Found problems: 4776
2021 AMC 12/AHSME Spring, 18
Let $f$ be a function defined on the set of positive rational numbers with the property that $f(a\cdot b)=f(a)+f(b)$ for all positive rational numbers $a$ and $b$. Suppose that $f$ also has the property that $f(p)=p$ for every prime number $p$. For which of the following numbers $x$ is $f(x)<0?$
$\textbf{(A) } \frac{17}{32} \qquad \textbf{(B) } \frac{11}{16} \qquad \textbf{(C) } \frac{7}{9} \qquad \textbf{(D) } \frac{7}{6} \qquad \textbf{(E) } \frac{25}{11}$
2025 AIME, 15
Let
\[f(x)=\frac{(x-18)(x-72)(x-98)(x-k)}{x}.\]
There exist exactly three positive real values of $k$ such that $f$ has a minimum at exactly two real values of $x$. Find the sum of these three values of $k$.
2001 Rioplatense Mathematical Olympiad, Level 3, 3
For every integer $n > 1$, the sequence $\left( {{S}_{n}} \right)$ is defined by ${{S}_{n}}=\left\lfloor {{2}^{n}}\underbrace{\sqrt{2+\sqrt{2+...+\sqrt{2}}}}_{n\ radicals} \right\rfloor $
where $\left\lfloor x \right\rfloor$ denotes the floor function of $x$. Prove that ${{S}_{2001}}=2\,{{S}_{2000}}+1$.
.
2009 Harvard-MIT Mathematics Tournament, 5
Compute \[\lim_{h\to 0}\dfrac{\sin(\frac{\pi}{3}+4h)-4\sin(\frac{\pi}{3}+3h)+6\sin(\frac{\pi}{3}+2h)-4\sin(\frac{\pi}{3}+h)+\sin(\frac{\pi}{3})}{h^4}.\]
2012 Macedonia National Olympiad, 3
Find all functions $f : \mathbb{R} \to \mathbb{Z}$ which satisfy the conditions:
$f(x+y) < f(x) + f(y)$
$f(f(x)) = \lfloor {x} \rfloor + 2$
2021 Thailand TSTST, 2
Let $f:\mathbb{R}^+\to\mathbb{R}^+$ be such that $$f(x+f(y))^2\geq f(x)\left(f(x+f(y))+f(y)\right)$$ for all $x,y\in\mathbb{R}^+$. Show that $f$ is [i]unbounded[/i], i.e. for each $M\in\mathbb{R}^+$, there exists $x\in\mathbb{R}^+$ such that $f(x)>M$.
2000 Miklós Schweitzer, 1
Prove that there exists a function $f\colon [\omega_1]^2 \rightarrow \omega _1$ such that
(i) $f(\alpha, \beta)< \mathrm{min}(\alpha, \beta)$ whenever $\mathrm{min}(\alpha,\beta)>0$; and
(ii) if $\alpha_0<\alpha_1<\ldots<\alpha_i<\ldots<\omega_1$ then $\sup\left\{ a_i \colon i<\omega \right\} =\sup \left\{ f(\alpha_i, \alpha_j)\colon i,j<\omega\right\}$.
2001 IMO Shortlist, 1
Let $ T$ denote the set of all ordered triples $ (p,q,r)$ of nonnegative integers. Find all functions $ f: T \rightarrow \mathbb{R}$ satisfying
\[ f(p,q,r) = \begin{cases} 0 & \text{if} \; pqr = 0, \\
1 + \frac{1}{6}(f(p + 1,q - 1,r) + f(p - 1,q + 1,r) & \\
+ f(p - 1,q,r + 1) + f(p + 1,q,r - 1) & \\
+ f(p,q + 1,r - 1) + f(p,q - 1,r + 1)) & \text{otherwise} \end{cases}
\]
for all nonnegative integers $ p$, $ q$, $ r$.
1990 USAMO, 1
A certain state issues license plates consisting of six digits (from 0 to 9). The state requires that any two license plates differ in at least two places. (For instance, the numbers 027592 and 020592 cannot both be used.) Determine, with proof, the maximum number of distinct license plates that the state can use.
2008 Moldova Team Selection Test, 2
Let $ a_1,\ldots,a_n$ be positive reals so that $ a_1\plus{}a_2\plus{}\ldots\plus{}a_n\le\frac n2$. Find the minimal value of
$ \sqrt{a_1^2\plus{}\frac1{a_2^2}}\plus{}\sqrt{a_2^2\plus{}\frac1{a_3^2}}\plus{}\ldots\plus{}\sqrt{a_n^2\plus{}\frac1{a_1^2}}$.
PEN K Problems, 18
Find all functions $f: \mathbb{Q}\to \mathbb{R}$ such that for all $x,y\in \mathbb{Q}$: \[f(xy)=f(x)f(y)-f(x+y)+1.\]
2008 SEEMOUS, Problem 1
Let $f:[1,\infty)\to(0,\infty)$ be a continuous function. Assume that for every $a>0$, the equation $f(x)=ax$ has at least one solution in the interval $[1,\infty)$.
(a) Prove that for every $a>0$, the equation $f(x)=ax$ has infinitely many solutions.
(b) Give an example of a strictly increasing continuous function $f$ with these properties.
2016 Poland - Second Round, 3
Determine, whether exists function $f$, which assigns each integer $k$, nonnegative integer $f(k)$ and meets the conditions:
$f(0) > 0$,
for each integer $k$ minimal number of the form $f(k - l) + f(l)$, where $l \in \mathbb{Z}$, equals $f(k)$.
2008 Bulgaria Team Selection Test, 3
Let $\mathbb{R}^{+}$ be the set of positive real numbers. Find all real numbers $a$ for which there exists a function $f :\mathbb{R}^{+} \to \mathbb{R}^{+}$ such that $3(f(x))^{2}=2f(f(x))+ax^{4}$, for all $x \in \mathbb{R}^{+}$.
2009 District Olympiad, 1
Let $ f:[0,\infty )\longrightarrow [0,\infty ) $ a nonincreasing function that satisfies the inequality:
$$ \int_0^x f(t)dt <1,\quad\forall x\ge 0. $$ Prove the following affirmations:
[b]a)[/b] $ \exists \lim_{x\to\infty} \int_0^x f(t)dt \in\mathbb{R} . $
[b]b)[/b] $ \lim_{x\to\infty} xf(x) =0. $
2014 Iran MO (2nd Round), 3
Let $ x,y,z $ be three non-negative real numbers such that \[x^2+y^2+z^2=2(xy+yz+zx). \] Prove that \[\dfrac{x+y+z}{3} \ge \sqrt[3]{2xyz}.\]
PEN K Problems, 13
Find all functions $f: \mathbb{Z}\to \mathbb{Z}$ such that for all $m\in \mathbb{Z}$: \[f(f(m))=m+1.\]
2022-IMOC, A3
Find all functions $f:\mathbb R\to \mathbb R$ such that $$xy(f(x+y)-f(x)-f(y))=2f(xy)$$ for all $x,y\in \mathbb R.$
[i]Proposed by USJL[/i]
2009 Today's Calculation Of Integral, 397
In $ xy$ plane, find the minimum volume of the solid by rotating the region boubded by the parabola $ y \equal{} x^2 \plus{} ax \plus{} b$ passing through the point $ (1,\ \minus{} 1)$ and the $ x$ axis about the $ x$ axis
2014 Iran Team Selection Test, 2
is there a function $f:\mathbb{N}\rightarrow \mathbb{N}$ such that
$i) \exists n\in \mathbb{N}:f(n)\neq n$
$ii)$ the number of divisors of $m$ is $f(n)$ if and only if the number of divisors of $f(m)$ is $n$
2010 Brazil Team Selection Test, 2
Let $f$ be a non-constant function from the set of positive integers into the set of positive integer, such that $a-b$ divides $f(a)-f(b)$ for all distinct positive integers $a$, $b$. Prove that there exist infinitely many primes $p$ such that $p$ divides $f(c)$ for some positive integer $c$.
[i]Proposed by Juhan Aru, Estonia[/i]
2009 Ukraine National Mathematical Olympiad, 4
Find all functions $f : \mathbb R \to \mathbb R$ such that
\[f\left(x+xy+f(y)\right)= \left( f(x)+\frac 12 \right) \left( f(y)+\frac 12 \right) \qquad \forall x,y \in \mathbb R.\]
2018 CMIMC Number Theory, 6
Let $\phi(n)$ denote the number of positive integers less than or equal to $n$ that are coprime to $n$. Find the sum of all $1<n<100$ such that $\phi(n)\mid n$.
2007 Today's Calculation Of Integral, 243
A cubic funtion $ y \equal{} ax^3 \plus{} bx^2 \plus{} cx \plus{} d\ (a\neq 0)$ intersects with the line $ y \equal{} px \plus{} q$ at $ x \equal{} \alpha ,\ \beta ,\ \gamma\ (\alpha < \beta < \gamma).$ Find the area of the region bounded by these graphs in terms of $ a,\ \alpha ,\ \beta ,\ \gamma$.
2011 Morocco National Olympiad, 3
Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that for all $x,y, \in \mathbb{R}$,
\[xf(x+xy)=xf(x)+f(x^{2})\cdot f(y).\]