Found problems: 4776
2006 IMC, 5
Let $a, b, c, d$ three strictly positive real numbers such that \[a^{2}+b^{2}+c^{2}=d^{2}+e^{2},\] \[a^{4}+b^{4}+c^{4}=d^{4}+e^{4}.\] Compare \[a^{3}+b^{3}+c^{3}\] with \[d^{3}+e^{3},\]
2000 National Olympiad First Round, 28
$$\begin{array}{ rlrlrl}
f_1(x)=&x^2+x & f_2(x)=&2x^2-x & f_3(x)=&x^2 +x \\
g_1(x)=&x-2 & g_2(x)=&2x \ \ & g_3(x)=&x+2 \\
\end{array}$$
If $h(x)=x$ can be get from $f_i$ and $g_i$ by using only addition, substraction, multiplication defined on those functions where $i\in\{1,2,3\}$, then $F_i=1$. Otherwise, $F_i=0$. What is $(F_1,F_2,F_3)$ ?
$ \textbf{(A)}\ (0,0,0)
\qquad\textbf{(B)}\ (0,0,1)
\qquad\textbf{(C)}\ (0,1,0)
\qquad\textbf{(D)}\ (0,1,1)
\qquad\textbf{(E)}\ \text{None}
$
1996 AIME Problems, 10
Find the smallest positive integer solution to $\tan 19x^\circ=\frac{\cos 96^\circ+\sin 96^\circ}{\cos 96^\circ-\sin 96^\circ}.$
2014 China Team Selection Test, 2
Given a fixed positive integer $a\geq 9$. Prove: There exist finitely many positive integers $n$, satisfying:
(1)$\tau (n)=a$
(2)$n|\phi (n)+\sigma (n)$
Note: For positive integer $n$, $\tau (n)$ is the number of positive divisors of $n$, $\phi (n)$ is the number of positive integers $\leq n$ and relatively prime with $n$, $\sigma (n)$ is the sum of positive divisors of $n$.
2017 IMO Shortlist, A4
A sequence of real numbers $a_1,a_2,\ldots$ satisfies the relation
$$a_n=-\max_{i+j=n}(a_i+a_j)\qquad\text{for all}\quad n>2017.$$
Prove that the sequence is bounded, i.e., there is a constant $M$ such that $|a_n|\leq M$ for all positive integers $n$.
1985 Traian Lălescu, 1.2
Is there a real interval $ I $ for which there exists a primitivable function $ f:I\longrightarrow I $ with the property that $ (f\circ f) (x)=-x, $ for all $ x\in I $ ?
2004 Putnam, B3
Determine all real numbers $a>0$ for which there exists a nonnegative continuous function $f(x)$ defined on $[0,a]$ with the property that the region
$R=\{(x,y): 0\le x\le a, 0\le y\le f(x)\}$
has perimeter $k$ units and area $k$ square units for some real number $k$.
2008 IMS, 2
Let $ f$ be an entire function on $ \mathbb C$ and $ \omega_1,\omega_2$ are complex numbers such that $ \frac {\omega_1}{\omega_2}\in{\mathbb C}\backslash{\mathbb Q}$. Prove that if for each $ z\in \mathbb C$, $ f(z) \equal{} f(z \plus{} \omega_1) \equal{} f(z \plus{} \omega_2)$ then $ f$ is constant.
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 $n$
2005 China Team Selection Test, 2
Let $n$ be a positive integer, and $x$ be a positive real number. Prove that $$\sum_{k=1}^{n} \left( x \left[\frac{k}{x}\right] - (x+1)\left[\frac{k}{x+1}\right]\right) \leq n,$$ where $[x]$ denotes the largest integer not exceeding $x$.
2025 Iran MO (2nd Round), 5
Find all functions $f:\mathbb{R}^+ \to \mathbb{R}$ such that for all $x,y,z>0$
$$
3(x^3+y^3+z^3)\geq f(x+y+z)\cdot f(xy+yz+xz) \geq (x+y+z)(xy+yz+xz).
$$
2010 Postal Coaching, 7
Does there exist a function $f : \mathbb{N} \rightarrow \mathbb{N}$ such that for every $n \ge 2$,
\[f (f (n - 1)) = f (n + 1) - f (n)?\]
2008 Purple Comet Problems, 15
For natural number $n$, define the function $f(n)$ to be the number you get by $f(n)$ adding the digits of the number $n$. For example, $f(16)=7$, $f(f(78))=6$, and $f(f(f(5978)))=2$.
Find the least natural number $n$ such that $f(f(f(n)))$ is not a one-digit number.
2015 AIME Problems, 13
Define the sequence $a_1,a_2,a_3,\ldots$ by $a_n=\sum_{k=1}^n\sin(k)$, where $k$ represents radian measure. Find the index of the $100$th term for which $a_n<0$.
2010 Today's Calculation Of Integral, 533
Let $ C$ be the circle with radius 1 centered on the origin. Fix the endpoint of the string with length $ 2\pi$ on the point $ A(1,\ 0)$ and put the other end point $ P$ on the point $ P_0(1,\ 2\pi)$. From this situation, when we twist the string around $ C$ by moving the point $ P$ in anti clockwise with the string streched tightly, find the length of the curve that the point $ P$ draws from sarting point $ P_0$ to reaching point $ A$.
PEN M Problems, 13
The sequence $\{x_{n}\}$ is defined by \[x_{0}\in [0, 1], \; x_{n+1}=1-\vert 1-2 x_{n}\vert.\] Prove that the sequence is periodic if and only if $x_{0}$ is irrational.
2013 Online Math Open Problems, 1
Determine the value of $142857 + 285714 + 428571 + 571428.$
[i]Proposed by Ray Li[/i]
1994 Hong Kong TST, 2
Given that, a function $f(n)$, defined on the natural numbers, satisfies the following conditions: (i)$f(n)=n-12$ if $n>2000$; (ii)$f(n)=f(f(n+16))$ if $n \leq 2000$.
(a) Find $f(n)$.
(b) Find all solutions to $f(n)=n$.
2002 SNSB Admission, 3
Classify up to homeomorphism the topological spaces of the support of functions that are real quadratic polynoms of three variables and and irreducible over the set of real numbers.
2010 Contests, 2
For a positive integer $n$, we define the function $f_n(x)=\sum_{k=1}^n |x-k|$ for all real numbers $x$. For any two-digit number $n$ (in decimal representation), determine the set of solutions $\mathbb{L}_n$ of the inequality $f_n(x)<41$.
[i](41st Austrian Mathematical Olympiad, National Competition, part 1, Problem 2)[/i]
2018 Poland - Second Round, 1
Determine all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ which satisfy conditions:
$f(x) + f(y) \ge xy$ for all real $x, y$ and
for each real $x$ exists real $y$, such that $f(x) + f(y) = xy$.
2007 Nicolae Coculescu, 3
Let $ F:\mathbb{R}\longrightarrow\mathbb{R} $ be a primitive with $ F(0)=0 $ of the function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ defined as $ f(x)=\sin (x^2) , $ and let be a sequence $ \left( a_n \right)_{n\ge 0} $ with $ a_0\in (0,1) $ and defined as $ a_{n}=a_{n-1}-F\left( a_{n-1} \right) . $
Calculate $ \lim_{n\to\infty } a_n\sqrt{n} . $
[i]Florian Dumitrel[/i]
2009 Germany Team Selection Test, 3
Find all functions $ f: \mathbb{R} \mapsto \mathbb{R}$ such that $ \forall x,y,z \in \mathbb{R}$ we have: If
\[ x^3 \plus{} f(y) \cdot x \plus{} f(z) \equal{} 0,\]
then
\[ f(x)^3 \plus{} y \cdot f(x) \plus{} z \equal{} 0.\]
2018 District Olympiad, 1
Find all strictly increasing functions $f : \mathbb{N} \to \mathbb{N} $ such that $\frac {f(x) + f(y)}{1 + f(x + y)}$ is a non-zero natural number, for all $x, y\in\mathbb{N}$.
2018 Korea Junior Math Olympiad, 1
Let $f$ be a quadratic function which satisfies the following condition. Find the value of $\frac{f(8)-f(2)}{f(2)-f(1)}$.
For two distinct real numbers $a,b$, if $f(a)=f(b)$, then $f(a^2-6b-1)=f(b^2+8)$.