Found problems: 4776
2010 Romania National Olympiad, 3
Let $f:\mathbb{R}\rightarrow [0,\infty)$. Prove that $f(x+y)\ge (y+1)f(x),\ (\forall)x\in \mathbb{R}$ if and only if the function $g:\mathbb{R}\rightarrow [0,\infty),\ g(x)=e^{-x}f(x),\ (\forall)x\in \mathbb{R}$ is increasing.
2007 JBMO Shortlist, 2
Let $ABCD$ be a convex quadrilateral with $\angle{DAC}= \angle{BDC}= 36^\circ$ , $\angle{CBD}= 18^\circ$ and $\angle{BAC}= 72^\circ$. The diagonals and intersect at point $P$ . Determine the measure of $\angle{APD}$.
2021 China Second Round Olympiad, Problem 5
Define the regions $M, N$ in the Cartesian Plane as follows:
\begin{align*}
M &= \{(x, y) \in \mathbb R^2 \mid 0 \leq y \leq \text{min}(2x, 3-x)\} \\
N &= \{(x, y) \in \mathbb R^2 \mid t \leq x \leq t+2 \}
\end{align*}
for some real number $t$. Denote the common area of $M$ and $N$ for some $t$ be $f(t)$. Compute the algebraic form of the function $f(t)$ for $0 \leq t \leq 1$.
[i](Source: China National High School Mathematics League 2021, Zhejiang Province, Problem 5)[/i]
2012 India IMO Training Camp, 3
Let $f:\mathbb{R}\longrightarrow \mathbb{R}$ be a function such that $f(x+y+xy)=f(x)+f(y)+f(xy)$ for all $x, y\in\mathbb{R}$. Prove that $f$ satisfies $f(x+y)=f(x)+f(y)$ for all $x, y\in\mathbb{R}$.
1977 Miklós Schweitzer, 9
Suppose that the components of he vector $ \textbf{u}=(u_0,\ldots,u_n)$ are real functions defined on the closed interval $ [a,b]$ with the property that every nontrivial linear combination of them has at most $ n$ zeros in $ [a,b]$. Prove that if $ \sigma$ is an increasing function on $ [a,b]$ and the rank of the operator \[ A(f)= \int_{a}^b \textbf{u}(x)f(x)d\sigma(x), \;f \in C[a,b]\ ,\] is $ r \leq n$, then $ \sigma$ has exactly $ r$ points of increase.
[i]E. Gesztelyi[/i]
2005 Today's Calculation Of Integral, 33
Evaluate
\[\int_{-\ln 2}^0\ \frac{dx}{\cos ^2 h x \cdot \sqrt{1-2a\tanh x +a^2}}\ (a>0)\]
2020 Bangladesh Mathematical Olympiad National, Problem 6
$f$ is a one-to-one function from the set of positive integers to itself such that $$f(xy) = f(x) × f(y)$$ Find the minimum possible value of $f(2020)$.
2009 Indonesia TST, 1
Ati has $ 7$ pots of flower, ordered in $ P_1,P_2,P_3,P_4,P_5,P_6,P_7$. She wants to rearrange the position of those pots to $ B_1,B_2,B_2,B_3,B_4,B_5,B_6,B_7$ such that for every positive integer $ n<7$, $ B_1,B_2,\dots,B_n$ is not the permutation of $ P_1,P_2,\dots,P_7$. In how many ways can Ati do this?
1989 China Team Selection Test, 1
Let $\mathbb{N} = \{1,2, \ldots\}.$ Does there exists a function $f: \mathbb{N} \mapsto \mathbb{N}$ such that $\forall n \in \mathbb{N},$ $f^{1989}(n) = 2 \cdot n$ ?
2016 Mathematical Talent Reward Programme, SAQ: P 5
Let $\mathbb{N}$ be the set of all positive integers. $f,g:\mathbb{N} \to \mathbb{N}$ be funtions such that $f$ is onto and $g$ is one-one and $f(n)\geq g(n)$ for all positive integers $n$. Prove that $f=g$.
2014 ELMO Shortlist, 3
We say a finite set $S$ of points in the plane is [i]very[/i] if for every point $X$ in $S$, there exists an inversion with center $X$ mapping every point in $S$ other than $X$ to another point in $S$ (possibly the same point).
(a) Fix an integer $n$. Prove that if $n \ge 2$, then any line segment $\overline{AB}$ contains a unique very set $S$ of size $n$ such that $A, B \in S$.
(b) Find the largest possible size of a very set not contained in any line.
(Here, an [i]inversion[/i] with center $O$ and radius $r$ sends every point $P$ other than $O$ to the point $P'$ along ray $OP$ such that $OP\cdot OP' = r^2$.)
[i]Proposed by Sammy Luo[/i]
2014 Harvard-MIT Mathematics Tournament, 10
For an integer $n$, let $f_9(n)$ denote the number of positive integers $d\leq 9$ dividing $n$. Suppose that $m$ is a positive integer and $b_1,b_2,\ldots,b_m$ are real numbers such that $f_9(n)=\textstyle\sum_{j=1}^mb_jf_9(n-j)$ for all $n>m$. Find the smallest possible value of $m$.
2007 Putnam, 3
Let $ k$ be a positive integer. Suppose that the integers $ 1,2,3,\dots,3k \plus{} 1$ are written down in random order. What is the probability that at no time during this process, the sum of the integers that have been written up to that time is a positive integer divisible by $ 3$ ? Your answer should be in closed form, but may include factorials.
2000 USA Team Selection Test, 4
Let $n$ be a positive integer. Prove that
\[ \binom{n}{0}^{-1} + \binom{n}{1}^{-1} + \cdots + \binom{n}{n}^{-1} = \frac{n+1}{2^{n+1}} \left( \frac{2}{1} + \frac{2^2}{2} + \cdots + \frac{2^{n+1}}{n+1} \right). \]
2016 India IMO Training Camp, 2
Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that $$f\left(x^3+f(y)\right)=x^2f(x)+y,$$for all $x,y\in\mathbb{R}.$ (Here $\mathbb{R}$ denotes the set of all real numbers.)
2021 JHMT HS, 1
The value of $x$ in the interval $[0, 2\pi]$ that minimizes the value of $x + 2\cos x$ can be written in the form $a\pi/b,$ where $a$ and $b$ are relatively prime positive integers. Compute $a + b.$
Today's calculation of integrals, 886
Find the functions $f(x),\ g(x)$ such that
$f(x)=e^{x}\sin x+\int_0^{\pi} ug(u)\ du$
$g(x)=e^{x}\cos x+\int_0^{\pi} uf(u)\ du$
2014 PUMaC Algebra B, 6
There is a sequence with $a(2)=0$, $a(3)=1$ and $a(n)=a\left(\left\lfloor\dfrac n2\right\rfloor\right)+a\left(\left\lceil\dfrac n2\right\rceil\right)$ for $n\geq 4$. Find $a(2014)$. [Note that $\left\lfloor\dfrac n2\right\rfloor$ and $\left\lceil\dfrac n2\right\rceil$ denote the floor function (largest integer $\leq\tfrac n2$) and the ceiling function (smallest integer $\geq\tfrac n2$), respectively.]
2001 Tournament Of Towns, 1
A bus that moves along a 100 km route is equipped with a computer, which predicts how much more time is needed to arrive at its final destination. This prediction is made on the assumption that the average speed of the bus in the remaining part of the route is the same as that in the part already covered. Forty minutes after the departure of the bus, the computer predicts that the remaining travelling time will be 1 hour. And this predicted time remains the same for the next 5 hours. Could this possibly occur? If so, how many kilometers did the bus cover when these 5 hours passed? (Average speed is the number of kilometers covered divided by the time it took to cover them.)
1995 IMC, 12
Suppose that $(f_{n})_{n=1}^{\infty}$ is a sequence of continuous functions on the interval $[0,1]$ such that
$$\int_{0}^{1}f_{m}(x)f_{n}(x) dx=
\begin{cases}
1& \text{if}\;n=m\\
0 & \text{if} \;n\ne m
\end{cases}$$
and $\sup\{|f_{n}(x)|: x\in [0,1]\, \text{and}\, n=1,2,\dots\}< \infty$.
Show that there exists no subsequence $(f_{n_{k}})$ of $(f_{n})$ such that $\lim_{k\to \infty}f_{n_{k}}(x)$ exist
for all $x\in [0,1]$.
2008 ITest, 100
Let $\alpha$ be a root of $x^6-x-1$, and call two polynomials $p$ and $q$ with integer coefficients $\textit{equivalent}$ if $p(\alpha)\equiv q(\alpha)\pmod3$. It is known that every such polynomial is equivalent to exactly one of $0,1,x,x^2,\ldots,x^{727}$. Find the largest integer $n<728$ for which there exists a polynomial $p$ such that $p^3-p-x^n$ is equivalent to $0$.
2024 CMI B.Sc. Entrance Exam, 2
$g(x) \colon \int_{10}^{x} \log_{10}(\log_{10}(t^2-1000t+10^{1000})) dt$
(a) Find the domain of $g(x)$
(b) Approximate the value of $g(1000)$
(c) Find $x \in [10, 1000]$ to maximize the slope of $g(x)$
(d) Find $x \in [10, 1000]$ to minimize the slope of $g(x)$
(e) Determine, if it exists, $\lim_{x \to \infty} \frac{\ln(x)}{g(x)}$
2015 IFYM, Sozopol, 2
Find all functions $f$ from positive integers to themselves such that:
1)$f(mn)=f(m)f(n)$ for all positive integers $m, n$
2)$\{1, 2, ..., n\}=\{f(1), f(2), ... f(n)\}$ is true for infinitely many positive integers $n$.
2013 India Regional Mathematical Olympiad, 6
For a natural number $n$, let $T(n)$ denote the number of ways we can place $n$ objects of weights $1,2,\cdots, n$ on a balance such that the sum of the weights in each pan is the same. Prove that $T(100) > T(99)$.
2020 Romanian Master of Mathematics, 4
Let $\mathbb N$ be the set of all positive integers. A subset $A$ of $\mathbb N$ is [i]sum-free[/i] if, whenever $x$ and $y$ are (not necessarily distinct) members of $A$, their sum $x+y$ does not belong to $A$. Determine all surjective functions $f:\mathbb N\to\mathbb N$ such that, for each sum-free subset $A$ of $\mathbb N$, the image $\{f(a):a\in A\}$ is also sum-free.
[i]Note: a function $f:\mathbb N\to\mathbb N$ is surjective if, for every positive integer $n$, there exists a positive integer $m$ such that $f(m)=n$.[/i]