Olympiad problems

2003 AMC 12-AHSME, 17

Tags: logarithm , function

If $ \log(xy^3)\equal{}1$ and $ \log(x^2y)\equal{}1$, what is $ \log(xy)$? $ \textbf{(A)}\ \minus{}\!\frac{1}{2} \qquad \textbf{(B)}\ 0 \qquad \textbf{(C)}\ \frac{1}{2} \qquad \textbf{(D)}\ \frac{3}{5} \qquad \textbf{(E)}\ 1$

2015 China Team Selection Test, 4

Tags: algebra , combinatorics , function

Let $n$ be a positive integer, let $f_1(x),\ldots,f_n(x)$ be $n$ bounded real functions, and let $a_1,\ldots,a_n$ be $n$ distinct reals. Show that there exists a real number $x$ such that $\sum^n_{i=1}f_i(x)-\sum^n_{i=1}f_i(x-a_i)<1$.

1996 Estonia Team Selection Test, 3

Tags: function , algebra , functional equation

Find all functions $f:\mathbb{R}\to\mathbb{R}$ which satisfy for all $x$: $(i)$ $f(x)=-f(-x);$ $(ii)$ $f(x+1)=f(x)+1;$ $(iii)$ $f\left( \frac{1}{x}\right)=\frac{1}{x^2}f(x)$ for $x\ne 0$

2014 Romania Team Selection Test, 2

Tags: function , algebra

Let $a$ be a real number in the open interval $(0,1)$. Let $n\geq 2$ be a positive integer and let $f_n\colon\mathbb{R}\to\mathbb{R}$ be defined by $f_n(x) = x+\frac{x^2}{n}$. Show that \[\frac{a(1-a)n^2+2a^2n+a^3}{(1-a)^2n^2+a(2-a)n+a^2}<(f_n \circ\ \cdots\ \circ f_n)(a)<\frac{an+a^2}{(1-a)n+a}\] where there are $n$ functions in the composition.

2016 CMIMC, 4

Tags: function , prime factorization , number theory

For some positive integer $n$, consider the usual prime factorization \[n = \displaystyle \prod_{i=1}^{k} p_{i}^{e_{i}}=p_1^{e_1}p_2^{e_2}\ldots p_k^{e_k},\] where $k$ is the number of primes factors of $n$ and $p_{i}$ are the prime factors of $n$. Define $Q(n), R(n)$ by \[ Q(n) = \prod_{i=1}^{k} p_{i}^{p_{i}} \text{ and } R(n) = \prod_{i=1}^{k} e_{i}^{e_{i}}. \] For how many $1 \leq n \leq 70$ does $R(n)$ divide $Q(n)$?

2002 Putnam, 1

Tags: function , derivative , limit , calculus

Let $k$ be a fixed positive integer. The $n$th derivative of $\tfrac{1}{x^k-1}$ has the form $\tfrac{P_n(x)}{(x^k-1)^{n+1}}$, where $P_n(x)$ is a polynomial. Find $P_n(1)$.

2013 China Girls Math Olympiad, 4

Tags: prime number , modular arithmetic , number theory , function , polynomial , algebra

Find the number of polynomials $f(x)=ax^3+bx$ satisfying both following conditions: (i) $a,b\in\{1,2,\ldots,2013\}$; (ii) the difference between any two of $f(1),f(2),\ldots,f(2013)$ is not a multiple of $2013$.

1997 Vietnam Team Selection Test, 1

Tags: induction , polynomial , relatively prime , number theory , function , algebra

The function $ f : \mathbb{N} \to \mathbb{Z}$ is defined by $ f(0) \equal{} 2$, $ f(1) \equal{} 503$ and $ f(n \plus{} 2) \equal{} 503f(n \plus{} 1) \minus{} 1996f(n)$ for all $ n \in\mathbb{N}$. Let $ s_1$, $ s_2$, $ \ldots$, $ s_k$ be arbitrary integers not smaller than $ k$, and let $ p(s_i)$ be an arbitrary prime divisor of $ f\left(2^{s_i}\right)$, ($ i \equal{} 1, 2, \ldots, k$). Prove that, for any positive integer $ t$ ($ t\le k$), we have $ 2^t \Big | \sum_{i \equal{} 1}^kp(s_i)$ if and only if $ 2^t | k$.

1990 China Team Selection Test, 2

Tags: algebra , function

Find all functions $f,g,h: \mathbb{R} \mapsto \mathbb{R}$ such that $f(x) - g(y) = (x-y) \cdot h(x+y)$ for $x,y \in \mathbb{R}.$

1999 Putnam, 4

Tags: calculus , derivative , limit , function

Let $f$ be a real function with a continuous third derivative such that $f(x)$, $f^\prime(x)$, $f^{\prime\prime}(x)$, $f^{\prime\prime\prime}(x)$ are positive for all $x$. Suppose that $f^{\prime\prime\prime}(x)\leq f(x)$ for all $x$. Show that $f^\prime(x)<2f(x)$ for all $x$.

2014 USAMO, 2

Tags: function , algebra

Let $\mathbb{Z}$ be the set of integers. Find all functions $f : \mathbb{Z} \rightarrow \mathbb{Z}$ such that \[xf(2f(y)-x)+y^2f(2x-f(y))=\frac{f(x)^2}{x}+f(yf(y))\] for all $x, y \in \mathbb{Z}$ with $x \neq 0$.

2015 Abels Math Contest (Norwegian MO) Final, 1b

Tags: algebra , function , functional equation

Find all functions $f : R \to R$ such that $x^2f(yf(x))= y^2f(x)f(f(x))$ for all real numbers $x$ and $y$.

2012 Bogdan Stan, 2

Tags: integration , find all functions , function , calculus

Find the continuous functions $ f:\left[ 0,\frac{1}{3} \right] \longrightarrow (0,\infty ) $ that satisfy the functional relation $$ 54\int_0^{1/3} f(x)dx +32\int_0^{1/3} \frac{dx}{\sqrt{x+f(x)}} =21. $$ [i]Cristinel Mortici[/i]

1975 Miklós Schweitzer, 4

Tags: real analysis , number theory , function

Prove that the set of rational-valued, multiplicative arithmetical functions and the set of complex rational-valued, multiplicative arithmetical functions form isomorphic groups with the convolution operation $ f \circ g$ defined by \[{ (f \circ g)(n)= %Error. "displatmath" is a bad command. \sum_{d|n} f(d)g(\frac nd}).\] (We call a complex number $ \textit{complex rational}$, if its real and imaginary parts are both rational.) [i]B. Csakany[/i]

1973 Putnam, A4

Tags: function , real analysis , root

How many zeroes does the function $f(x)=2^x -1 -x^2 $ have on the real line?

2002 All-Russian Olympiad, 3

Tags: trigonometry , inequalities , function

Prove that if $0<x<\frac{\pi}{2}$ and $n>m$, where $n$,$m$ are natural numbers, \[ 2 \left| \sin^n x - \cos^n x \right| \le 3 \left| \sin^m x - \cos^m x \right|.\]

2007 Nicolae Păun, 3

Tags: polynom , function , algebra

Let $ a,b,c,d $ be four real numbers such that $ |ax^3+bx^2+cx+d|\le 1,\forall x\in [0,1] . $ Prove that $ |dx^2+cx^2+bx+a|\le 9/2,\forall x\in [0,1] . $ [i]Lavinia Savu[/i]

2003 Balkan MO, 3

Tags: algebra , function

Find all functions $f: \mathbb{Q}\to\mathbb{R}$ which fulfill the following conditions: a) $f(1)+1>0$; b) $f(x+y) -xf(y) -yf(x) = f(x)f(y) -x-y +xy$, for all $x,y\in\mathbb{Q}$; c) $f(x) = 2f(x+1) +x+2$, for every $x\in\mathbb{Q}$.

2016 Iran MO (3rd Round), 3

Tags: algebra , functional equation , function

Find all functions $f:\mathbb {R}^{+} \rightarrow \mathbb {R}^{+} $ such that for all positive real numbers $x,y:$ $$f(y)f(x+f(y))=f(x)f(xy)$$

2003 Romania National Olympiad, 2

Tags: function , integration , real analysis , definite integral , titu andreescu

Let be an odd natural number $ n\ge 3. $ Find all continuous functions $ f:[0,1]\longrightarrow\mathbb{R} $ that satisfy the following equalities. $$ \int_0^1 \left( f\left(\sqrt[k]{x}\right) \right)^{n-k} dx=k/n,\quad\forall k\in\{ 1,2,\ldots ,n-1\} $$ [i]Titu Andreescu[/i]

2011 ISI B.Math Entrance Exam, 4

Tags: inequalities , function , algebra

Let $t_1 < t_2 < t_3 < \cdots < t_{99}$ be real numbers. Consider a function $f: \mathbb{R} \to \mathbb{R}$ given by $f(x)=|x-t_1|+|x-t_2|+...+|x-t_{99}|$ . Show that $f(x)$ will attain minimum value at $x=t_{50}$.

2003 Alexandru Myller, 3

Tags: invariant , function , analytic geometry , algebra

Let $ S $ be the first quadrant and $ T:S\longrightarrow S $ be a transformation that takes the reciprocal of the coordinates of the points that belong to its domain. Define an [i]S-line[/i] to be the intersection of a line with $ S. $ [b]a)[/b] Show that the fixed points of $ T $ lie on any fixed S-line of $ T. $ [b]b)[/b] Find all fixed S-lines of $ T. $ [i]Gabriel Popa[/i]

1994 Flanders Math Olympiad, 4

Tags: algebra , domain , quadratic , function

Let $(f_i)$ be a sequence of functions defined by: $f_1(x)=x, f_n(x) = \sqrt{f_{n-1}(x)}-\dfrac14$. ($n\in \mathbb{N}, n\ge2$) (a) Prove that $f_n(x) \le f_{n-1}(x)$ for all x where both functions are defined. (b) Find for each $n$ the points of $x$ inside the domain for which $f_n(x)=x$.

2005 AIME Problems, 13

Tags: function , quadratic , vieta , polynomial , algebra

Let $P(x)$ be a polynomial with integer coefficients that satisfies $P(17)=10$ and $P(24)=17$. Given that $P(n)=n+3$ has two distinct integer solutions $n_1$ and $n_2$, find the product $n_1\cdot n_2$.

2010 ELMO Shortlist, 1

Tags: function , number theory , inequalities

For a positive integer $n$, let $\mu(n) = 0$ if $n$ is not squarefree and $(-1)^k$ if $n$ is a product of $k$ primes, and let $\sigma(n)$ be the sum of the divisors of $n$. Prove that for all $n$ we have \[\left|\sum_{d|n}\frac{\mu(d)\sigma(d)}{d}\right| \geq \frac{1}{n}, \] and determine when equality holds. [i]Wenyu Cao.[/i]