Found problems: 4776
2023 India IMO Training Camp, 3
Let $Q$ be a set of prime numbers, not necessarily finite. For a positive integer $n$ consider its prime factorization: define $p(n)$ to be the sum of all the exponents and $q(n)$ to be the sum of the exponents corresponding only to primes in $Q$. A positive integer $n$ is called [i]special[/i] if $p(n)+p(n+1)$ and $q(n)+q(n+1)$ are both even integers. Prove that there is a constant $c>0$ independent of the set $Q$ such that for any positive integer $N>100$, the number of special integers in $[1,N]$ is at least $cN$.
(For example, if $Q=\{3,7\}$, then $p(42)=3$, $q(42)=2$, $p(63)=3$, $q(63)=3$, $p(2022)=3$, $q(2022)=1$.)
2010 Gheorghe Vranceanu, 2
Let be a natural number $ n, $ a nonzero number $ \alpha, \quad n $ numbers $ a_1,a_2,\ldots ,a_n $ and $ n+1 $ functions $ f_0,f_1,f_2,\ldots ,f_n $ such that $ f_0=\alpha $ and the rest are defined recursively as
$$ f_k (x)=a_k+\int_0^x f_{k-1} (x)dx . $$
Prove that if all these functions are everywhere nonnegative, then the sum of all these functions is everywhere nonnegative.
2010 Today's Calculation Of Integral, 638
Let $(a,\ b)$ be a point on the curve $y=\frac{x}{1+x}\ (x\geq 0).$ Denote $U$ the volume of the figure enclosed by the curve , the $x$ axis and the line $x=a$, revolved around the the $x$ axis and denote $V$ the volume of the figure enclosed by the curve , the $y$ axis and th line $y=b$, revolved around the $y$ axis. What's the relation of $U$ and $V?$
1978 Chuo university entrance exam/Science and Technology
2005 Iran MO (3rd Round), 4
Suppose $P,Q\in \mathbb R[x]$ that $deg\ P=deg\ Q$ and $PQ'-QP'$ has no real root. Prove that for each $\lambda \in \mathbb R$ number of real roots of $P$ and $\lambda P+(1-\lambda)Q$ are equal.
2014 AMC 12/AHSME, 21
For every real number $x$, let $\lfloor x\rfloor$ denote the greatest integer not exceeding $x$, and let \[f(x)=\lfloor x\rfloor(2014^{x-\lfloor x\rfloor}-1).\] The set of all numbers $x$ such that $1\leq x<2014$ and $f(x)\leq 1$ is a union of disjoint intervals. What is the sum of the lengths of those intervals?
$\textbf{(A) }1\qquad
\textbf{(B) }\dfrac{\log 2015}{\log 2014}\qquad
\textbf{(C) }\dfrac{\log 2014}{\log 2013}\qquad
\textbf{(D) }\dfrac{2014}{2013}\qquad
\textbf{(E) }2014^{\frac1{2014}}\qquad$
2021 Miklós Schweitzer, 3
Let $I \subset \mathbb{R}$ be a nonempty open interval and let $f: I \cap \mathbb{Q} \to \mathbb{R}$ be a function such that for all $x, y \in I \cap \mathbb{Q}$,
\[ 4f\left(\frac{3x + y}{4}\right)+ 4f\left(\frac{x + 3y}{4}\right) \le f(x) + 6f\left(\frac{x + y}{2}\right)+ f(y). \] Show that $f$ can be continuously extended to $I$.
2000 IMC, 6
Let $f: \mathbb{R}\rightarrow ]0,+\infty[$ be an increasing differentiable function with $\lim_{x\rightarrow+\infty}f(x)=+\infty$ and $f'$ is bounded, and let $F(x)=\int^x_0 f(t) dt$.
Define the sequence $(a_n)$ recursively by $a_0=1,a_{n+1}=a_n+\frac1{f(a_n)}$
Define the sequence $(b_n)$ by $b_n=F^{-1}(n)$.
Prove that $\lim_{x\rightarrow+\infty}(a_n-b_n)=0$.
2013 Stars Of Mathematics, 1
Let $\mathcal{F}$ be the family of bijective increasing functions $f\colon [0,1] \to [0,1]$, and let $a \in (0,1)$. Determine the best constants $m_a$ and $M_a$, such that for all $f \in \mathcal{F}$ we have
\[m_a \leq f(a) + f^{-1}(a) \leq M_a.\]
[i](Dan Schwarz)[/i]
2022 Iran-Taiwan Friendly Math Competition, 2
Find all functions $f:\mathbb{R} \rightarrow \mathbb{R}$ such that:
$\bullet$ $f(x)<2$ for all $x\in (0,1)$;
$\bullet$ for all real numbers $x,y$ we have:
$$max\{f(x+y),f(x-y)\}=f(x)+f(y)$$
Proposed by Navid Safaei
2006 Federal Competition For Advanced Students, Part 2, 1
Let $ N$ be a positive integer. How many non-negative integers $ n \le N$ are there that have an integer multiple, that only uses the digits $ 2$ and $ 6$ in decimal representation?
2010 Today's Calculation Of Integral, 666
Let $f(x)$ be a function defined in $0<x<\frac{\pi}{2}$ satisfying:
(i) $f\left(\frac{\pi}{6}\right)=0$
(ii) $f'(x)\tan x=\int_{\frac{\pi}{6}}^x \frac{2\cos t}{\sin t}dt$.
Find $f(x)$.
[i]1987 Sapporo Medical University entrance exam[/i]
2005 China Second Round Olympiad, 2
Assume that positive numbers $a, b, c, x, y, z$ satisfy $cy + bz = a$, $az + cx = b$, and $bx + ay = c$. Find the minimum value of the function \[ f(x, y, z) = \frac{x^2}{x+1} + \frac {y^2}{y+1} + \frac{z^2}{z+1}. \]
2003 China Team Selection Test, 1
Find all functions $f: \mathbb{Z}^+\to \mathbb{R}$, which satisfies $f(n+1)\geq f(n)$ for all $n\geq 1$ and $f(mn)=f(m)f(n)$ for all $(m,n)=1$.
2020 Italy National Olympiad, #5
Le $S$ be the set of positive integers greater than or equal to $2$. A function $f: S\rightarrow S$ is italian if $f$ satifies all the following three conditions:
1) $f$ is surjective
2) $f$ is increasing in the prime numbers(that is, if $p_1<p_2$ are prime numbers, then $f(p_1)<f(p_2)$)
3) For every $n\in S$ the number $f(n)$ is the product of $f(p)$, where $p$ varies among all the primes which divide $n$ (For instance, $f(360)=f(2^3\cdot 3^2\cdot 5)=f(2)\cdot f(3)\cdot f(5)$).
Determine the maximum and the minimum possible value of $f(2020)$, when $f$ varies among all italian functions.
2007 Harvard-MIT Mathematics Tournament, 6
The elliptic curve $y^2=x^3+1$ is tangent to a circle centered at $(4,0)$ at the point $(x_0,y_0)$. Determine the sum of all possible values of $x_0$.
2010 Iran Team Selection Test, 12
Prove that for each natural number $m$, there is a natural number $N$ such that for each $b$ that $2\leq b\leq1389$ sum of digits of $N$ in base $b$ is larger than $m$.
2011 Iran MO (3rd Round), 4
For positive real numbers $a,b$ and $c$ we have $a+b+c=3$. Prove
$\frac{a}{1+(b+c)^2}+\frac{b}{1+(a+c)^2}+\frac{c}{1+(a+b)^2}\le \frac{3(a^2+b^2+c^2)}{a^2+b^2+c^2+12abc}$.
[i]proposed by Mohammad Ahmadi[/i]
2003 China Team Selection Test, 3
Suppose $A\subset \{(a_1,a_2,\dots,a_n)\mid a_i\in \mathbb{R},i=1,2\dots,n\}$. For any $\alpha=(a_1,a_2,\dots,a_n)\in A$ and $\beta=(b_1,b_2,\dots,b_n)\in A$, we define
\[ \gamma(\alpha,\beta)=(|a_1-b_1|,|a_2-b_2|,\dots,|a_n-b_n|), \] \[ D(A)=\{\gamma(\alpha,\beta)\mid\alpha,\beta\in A\}. \] Please show that $|D(A)|\geq |A|$.
1967 IMO Shortlist, 3
The function $\varphi(x,y,z)$ defined for all triples $(x,y,z)$ of real numbers, is such that there are two functions $f$ and $g$ defined for all pairs of real numbers, such that
\[\varphi(x,y,z) = f(x+y,z) = g(x,y+z)\]
for all real numbers $x,y$ and $z.$ Show that there is a function $h$ of one real variable, such that
\[\varphi(x,y,z) = h(x+y+z)\]
for all real numbers $x,y$ and $z.$
2025 Romania National Olympiad, 3
Prove that, for a function $f \colon \mathbb{R} \to \mathbb{R}$, the following $2$ statements are equivalent:
a) $f$ is differentiable, with continuous first derivative.
b) For any $a\in\mathbb{R}$ and for any two sequences $(x_n)_{n\geq 1},(y_n)_{n\geq 1}$, convergent to $a$, such that $x_n \neq y_n$ for any positive integer $n$, the sequence $\left(\frac{f(x_n)-f(y_n)}{x_n-y_n}\right)_{n\geq 1}$ is convergent.
1973 Miklós Schweitzer, 6
If $ f$ is a nonnegative, continuous, concave function on the closed interval $ [0,1]$ such that $ f(0)=1$, then \[ \int_0^1 xf(x)dx \leq \frac 23 \left[ %Error. "diaplaymath" is a bad command.
\int_0^1 f(x)dx \right]^2.\]
[i]Z. Daroczy[/i]
2016 Balkan MO, 1
Find all injective functions $f: \mathbb R \rightarrow \mathbb R$ such that for every real number $x$ and every positive integer $n$,$$ \left|\sum_{i=1}^n i\left(f(x+i+1)-f(f(x+i))\right)\right|<2016$$
[i](Macedonia)[/i]
2010 Saudi Arabia BMO TST, 4
Let $f : N \to [0, \infty)$ be a function satisfying the following conditions:
a) $f(4)=2$
b) $\frac{1}{f( 0 ) + f( 1)} + \frac{1}{f( 1 ) + f( 2 )} + ... + \frac{1}{f (n ) + f(n + 1) }= f ( n + 1)$ for all integers $n \ge 0$.
Find $f(n)$ in closed form.
1991 Dutch Mathematical Olympiad, 3
A real function $ f$ satisfies $ 4f(f(x))\minus{}2f(x)\minus{}3x\equal{}0$ for all real numbers $ x$. Prove that $ f(0)\equal{}0$.
2011 USAMTS Problems, 2
Four siblings are sitting down to eat some mashed potatoes for lunch: Ethan has 1 ounce of mashed potatoes, Macey has 2 ounces, Liana has 4 ounces, and Samuel has 8 ounces. This is not fair. A blend consists of choosing any two children at random, combining their plates of mashed potatoes, and then giving each of those two children half of the combination. After the children's father performs four blends consecutively, what is the probability that the four children will all have the same amount of mashed potatoes?