Found problems: 4776
2013 Brazil Team Selection Test, 2
Determine all positive integers $n$ for which $\dfrac{n^2+1}{[\sqrt{n}]^2+2}$ is an integer. Here $[r]$ denotes the greatest integer less than or equal to $r$.
2019 Macedonia National Olympiad, 4
Determine all functions $f: \mathbb {N} \to \mathbb {N}$ such that
$n!\hspace{1mm} +\hspace{1mm} f(m)!\hspace{1mm} |\hspace{1mm} f(n)!\hspace{1mm} +\hspace{1mm} f(m!)$,
for all $m$, $n$ $\in$ $\mathbb{N}$.
2017 Serbia Team Selection Test, 3
A function $f:\mathbb{N} \rightarrow \mathbb{N} $ is called nice if $f^a(b)=f(a+b-1)$, where $f^a(b)$ denotes $a$ times applied function $f$.
Let $g$ be a nice function, and an integer $A$ exists such that $g(A+2018)=g(A)+1$.
a) Prove that $g(n+2017^{2017})=g(n)$ for all $n \geq A+2$.
b) If $g(A+1) \neq g(A+1+2017^{2017})$ find $g(n)$ for $n <A$.
2008 ISI B.Stat Entrance Exam, 1
Of all triangles with given perimeter, find the triangle with the maximum area. Justify your answer
2010 Today's Calculation Of Integral, 623
Find the continuous function satisfying the following equation.
\[\int_0^x f(t)dt+\int_0^x tf(x-t)dt=e^{-x}-1.\]
[i]1978 Shibaura Institute of Technology entrance exam[/i]
2014 Taiwan TST Round 1, 1
Find all increasing functions $f$ from the nonnegative integers to the integers satisfying $f(2)=7$ and \[ f(mn) = f(m) + f(n) + f(m)f(n) \] for all nonnegative integers $m$ and $n$.
2012 NIMO Problems, 8
The positive integer-valued function $f(n)$ satisfies $f(f(n)) = 4n$ and $f(n + 1) > f(n) > 0$ for all positive integers $n$. Compute the number of possible 16-tuples $(f(1), f(2), f(3), \dots, f(16))$.
[i]Proposed by Lewis Chen[/i]
2003 IMC, 5
Let $g:[0,1]\rightarrow \mathbb{R}$ be a continuous function and let $f_{n}:[0,1]\rightarrow \mathbb{R}$ be a
sequence of functions defined by $f_{0}(x)=g(x)$ and
$$f_{n+1}(x)=\frac{1}{x}\int_{0}^{x}f_{n}(t)dt.$$
Determine $\lim_{n\to \infty}f_{n}(x)$ for every $x\in (0,1]$.
2023 Philippine MO, 8
Let $\mathcal{S}$ be the set of all points in the plane. Find all functions $f : \mathcal{S} \rightarrow \mathbb{R}$ such that for all nondegenerate triangles $ABC$ with orthocenter $H$, if $f(A) \leq f(B) \leq f(C)$, then $$f(A) + f(C) = f(B) + f(H).$$
2023 Switzerland Team Selection Test, 12
Let $m,n \geqslant 2$ be integers, let $X$ be a set with $n$ elements, and let $X_1,X_2,\ldots,X_m$ be pairwise distinct non-empty, not necessary disjoint subset of $X$. A function $f \colon X \to \{1,2,\ldots,n+1\}$ is called [i]nice[/i] if there exists an index $k$ such that \[\sum_{x \in X_k} f(x)>\sum_{x \in X_i} f(x) \quad \text{for all } i \ne k.\] Prove that the number of nice functions is at least $n^n$.
2014 USAMTS Problems, 3a:
A group of people is lined up in [i]almost-order[/i] if, whenever person $A$ is to the left of person $B$ in the line, $A$ is not more than $8$ centimeters taller than $B$. For example, five people with heights $160, 165, 170, 175$, and $180$ centimeters could line up in almost-order with heights (from left-to-right) of $160, 170, 165, 180, 175$ centimeters.
(a) How many different ways are there to line up $10$ people in [i]almost-order[/i] if their heights are $140, 145, 150, 155,$ $160,$ $165,$ $170,$ $175,$ $180$, and $185$ centimeters?
2010 Purple Comet Problems, 20
Suppose that $f$ is a function such that $3f(x)- 5xf \left(\frac{1}{x}\right)= x - 7$ for all non-zero real numbers $x.$ Find $f(2010).$
2023 Israel TST, P1
Find all functions $f:\mathbb{R}\to \mathbb{R}$ such that for all $x, y\in \mathbb{R}$ the following holds:
\[f(x)+f(y)=f(xy)+f(f(x)+f(y))\]
1979 Miklós Schweitzer, 8
Let $ K_n(n=1,2,\ldots)$ be periodical continuous functions of period $ 2 \pi$, and write \[ k_n(f;x)= \int_0^{2\pi}f(t)K_n(x-t)dt .\] Prove that the following statements are equivalent:
(i) $ \int_0^{2\pi}|k_n(f;x)-f(x)|dx \rightarrow 0 \;(n \rightarrow \infty)$ for all $ f \in \mathcal{L}_1[0,2 \pi]$.
(ii) $ k_n(f;0) \rightarrow f(0)$ for all continuous, $ 2 \pi$-periodic functions $ f$.
[i]V. Totik[/i]
2014 Singapore Senior Math Olympiad, 4
Find the smallest number among the following numbers:
$ \textbf{(A) }\sqrt{55}-\sqrt{52}\qquad\textbf{(B) }\sqrt{56}-\sqrt{53}\qquad\textbf{(C) }\sqrt{77}-\sqrt{74}\qquad\textbf{(D) }\sqrt{88}-\sqrt{85}\qquad\textbf{(E) }\sqrt{70}-\sqrt{67} $
2005 Harvard-MIT Mathematics Tournament, 10
Let $ f : \mathbf{R} \to \mathbf{R} $ be a smooth function such that $f'(x)=f(1-x)$ for all $x$ and $f(0)=1$. Find $f(1)$.
2017 Romanian Masters In Mathematics, 4
In the Cartesian plane, let $G_1$ and $G_2$ be the graphs of the quadratic functions $f_1(x) = p_1x^2 + q_1x + r_1$ and $f_2(x) = p_2x^2 + q_2x + r_2$, where $p_1 > 0 > p_2$. The graphs $G_1$ and $G_2$ cross at distinct points $A$ and $B$. The four tangents to $G_1$ and $G_2$ at $A$ and $B$ form a convex quadrilateral which has an inscribed circle. Prove that the graphs $G_1$ and $G_2$ have the same axis of symmetry.
2005 National Olympiad First Round, 15
For how many positive real numbers $a$ has the equation $a^2x^2 + ax+1-7a^2 = 0$ two distinct integer roots?
$
\textbf{(A)}\ 1
\qquad\textbf{(B)}\ 2
\qquad\textbf{(C)}\ 3
\qquad\textbf{(D)}\ \text{Infinitely many}
\qquad\textbf{(E)}\ \text{None of above}
$
1946 Putnam, A1
Suppose that the function $f(x)=a x^2 +bx+c$, where $a,b,c$ are real, satisfies the condition $|f(x)|\leq 1$ for $|x|\leq1$. Prove that $|f'(x)|\leq 4$ for $|x|\leq1$.
2016 Azerbaijan Team Selection Test, 3
Prove that there does not exist a function $f : \mathbb R^+\to\mathbb R^+$ such that \[f(f(x)+y)=f(x)+3x+yf(y)\] for all positive reals $x,y$.
1992 Iran MO (2nd round), 3
Let $X \neq \varnothing$ be a finite set and let $f: X \to X$ be a function such that for every $x \in X$ and a fixed prime $p$ we have $f^p(x)=x.$ Let $Y=\{x \in X | f(x) \neq x\}.$ Prove that the number of the members of the set $Y$ is divisible by $p.$
[i]Note.[/i] ${f^p(x)=x = \underbrace{f(f(f(\cdots ((f}_{ p \text{ times}}(x) ) \cdots )))} .$
2019 Iran Team Selection Test, 5
Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that for all $x,y\in \mathbb{R}$:
$$f\left(f(x)^2-y^2\right)^2+f(2xy)^2=f\left(x^2+y^2\right)^2$$
[i]Proposed by Ali Behrouz - Mojtaba Zare Bidaki[/i]
2018 Switzerland - Final Round, 5
Does there exist any function $f: \mathbb{R}^+ \to \mathbb{R}$ such that for every positive real number $x,y$ the following is true :
$$f(xf(x)+yf(y)) = xy$$
2018 Olympic Revenge, 5
Let $p$ a positive prime number and $\mathbb{F}_{p}$ the set of integers $mod \ p$. For $x\in \mathbb{F}_{p}$, define $|x|$ as the cyclic distance of $x$ to $0$, that is, if we represent $x$ as an integer between $0$ and $p-1$, $|x|=x$ if $x<\frac{p}{2}$, and $|x|=p-x$ if $x>\frac{p}{2}$ . Let $f: \mathbb{F}_{p} \rightarrow \mathbb{F}_{p}$ a function such that for every $x,y \in \mathbb{F}_{p}$
\[ |f(x+y)-f(x)-f(y)|<100 \]
Prove that exist $m \in \mathbb{F}_{p}$ such that for every $x \in \mathbb{F}_{p}$
\[ |f(x)-mx|<1000 \]
2004 Nicolae Coculescu, 4
Let be a function satisfying [url=http://mathworld.wolfram.com/CauchyFunctionalEquation.html]Cauchy's functional equation,[/url] and having the property that it's monotonic on a real interval. Prove that this function is globally monotonic.
[i]Florian Dumitrel[/i]