Olympiad problems

2009 AIME Problems, 7

Define $ n!!$ to be $ n(n\minus{}2)(n\minus{}4)\ldots3\cdot1$ for $ n$ odd and $ n(n\minus{}2)(n\minus{}4)\ldots4\cdot2$ for $ n$ even. When $ \displaystyle \sum_{i\equal{}1}^{2009} \frac{(2i\minus{}1)!!}{(2i)!!}$ is expressed as a fraction in lowest terms, its denominator is $ 2^ab$ with $ b$ odd. Find $ \displaystyle \frac{ab}{10}$.

1982 IMO Longlists, 34

Tags: algebra , function

Let $M$ be the set of all functions $f$ with the following properties: [b](i)[/b] $f$ is defined for all real numbers and takes only real values. [b](ii)[/b] For all $x, y \in \mathbb R$ the following equality holds: $f(x)f(y) = f(x + y) + f(x - y).$ [b](iii)[/b] $f(0) \neq 0.$ Determine all functions $f \in M$ such that [b](a)[/b] $f(1)=\frac 52$, [b](b)[/b] $f(1)= \sqrt 3$.

2019 Thailand TSTST, 3

Tags: functional equation , algebra , function

Find all function $f:\mathbb{Z}\to\mathbb{Z}$ satisfying $\text{(i)}$ $f(f(m)+n)+2m=f(n)+f(3m)$ for every $m,n\in\mathbb{Z}$, $\text{(ii)}$ there exists a $d\in\mathbb{Z}$ such that $f(d)-f(0)=2$, and $\text{(iii)}$ $f(1)-f(0)$ is even.

1978 IMO Longlists, 6

Tags: geometry , polynomial , function , trigonometry , algebra

Prove that for all $X > 1$, there exists a triangle whose sides have lengths $P_1(X) = X^4+X^3+2X^2+X+1, P_2(X) = 2X^3+X^2+2X+1$, and $P_3(X) = X^4-1$. Prove that all these triangles have the same greatest angle and calculate it.

2000 China Team Selection Test, 3

Tags: algebra , function

Let $n$ be a positive integer. Denote $M = \{(x, y)|x, y \text{ are integers }, 1 \leq x, y \leq n\}$. Define function $f$ on $M$ with the following properties: [b]a.)[/b] $f(x, y)$ takes non-negative integer value; [b] b.)[/b] $\sum^n_{y=1} f(x, y) = n - 1$ for $1 \eq x \leq n$; [b]c.)[/b] If $f(x_1, y_1)f(x2, y2) > 0$, then $(x_1 - x_2)(y_1 - y_2) \geq 0.$ Find $N(n)$, the number of functions $f$ that satisfy all the conditions. Give the explicit value of $N(4)$.

Today's calculation of integrals, 867

Tags: algebra , quadratic , derivative , linear equation , integration , function , calculus

Express $\int_0^2 f(x)dx$ for any quadratic functions $f(x)$ in terms of $f(0),\ f(1)$ and $f(2).$

2013 Stars Of Mathematics, 1

Tags: algebra , functional inequality , function

Let $\mathcal{F}$ be the family of bijective increasing functions $f\colon [0,1] \to [0,1]$, and let $a \in (0,1)$. Determine the best constants $m_a$ and $M_a$, such that for all $f \in \mathcal{F}$ we have \[m_a \leq f(a) + f^{-1}(a) \leq M_a.\] [i](Dan Schwarz)[/i]

2000 IMC, 6

Tags: integration , limit , real analysis , function

Let $f: \mathbb{R}\rightarrow ]0,+\infty[$ be an increasing differentiable function with $\lim_{x\rightarrow+\infty}f(x)=+\infty$ and $f'$ is bounded, and let $F(x)=\int^x_0 f(t) dt$. Define the sequence $(a_n)$ recursively by $a_0=1,a_{n+1}=a_n+\frac1{f(a_n)}$ Define the sequence $(b_n)$ by $b_n=F^{-1}(n)$. Prove that $\lim_{x\rightarrow+\infty}(a_n-b_n)=0$.

1976 Miklós Schweitzer, 3

Tags: number theory , logarithm , function , inequalities

Let $ H$ denote the set of those natural numbers for which $ \tau(n)$ divides $ n$, where $ \tau(n)$ is the number of divisors of $ n$. Show that a) $ n! \in H$ for all sufficiently large $ n$, b)$ H$ has density $ 0$. [i]P. Erdos[/i]

2007 Nicolae Păun, 4

Tags: real analysis , monotone , topology , darboux s property , pathological , function

Construct a function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ having the following properties: $ \text{(i)} f $ is not monotonic on any real interval. $ \text{(ii)} f $ has Darboux property (intermediate value property) on any real interval. $ \text{(iii)} f(x)\leqslant f\left( x+1/n \right) ,\quad \forall x\in\mathbb{R} ,\quad \forall n\in\mathbb{N} $ [i]Alexandru Cioba[/i]

1990 AMC 12/AHSME, 12

Tags: function

Let $f$ be the function defined by $f(x)=ax^2-\sqrt2$ for some positive $a$. If $f(f(\sqrt2 ))=-\sqrt 2$, then $a=$ $\text{(A)} \ \frac{2-\sqrt2}{2} \qquad \text{(B)} \ \frac12 \qquad \text{(C)} \ 2-\sqrt2 \qquad \text{(D)} \ \frac{\sqrt{2}}{2} \qquad \text{(E)} \ \frac{2+\sqrt2}{2}$

2019 Taiwan TST Round 3, 2

Tags: algebra , function

Determine all functions $f:(0,\infty)\to\mathbb{R}$ satisfying $$\left(x+\frac{1}{x}\right)f(y)=f(xy)+f\left(\frac{y}{x}\right)$$ for all $x,y>0$.

1978 IMO Longlists, 33

Tags: algebra , logarithm , function , inequalities

A sequence $(a_n)^{\infty}_0$ of real numbers is called [i]convex[/i] if $2a_n\le a_{n-1}+a_{n+1}$ for all positive integers $n$. Let $(b_n)^{\infty}_0$ be a sequence of positive numbers and assume that the sequence $(\alpha^nb_n)^{\infty}_0$ is convex for any choice of $\alpha > 0$. Prove that the sequence $(\log b_n)^{\infty}_0$ is convex.

2013 Baltic Way, 20

Tags: number theory , algebra , limit , function , polynomial

Find all polynomials $f$ with non-negative integer coefficients such that for all primes $p$ and positive integers $n$ there exist a prime $q$ and a positive integer $m$ such that $f(p^n)=q^m$.

2006 District Olympiad, 1

Tags: algebra , function , inequalities

Let $x,y,z$ be positive real numbers. Prove the following inequality: \[ \frac 1{x^2+yz} + \frac 1{y^2+zx } + \frac 1{z^2+xy} \leq \frac 12 \left( \frac 1{xy} + \frac 1{yz} + \frac 1{zx} \right). \]

2007 Today's Calculation Of Integral, 227

Tags: integration , calculus , trigonometry , function , calculus computations

Evaluate $ \frac{1}{\displaystyle \int _0^{\frac{\pi}{2}} \cos ^{2006}x \cdot \sin 2008 x\ dx}$

2013 ELMO Shortlist, 6

Tags: number theory , function

Let $\mathbb N$ denote the set of positive integers, and for a function $f$, let $f^k(n)$ denote the function $f$ applied $k$ times. Call a function $f : \mathbb N \to \mathbb N$ [i]saturated[/i] if \[ f^{f^{f(n)}(n)}(n) = n \] for every positive integer $n$. Find all positive integers $m$ for which the following holds: every saturated function $f$ satisfies $f^{2014}(m) = m$. [i]Proposed by Evan Chen[/i]

2013 Bogdan Stan, 2

Tags: floor function , function , injectivity , surjectivity , bijectivity , integer part , algebra

Consider the parametric function $ f_k:\mathbb{R}\longrightarrow\mathbb{R}, f(x)=x+k\lfloor x \rfloor . $ [b]a)[/b] For which integer values of $ k $ the above function is injective? [b]b)[/b] For which integer values of $ k $ the above function is surjective? [b]c)[/b] Given two natural numbers $ n,m, $ create two bijective functions: $$ \phi : f_m (\mathbb{R} )\cap [0,\infty )\longrightarrow f_n(\mathbb{R})\cap [0,\infty ) $$ $$ \psi : \left(\mathbb{R}\setminus f_m (\mathbb{R})\right)\cap [0,\infty )\longrightarrow\left(\mathbb{R}\setminus f_n (\mathbb{R})\right)\cap [0,\infty ) $$ [i]Cristinel Mortici[/i]

2012 Today's Calculation Of Integral, 791

Tags: algebra , inequalities , analytic geometry , integration , function , domain , calculus

Let $S$ be the domain in the coordinate plane determined by two inequalities: \[y\geq \frac 12x^2,\ \ \frac{x^2}{4}+4y^2\leq \frac 18.\] Denote by $V_1$ the volume of the solid by a rotation of $S$ about the $x$-axis and by $V_2$, by a rotation of $S$ about the $y$-axis. (1) Find the values of $V_1,\ V_2$. (2) Compare the size of the value of $\frac{V_2}{V_1}$ and 1.

2008 Canada National Olympiad, 2

Tags: algebra , function , functional equation

Determine all functions $ f$ defined on the set of rational numbers that take rational values for which \[ f(2f(x) \plus{} f(y)) \equal{} 2x \plus{} y, \] for each $ x$ and $ y$.

2008 Harvard-MIT Mathematics Tournament, 10

Tags: calculus , function , integration

Evaluate the infinite sum \[\sum_{n \equal{} 0}^\infty \binom{2n}{n}\frac {1}{5^n}.\]

1964 Dutch Mathematical Olympiad, 4

Tags: functional equation , algebra , function

The function $ƒ$ is defined at $[0,1]$, and $f\{f(x)\} = ƒ(x)$. $\exists _{c\in [0,1]} \left[f(c) =\frac12 \right]$ Determine $f\left(\frac12 \right).$ $\forall _{t\in [0,1]}\exists _{s\in [0,1]}[f(s) = t]$. Determine $f$. Prove that the function $g$, with $g(x) = x$,$0 \le x \le k$, $g(x) = k$, $k \le x \le 1$ satisfies the relation $g\{g(x)\} = g(x)$.

2003 IMO Shortlist, 6

Tags: calculus , inequalities , function

Let $n$ be a positive integer and let $(x_1,\ldots,x_n)$, $(y_1,\ldots,y_n)$ be two sequences of positive real numbers. Suppose $(z_2,\ldots,z_{2n})$ is a sequence of positive real numbers such that $z_{i+j}^2 \geq x_iy_j$ for all $1\le i,j \leq n$. Let $M=\max\{z_2,\ldots,z_{2n}\}$. Prove that \[ \left( \frac{M+z_2+\dots+z_{2n}}{2n} \right)^2 \ge \left( \frac{x_1+\dots+x_n}{n} \right) \left( \frac{y_1+\dots+y_n}{n} \right). \] [hide="comment"] [i]Edited by Orl.[/i] [/hide] [i]Proposed by Reid Barton, USA[/i]

2018 ELMO Shortlist, 1

Tags: functional equation , algebra , function

Let $f:\mathbb{R}\to\mathbb{R}$ be a bijective function. Does there always exist an infinite number of functions $g:\mathbb{R}\to\mathbb{R}$ such that $f(g(x))=g(f(x))$ for all $x\in\mathbb{R}$? [i]Proposed by Daniel Liu[/i]

1952 AMC 12/AHSME, 13

Tags: parabola , conic , quadratic , calculus , function

The function $ x^2 \plus{} px \plus{} q$ with $ p$ and $ q$ greater than zero has its minimum value when: $ \textbf{(A)}\ x \equal{} \minus{} p \qquad\textbf{(B)}\ x \equal{} \frac {p}{2} \qquad\textbf{(C)}\ x \equal{} \minus{} 2p \qquad\textbf{(D)}\ x \equal{} \frac {p^2}{4q} \qquad\textbf{(E)}\ x \equal{} \frac { \minus{} p}{2}$