, infinitely many of which are greater than $1$. Prove that for every positive integer $N$ at least half of the numbers $\lfloor \alpha\rfloor, \lfloor 2\alpha\rfloor, \ldots, \lfloor N\alpha\rfloor$ are even. [i]Proposed by Géza Kós, Budapest[/i]

1993 Polish MO Finals, 2

Tags: function , continued fraction , 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 $x \not = 0$.

2009 IberoAmerican, 5

Tags: number theory , induction , algebra , search , algorithm , continued fraction , euclidean algorithm

Consider the sequence $ \{a_n\}_{n\geq1}$ defined as follows: $ a_1 \equal{} 1$, $ a_{2k} \equal{} 1 \plus{} a_k$ and $ a_{2k \plus{} 1} \equal{} \frac {1}{a_{2k}}$ for every $ k\geq 1$. Prove that every positive rational number appears on the sequence $ \{a_n\}$ exactly once.

1950 Miklós Schweitzer, 2

Tags: number theory , continued fraction

Show that there exists a positive constant $ c$ with the following property: To every positive irrational $ \alpha$, there can be found infinitely many fractions $ \frac{p}{q}$ with $ (p,q)\equal{}1$ satisfying $ \left|\alpha\minus{}\frac{p}{q}\right|\le \frac{c}{q^2}$

2011 District Olympiad, 1

Tags: limit , continued fraction , real analysis , function

a) Prove that $\{x+y\}-\{y\}$ can only be equal to $\{x\}$ or $\{x\}-1$ for any $x,y\in \mathbb{R}$. b) Let $\alpha\in \mathbb{R}\backslash \mathbb{Q}$. Denote $a_n=\{n\alpha\}$ for all $n\in \mathbb{N}^*$ and define the sequence $(x_n)_{n\ge 1}$ by \[x_n=(a_2-a_1)(a_3-a_2)\cdot \ldots \cdot (a_{n+1}-a_n)\] Prove that the sequence $(x_n)_{n\ge 1}$ is convergent and find it's limit.

2009 Turkey Team Selection Test, 1

Tags: function , algebra , functional equation , algorithm , euclidean algorithm , continued fraction , number theory

Find all $ f: Q^ \plus{} \to\ Z$ functions that satisfy $ f \left(\frac {1}{x} \right) \equal{} f(x)$ and $ (x \plus{} 1)f(x \minus{} 1) \equal{} xf(x)$ for all rational numbers that are bigger than 1.