Found problems: 4776
2013 District Olympiad, 3
Take the function $f:\mathbb{R}\to \mathbb{R}$, $f\left( x \right)=ax,x\in \mathbb{Q},f\left( x \right)=bx,x\in \mathbb{R}\backslash \mathbb{Q}$, where $a$ and $b$ are two real numbers different from 0.
Prove that $f$ is injective if and only if $f$ is surjective.
2013 Romanian Master of Mathematics, 4
Suppose two convex quadrangles in the plane $P$ and $P'$, share a point $O$ such that, for every line $l$ trough $O$, the segment along which $l$ and $P$ meet is longer then the segment along which $l$ and $P'$ meet. Is it possible that the ratio of the area of $P'$ to the area of $P$ is greater then $1.9$?
2003 USA Team Selection Test, 4
Let $\mathbb{N}$ denote the set of positive integers. Find all functions $f: \mathbb{N} \to \mathbb{N}$ such that \[ f(m+n)f(m-n) = f(m^2) \] for $m,n \in \mathbb{N}$.
2009 International Zhautykov Olympiad, 1
On the plane, a Cartesian coordinate system is chosen. Given points $ A_1,A_2,A_3,A_4$ on the parabola $ y \equal{} x^2$, and points $ B_1,B_2,B_3,B_4$ on the parabola $ y \equal{} 2009x^2$. Points $ A_1,A_2,A_3,A_4$ are concyclic, and points $ A_i$ and $ B_i$ have equal abscissas for each $ i \equal{} 1,2,3,4$.
Prove that points $ B_1,B_2,B_3,B_4$ are also concyclic.
2018 Mathematical Talent Reward Programme, MCQ: P 2
$\lim _{x \rightarrow 0^{+}} \frac{[x]}{\tan x}$ where $[x]$ is the greatest integer function
[list=1]
[*] -1
[*] 0
[*] 1
[*] Does not exists
[/list]
2010 Today's Calculation Of Integral, 575
For a function $ f(x)\equal{}\int_x^{\frac{\pi}{4}\minus{}x} \log_4 (1\plus{}\tan t)dt\ \left(0\leq x\leq \frac{\pi}{8}\right)$, answer the following questions.
(1) Find $ f'(x)$.
(2) Find the $ n$ th term of the sequence $ a_n$ such that $ a_1\equal{}f(0),\ a_{n\plus{}1}\equal{}f(a_n)\ (n\equal{}1,\ 2,\ 3,\ \cdots)$.
2013 NIMO Problems, 6
Let $ABC$ be a triangle with $AB = 42$, $AC = 39$, $BC = 45$. Let $E$, $F$ be on the sides $\overline{AC}$ and $\overline{AB}$ such that $AF = 21, AE = 13$. Let $\overline{CF}$ and $\overline{BE}$ intersect at $P$, and let ray $AP$ meet $\overline{BC}$ at $D$. Let $O$ denote the circumcenter of $\triangle DEF$, and $R$ its circumradius. Compute $CO^2-R^2$.
[i]Proposed by Yang Liu[/i]
2012 Postal Coaching, 3
Given an integer $n\ge 2$, prove that
\[\lfloor \sqrt n \rfloor + \lfloor \sqrt[3]n\rfloor + \cdots +\lfloor \sqrt[n]n\rfloor = \lfloor \log_2n\rfloor + \lfloor \log_3n\rfloor + \cdots +\lfloor \log_nn\rfloor\].
[hide="Edit"] Thanks to shivangjindal for pointing out the mistake (and sorry for the late edit)[/hide]
2019 Olympic Revenge, 5
Define $f: \mathbb{N} \rightarrow \mathbb{N}$ by $$f(n) = \sum \frac{(1+\sum_{i=1}^{n} t_i)!}{(1+t_1) \cdot \prod_{i=1}^{n} (t_i!) }$$
where the sum runs through all $n$-tuples such that $\sum_{j=1}^{n}j \cdot t_j=n$ and $t_j \ge 0$ for all $1 \le j \le n$.
Given a prime $p$ greater than $3$, prove that $$\sum_{1 \le i < j <k \le p-1 } \frac{f(i)}{i \cdot j \cdot k} \equiv \sum_{1 \le i < j <k \le p-1 } \frac{2^i}{i \cdot j \cdot k} \pmod{p}.$$
1995 Niels Henrik Abels Math Contest (Norwegian Math Olympiad) Round 2, 1
The numbers from 1 to 1996 are written down ------ 12345678910111213.... How many zeros are written?
A. 489
B. 699
C. 796
D. 996
E. None of these
2003 Alexandru Myller, 4
Find the differentiable functions $ f:\mathbb{R}_{\ge 0 }\longrightarrow\mathbb{R} $ that verify $ f(0)=0 $ and
$$ f'(x)=1/3\cdot f'\left( x/3 \right) +2/3\cdot f'\left( 2x/3 \right) , $$
for any nonnegative real number $ x. $
2006 Romania National Olympiad, 4
Let $a,b,c \in \left[ \frac 12, 1 \right]$. Prove that \[ 2 \leq \frac{ a+b}{1+c} + \frac{ b+c}{1+a} + \frac{ c+a}{1+b} \leq 3 . \]
[i]selected by Mircea Lascu[/i]
2019 China Team Selection Test, 5
Determine all functions $f: \mathbb{Q} \to \mathbb{Q}$ such that
$$f(2xy + \frac{1}{2}) + f(x-y) = 4f(x)f(y) + \frac{1}{2}$$
for all $x,y \in \mathbb{Q}$.
2009 Vietnam Team Selection Test, 1
Let $ a,b,c$ be positive numbers.Find $ k$ such that:
$ (k \plus{} \frac {a}{b \plus{} c})(k \plus{} \frac {b}{c \plus{} a})(k \plus{} \frac {c}{a \plus{} b}) \ge (k \plus{} \frac {1}{2})^3$
2012 Germany Team Selection Test, 3
Determine all pairs $(f,g)$ of functions from the set of real numbers to itself that satisfy \[g(f(x+y)) = f(x) + (2x + y)g(y)\] for all real numbers $x$ and $y$.
[i]Proposed by Japan[/i]
2016 Postal Coaching, 4
Find a real function $f : [0,\infty)\to \mathbb R$ such that $f(2x+1) = 3f(x)+5$, for all $x$ in $[0,\infty)$.
2008 Balkan MO Shortlist, A2
Is there a sequence $ a_1,a_2,\ldots$ of positive reals satisfying simoultaneously the following inequalities for all positive integers $ n$:
a) $ a_1\plus{}a_2\plus{}\ldots\plus{}a_n\le n^2$
b) $ \frac1{a_1}\plus{}\frac1{a_2}\plus{}\ldots\plus{}\frac1{a_n}\le2008$?
2004 Romania National Olympiad, 1
Find all continuous functions $f : \mathbb R \to \mathbb R$ such that for all $x \in \mathbb R$ and for all $n \in \mathbb N^{\ast}$ we have \[ n^2 \int_{x}^{x + \frac{1}{n}} f(t) \, dt = n f(x) + \frac12 . \]
[i]Mihai Piticari[/i]
2024 IMC, 6
Prove that for any function $f:\mathbb{Q} \to \mathbb{Z}$, there exist $a,b,c \in \mathbb{Q}$ such that $a<b<c$, $f(b) \ge f(a)$ and $f(b) \ge f(c)$.
2008 Romanian Master of Mathematics, 2
Prove that every bijective function $ f: \mathbb{Z}\rightarrow\mathbb{Z}$ can be written in the way $ f\equal{}u\plus{}v$ where $ u,v: \mathbb{Z}\rightarrow\mathbb{Z}$ are bijective functions.
2003 Iran MO (2nd round), 2
In a village, there are $n$ houses with $n>2$ and all of them are not collinear. We want to generate a water resource in the village. For doing this, point $A$ is [i]better[/i] than point $B$ if the sum of the distances from point $A$ to the houses is less than the sum of the distances from point $B$ to the houses. We call a point [i]ideal[/i] if there doesn’t exist any [i]better[/i] point than it. Prove that there exist at most $1$ [i]ideal[/i] point to generate the resource.
2007 Today's Calculation Of Integral, 234
For $ x\geq 0,$ define a function $ f(x)\equal{}\sin \left(\frac{n\pi}{4}\right)\sin x\ (n\pi \leq x<(n\plus{}1)\pi )\ (n\equal{}0,\ 1,\ 2,\ \cdots)$.
Evaluate $ \int_0^{100\pi } f(x)\ dx.$
2007 ISI B.Math Entrance Exam, 7
Let $ 0\leq \theta\leq \frac{\pi}{2}$ . Prove that $\sin \theta \geq \frac{2\theta}{\pi}$.
1996 IMC, 7
Prove that if $f:[0,1]\rightarrow[0,1]$ is a continuous function, then the sequence of iterates $x_{n+1}=f(x_{n})$ converges if and only if
$$\lim_{n\to \infty}(x_{n+1}-x_{n})=0$$
1998 Austrian-Polish Competition, 3
Find all pairs of real numbers $(x, y)$ satisfying the following system of
equations
$2-x^{3}=y, 2-y^{3}=x$.