Found problems: 4776
2007 Harvard-MIT Mathematics Tournament, 8
Let $A \text{ :}= \mathbb{Q}\setminus \{0,1\}$ denote the set of all rationals other than $0$ and $1$. A function $f:A\to \mathbb{R}$ has the property that for all $x\in A$, \[f(x)+f\left(1-\dfrac{1}{x}\right)=\log |x|.\] Compute the value of $f(2007)$.
2012 Harvard-MIT Mathematics Tournament, 1
Let $f$ be the function such that
\[f(x)=\begin{cases}2x & \text{if }x\leq \frac{1}{2}\\2-2x & \text{if }x>\frac{1}{2}\end{cases}\]
What is the total length of the graph of $\underbrace{f(f(\ldots f}_{2012\text{ }f's}(x)\ldots))$ from $x=0$ to $x=1?$
1993 AMC 12/AHSME, 24
A box contains $3$ shiny pennies and $4$ dull pennies. One by one, pennies are drawn at random from the box and not replaced. If the probability is $\frac{a}{b}$ that it will take more than four draws until the third shiny penny appears and $\frac{a}{b}$ is in lowest terms, then $a+b=$
$ \textbf{(A)}\ 11 \qquad\textbf{(B)}\ 20 \qquad\textbf{(C)}\ 35 \qquad\textbf{(D)}\ 58 \qquad\textbf{(E)}\ 66 $
2016 IFYM, Sozopol, 8
For a quadratic trinomial $f(x)$ and the different numbers $a$ and $b$ it is known that $f(a)=b$ and $f(b)=a$. We call such $a$ and $b$ [i]conjugate[/i] for $f(x)$. Prove that $f(x)$ has no other [i]conjugate[/i] numbers.
2008 Alexandru Myller, 4
Let be a function $ f:\mathbb{R}\rightarrow\mathbb{R} $ satisfying the following properties:
$ \text{(i)} $ is continuous on the rational numbers.
$ \text{(ii)} f(x)<f\left( x+\frac{1}{n}\right) , $ for any real $ x $ and natural $ n. $
Prove that $ f $ is increasing.
[i]Gabriel Mârşanu, Mihai Piticari[/i]
2007 Romania National Olympiad, 1
Let $\mathcal{F}$ be the set of functions $f: [0,1]\to\mathbb{R}$ that are differentiable, with continuous derivative, and $f(0)=0$, $f(1)=1$. Find the minimum of $\int_{0}^{1}\sqrt{1+x^{2}}\cdot \big(f'(x)\big)^{2}\ dx$ (where $f\in\mathcal{F}$) and find all functions $f\in\mathcal{F}$ for which this minimum is attained.
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In the contest, this was the b) point of the problem. The a) point was simply ``Prove the Cauchy inequality in integral form''.
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2000 Putnam, 6
Let $f(x)$ be a polynomial with integer coefficients. Define a sequence $a_0, a_1, \cdots $ of integers such that $a_0=0$ and $a_{n+1}=f(a_n)$ for all $n \ge 0$. Prove that if there exists a positive integer $m$ for which $a_m=0$ then either $a_1=0$ or $a_2=0$.
1985 AIME Problems, 10
How many of the first 1000 positive integers can be expressed in the form
\[ \lfloor 2x \rfloor + \lfloor 4x \rfloor + \lfloor 6x \rfloor + \lfloor 8x \rfloor, \]
where $x$ is a real number, and $\lfloor z \rfloor$ denotes the greatest integer less than or equal to $z$?
1985 IMO Shortlist, 9
Determine the radius of a sphere $S$ that passes through the centroids of each face of a given tetrahedron $T$ inscribed in a unit sphere with center $O$. Also, determine the distance from $O$ to the center of $S$ as a function of the edges of $T.$
2009 IMO, 3
Suppose that $ s_1,s_2,s_3, \ldots$ is a strictly increasing sequence of positive integers such that the sub-sequences \[s_{s_1},\, s_{s_2},\, s_{s_3},\, \ldots\qquad\text{and}\qquad s_{s_1+1},\, s_{s_2+1},\, s_{s_3+1},\, \ldots\] are both arithmetic progressions. Prove that the sequence $ s_1, s_2, s_3, \ldots$ is itself an arithmetic progression.
[i]Proposed by Gabriel Carroll, USA[/i]
2010 Slovenia National Olympiad, 4
Let $x,y$ and $z$ be real numbers such that $0 \leq x,y,z \leq 1.$ Prove that
\[xyz+(1-x)(1-y)(1-z) \leq 1.\]
When does equality hold?
1962 Miklós Schweitzer, 5
Let $ f$ be a finite real function of one variable. Let $ \overline{D}f$ and $ \underline{D}f$ be its upper and lower derivatives, respectively, that is, \[ \overline{D}f\equal{}\limsup_{{h,k\rightarrow 0}_{{h,k \geq 0}_{h\plus{}k>0}}} \frac{f(x\plus{}h)\minus{}f(x\minus{}k)}{h\plus{}k}\] ,
\[ \underline{D}f\equal{}\liminf_{{h,k\rightarrow 0}_{{h,k \geq 0}_{h\plus{}k>0}}} \frac{f(x\plus{}h)\minus{}f(x\minus{}k)}{h\plus{}k}.\] Show that $ \overline{D}f$ and $ \underline{D}f$ are Borel-measurable functions. [A. Csaszar]
1998 Greece National Olympiad, 4
Let a function $g:\mathbb{N}_0\to\mathbb{N}_0$ satisfy $g(0)=0$ and $g(n)=n-g(g(n-1))$ for all $n\ge 1$. Prove that:
a) $g(k)\ge g(k-1)$ for any positive integer $k$.
b) There is no $k$ such that $g(k-1)=g(k)=g(k+1)$.
1985 Traian Lălescu, 2.1
Let $ f:\mathbb{R}\longrightarrow\mathbb{R} $ be a bounded function in some neighbourhood of $ 0, $ such that there are three real numbers $ a>0, b>1, c $ with the property that
$$ f(ax)=bf(x)+c,\quad\forall x\in\mathbb{R} . $$
Show that $ f $ is continuous at $ 0 $ if and only if $ c=0. $
1999 Bosnia and Herzegovina Team Selection Test, 3
Let $f : [0,1] \rightarrow \mathbb{R}$ be injective function such that $f(0)+f(1)=1$. Prove that exists $x_1$, $x_2 \in [0,1]$, $x_1 \neq x_2$ such that $2f(x_1)<f(x_2)+\frac{1}{2}$. After that state at least one generalization of this result
2007 Grigore Moisil Intercounty, 3
Find the natural numbers $ a $ that have the property that there exists a function $ f:\mathbb{N}\longrightarrow\mathbb{N} $ such that $ f(f(n))=a+n, $ for any natural number $ n, $ and the function $ g:\mathbb{N}\longrightarrow\mathbb{N} $ defined as $ g(n)=f(n)-n $ is injective.
2015 USA TSTST, 5
Let $\varphi(n)$ denote the number of positive integers less than $n$ that are relatively prime to $n$. Prove that there exists a positive integer $m$ for which the equation $\varphi(n)=m$ has at least $2015$ solutions in $n$.
[i]Proposed by Iurie Boreico[/i]
2021 Turkey Team Selection Test, 8
Let \(c\) be a real number. For all \(x\) and \(y\) real numbers we have,
\[f(x-f(y))=f(x-y)+c(f(x)-f(y))\]
and \(f(x)\) is not constant.
\(a)\) Find all possible values of \(c\).
\(b)\) Can \(f\) be periodic?
2010 Brazil Team Selection Test, 4
Find all functions $f$ from the set of real numbers into the set of real numbers which satisfy for all $x$, $y$ the identity \[ f\left(xf(x+y)\right) = f\left(yf(x)\right) +x^2\]
[i]Proposed by Japan[/i]
2007 Korea Junior Math Olympiad, 6
Let $T = \{1,2,...,10\}$. Find the number of bijective functions $f : T\to T$ that satis es the following for all $x \in T$:
$f(f(x)) = x$
$|f(x) - x| \ge 2$
1987 IMO Longlists, 7
Let $f : (0,+\infty) \to \mathbb R$ be a function having the property that $f(x) = f\left(\frac{1}{x}\right)$ for all $x > 0.$ Prove that there exists a function $u : [1,+\infty) \to \mathbb R$ satisfying $u\left(\frac{x+\frac 1x }{2} \right) = f(x)$ for all $x > 0.$
1988 Flanders Math Olympiad, 4
Be $R$ a positive real number. If $R, 1, R+\frac12$ are triangle sides, call $\theta$ the angle between $R$ and $R+\frac12$ (in rad).
Prove $2R\theta$ is between $1$ and $\pi$.
2005 District Olympiad, 1
Let $a,b>1$ be two real numbers. Prove that $a>b$ if and only if there exists a function $f: (0,\infty)\to\mathbb{R}$ such that
i) the function $g:\mathbb{R}\to\mathbb{R}$, $g(x)=f(a^x)-x$ is increasing;
ii) the function $h:\mathbb{R}\to\mathbb{R}$, $h(x)=f(b^x)-x$ is decreasing.
2007 IMAC Arhimede, 2
Let $ABCD$ be a parallelogram that is not rhombus. We draw the symmetrical half-line of $(DC$ with respect to line $BD$. Similarly we draw the symmetrical half- line of $(AB$ with respect to $AC$. These half- lines intersect each other in $P$. If $\frac{AP}{DP}= q$ find the value of $\frac{AC}{BD}$ in function of $q$.
2013 China Girls Math Olympiad, 1
Let $A$ be the closed region bounded by the following three lines in the $xy$ plane: $x=1, y=0$ and $y=t(2x-t)$, where $0<t<1$. Prove that the area of any triangle inside the region $A$, with two vertices $P(t,t^2)$ and $Q(1,0)$, does not exceed $\frac{1}{4}.$