Olympiad problems

2010 Today's Calculation Of Integral, 549

Tags: calculus , integration , function , algebra , polynomial , limit , calculus computations

Let $ f(x)$ be a function defined on $ [0,\ 1]$. For $ n=1,\ 2,\ 3,\ \cdots$, a polynomial $ P_n(x)$ is defined by $ P_n(x)=\sum_{k=0}^n {}_nC{}_k f\left(\frac{k}{n}\right)x^k(1-x)^{n-k}$. Prove that $ \lim_{n\to\infty} \int_0^1 P_n(x)dx=\int_0^1 f(x)dx$.

1987 Bulgaria National Olympiad, Problem 6

Tags: triangle , function , geometry , geometric transformation

Let $\Delta$ be the set of all triangles inscribed in a given circle, with angles whose measures are integer numbers of degrees different than $45^\circ,90^\circ$ and $135^\circ$. For each triangle $T\in\Delta$, $f(T)$ denotes the triangle with vertices at the second intersection points of the altitudes of $T$ with the circle. (a) Prove that there exists a natural number $n$ such that for every triangle $T\in\Delta$, among the triangles $T,f(T),\ldots,f^n(T)$ (where $f^0(T)=T$ and $f^k(T)=f(f^{k-1}(T))$) at least two are equal. (b) Find the smallest $n$ with the property from (a).

2011 District Olympiad, 1

Tags: function , limit , continued fraction , real analysis

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.

1995 Abels Math Contest (Norwegian MO), 1a

Tags: algebra , function

Let a function $f$ satisfy $f(1) = 1$ and $f(1)+ f(2)+...+ f(n) = n^2f(n)$ for all $n \in N$. Determine $f(1995)$.

1980 IMO Shortlist, 7

Tags: function , algebra , functional equation

The function $f$ is defined on the set $\mathbb{Q}$ of all rational numbers and has values in $\mathbb{Q}$. It satisfies the conditions $f(1) = 2$ and $f(xy) = f(x)f(y) - f(x+y) + 1$ for all $x,y \in \mathbb{Q}$. Determine $f$.

2008 District Olympiad, 3

Tags: function , parameterization , euler , algebra

For any real $ a$ define $ f_a : \mathbb{R} \rightarrow \mathbb{R}^2$ by the law $ f_a(t) \equal{} \left( \sin(t), \cos(at) \right)$. a) Prove that $ f_{\pi}$ is not periodic. b) Determine the values of the parameter $ a$ for which $ f_a$ is periodic. [b]Remark[/b]. L. Euler proved in $ 1737$ that $ \pi$ is irrational.

2002 Germany Team Selection Test, 1

Tags: function , quadratic , algebra

Let $P$ denote the set of all ordered pairs $ \left(p,q\right)$ of nonnegative integers. Find all functions $f: P \rightarrow \mathbb{R}$ satisfying \[ f(p,q) \equal{} \begin{cases} 0 & \text{if} \; pq \equal{} 0, \\ 1 \plus{} \frac{1}{2} f(p+1,q-1) \plus{} \frac{1}{2} f(p-1,q+1) & \text{otherwise} \end{cases} \] Compare IMO shortlist problem 2001, algebra A1 for the three-variable case.

2022 Romania National Olympiad, P3

Tags: function , algebra

Determine all functions $f:\mathbb{R}\to\mathbb{R}$ for which there exists a function $g:\mathbb{R}\to\mathbb{R}$ such that $f(x)+f(y)=\lfloor g(x+y)\rfloor$ for all real numbers $x$ and $y$. [i]Emil Vasile[/i]

2009 AMC 12/AHSME, 23

Tags: quadratic , function , analytic geometry , geometric transformation , rotation , algebra , geometry

Functions $ f$ and $ g$ are quadratic, $ g(x) \equal{} \minus{} f(100 \minus{} x)$, and the graph of $ g$ contains the vertex of the graph of $ f$. The four $ x$-intercepts on the two graphs have $ x$-coordinates $ x_1$, $ x_2$, $ x_3$, and $ x_4$, in increasing order, and $ x_3 \minus{} x_2 \equal{} 150$. The value of $ x_4 \minus{} x_1$ is $ m \plus{} n\sqrt p$, where $ m$, $ n$, and $ p$ are positive integers, and $ p$ is not divisible by the square of any prime. What is $ m \plus{} n \plus{} p$? $ \textbf{(A)}\ 602\qquad \textbf{(B)}\ 652\qquad \textbf{(C)}\ 702\qquad \textbf{(D)}\ 752\qquad \textbf{(E)}\ 802$

1982 National High School Mathematics League, 5

Tags: function

For any$\varphi\in(0,\frac{\pi}{2})$, we have $\text{(A)}\sin\sin\varphi<\cos\varphi<\cos\cos\varphi\qquad\text{(B)}\sin\sin\varphi>\cos\varphi>\cos\cos\varphi$ $\text{(C)}\sin\cos\varphi>\cos\varphi>\cos\sin\varphi\qquad\text{(D)}\sin\cos\varphi<\cos\varphi<\cos\sin\varphi$

2005 Iran Team Selection Test, 1

Tags: function , number theory

Find all $f : N \longmapsto N$ that there exist $k \in N$ and a prime $p$ that: $\forall n \geq k \ f(n+p)=f(n)$ and also if $m \mid n$ then $f(m+1) \mid f(n)+1$

1962 Putnam, A2

Tags: function , algebra , domain

Find every real-valued function $f$ whose domain is an interval $I$ (finite or infinite) having $0$ as a left-hand endpoint, such that for every positive $x\in I$ the average of $f$ over the closed interval $[0,x]$ is equal to $\sqrt{ f(0) f(x)}.$

2009 Harvard-MIT Mathematics Tournament, 3

Tags: calculus , function , integration , logarithm

Compute $e^A$ where $A$ is defined as \[\int_{3/4}^{4/3}\dfrac{2x^2+x+1}{x^3+x^2+x+1}dx.\]

2019 Canadian Mathematical Olympiad Qualification, 1

Tags: algebra , function , injective function , injective , functional equation

A function $f$ is called injective if when $f(n) = f(m)$, then $n = m$. Suppose that $f$ is injective and $\frac{1}{f(n)}+\frac{1}{f(m)}=\frac{4}{f(n) + f(m)}$. Prove $m = n$

2005 Germany Team Selection Test, 2

Tags: polynomial , function , power of 2 , algebra

For any positive integer $ n$, prove that there exists a polynomial $ P$ of degree $ n$ such that all coeffients of this polynomial $ P$ are integers, and such that the numbers $ P\left(0\right)$, $ P\left(1\right)$, $ P\left(2\right)$, ..., $ P\left(n\right)$ are pairwisely distinct powers of $ 2$.

2007 Gheorghe Vranceanu, 3

Tags: function , partition , algebra , inequalities

Let be a function $ s:\mathbb{N}^2\longrightarrow \mathbb{N} $ that sends $ (m,n) $ to the number of solutions in $ \mathbb{N}^n $ of the equation: $$ x_1+x_2+\cdots +x_n=m $$ [b]1)[/b] Prove that: $$ s(m+1,n+1)=s(m,n)+s(m,n+1) =\prod_{r=1}^n\frac{m-r+1}{r} ,\quad\forall m,n\in\mathbb{N} $$ [b]2)[/b] Find $ \max\left\{ a_1a_2\cdots a_{20}\bigg| a_1+a_2+\cdots +a_{20}=2007, a_1,a_2,\ldots a_{20}\in\mathbb{N} \right\} . $

2020 Miklós Schweitzer, 8

Tags: algebra , functional equation , number theory , function

Let $\mathbb{F}_{p}$ denote the $p$-element field for a prime $p>3$ and let $S$ be the set of functions from $\mathbb{F}_{p}$ to $\mathbb{F}_{p}$. Find all functions $D\colon S\to S$ satisfying \[D(f\circ g)=(D(f)\circ g)\cdot D(g)\] for all $f,g \in {S}$. Here, $\circ$ denotes the function composition, so $(f\circ g)(x)$ is the function $f(g(x))$, and $\cdot$ denotes multiplication, so $(f\cdot g)=f(x)g(x)$.

2019 LIMIT Category B, Problem 8

Tags: function

If $f(x)=\cos(x)-1+\frac{x^2}2$, then $\textbf{(A)}~f(x)\text{ is an increasing function on the real line}$ $\textbf{(B)}~f(x)\text{ is a decreasing function on the real line}$ $\textbf{(C)}~f(x)\text{ is increasing on }-\infty<x\le0\text{ and decreasing on }0\le x<\infty$ $\textbf{(D)}~f(x)\text{ is decreasing on }-\infty<x\le0\text{ and increasing on }0\le x<\infty$

1996 Poland - Second Round, 3

Tags: function , inequalities

$a,b,c \geq-3/4$ and $a+b+c=1$. Show that: $\frac{a}{1+a^{2}}+\frac{b}{1+b^{2}}+\frac{c}{1+c^{2}}\leq \frac{9}{10}$

1998 Romania National Olympiad, 1

Tags: calculus , integration , function , real analysis

Suppose that $a,b\in\mathbb{R}^+$ which $a+b<1$ and $f:[0,+\infty) \rightarrow [0,+\infty) $ be the increasing function s.t. $\forall x\geq 0 ,\int _0^x f(t)dt=\int _0^{ax} f(t)dt+\int _0^{bx} f(t)dt$. Prove that $\forall x\geq 0 , f(x)=0$

2023 India National Olympiad, 3

Tags: function , number theory

Let $\mathbb N$ denote the set of all positive integers. Find all real numbers $c$ for which there exists a function $f:\mathbb N\to \mathbb N$ satisfying: [list] [*] for any $x,a\in\mathbb N$, the quantity $\frac{f(x+a)-f(x)}{a}$ is an integer if and only if $a=1$; [*] for all $x\in \mathbb N$, we have $|f(x)-cx|<2023$. [/list] [i]Proposed by Sutanay Bhattacharya[/i]

2015 Romania Team Selection Tests, 3

Tags: least common multiple , function , number theory , rip

If $k$ and $n$ are positive integers , and $k \leq n$ , let $M(n,k)$ denote the least common multiple of the numbers $n , n-1 , \ldots , n-k+1$.Let $f(n)$ be the largest positive integer $ k \leq n$ such that $M(n,1)<M(n,2)<\ldots <M(n,k)$ . Prove that : [b](a)[/b] $f(n)<3\sqrt{n}$ for all positive integers $n$ . [b](b)[/b] If $N$ is a positive integer , then $f(n) > N$ for all but finitely many positive integers $n$.

1974 Miklós Schweitzer, 4

Tags: superior algebra , ring theory , function , induction , number theory , prime number

Let $ R$ be an infinite ring such that every subring of $ R$ different from $ \{0 \}$ has a finite index in $ R$. (By the index of a subring, we mean the index of its additive group in the additive group of $ R$.) Prove that the additive group of $ R$ is cyclic. [i]L. Lovasz, J. Pelikan[/i]

1989 IMO Longlists, 15

Tags: function , rational function , algebra

A sequence $ a_1, a_2, a_3, \ldots$ is defined recursively by $ a_1 \equal{} 1$ and $ a_{2^k\plus{}j} \equal{} \minus{}a_j$ $ (j \equal{} 1, 2, \ldots, 2^k).$ Prove that this sequence is not periodic.

2021 China Team Selection Test, 4

Tags: number theory , totient function , function

Find all functions $f: \mathbb{Z}^+\rightarrow \mathbb{Z}^+$ such that for all positive integers $m,n$ with $m\ge n$, $$f(m\varphi(n^3)) = f(m)\cdot \varphi(n^3).$$ Here $\varphi(n)$ denotes the number of positive integers coprime to $n$ and not exceeding $n$.