Found problems: 4776
2006 China Team Selection Test, 2
$x_{1}, x_{2}, \cdots, x_{n}$ are positive numbers such that $\sum_{i=1}^{n}x_{i}= 1$. Prove that \[\left( \sum_{i=1}^{n}\sqrt{x_{i}}\right) \left( \sum_{i=1}^{n}\frac{1}{\sqrt{1+x_{i}}}\right) \leq \frac{n^{2}}{\sqrt{n+1}}\]
2002 Poland - Second Round, 1
Prove that all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ satisfying, for all real $x$,
\[ f(x)=f(2x)=f(1-x)\]
are periodic.
2006 China Team Selection Test, 3
Given $n$ real numbers $a_1$, $a_2$ $\ldots$ $a_n$. ($n\geq 1$). Prove that there exists real numbers $b_1$, $b_2$ $\ldots$ $b_n$ satisfying:
(a) For any $1 \leq i \leq n$, $a_i - b_i$ is a positive integer.
(b)$\sum_{1 \leq i < j \leq n} (b_i - b_j)^2 \leq \frac{n^2-1}{12}$
1992 IMO Longlists, 24
[i](a)[/i] Show that there exists exactly one function $ f : \mathbb Q^+ \to \mathbb Q^+$ satisfying the following conditions:
[b](i)[/b] if $0 < q < \frac 12$, then $f(q)=1+f \left( \frac{q}{1-2q} \right);$
[b](ii)[/b] if $1 < q \leq 2$, then $f(q) = 1+f(q + 1);$
[b](iii)[/b] $f(q)f(1/q) = 1$ for all $q \in \mathbb Q^+.$
[i](b)[/i] Find the smallest rational number $q \in \mathbb Q^+$ such that $f(q) = \frac{19}{92}.$
2009 CHKMO, 2
Let $ n>4$ be a positive integer such that $ n$ is composite (not a prime) and divides $ \varphi (n) \sigma (n) \plus{}1$, where $ \varphi (n)$ is the Euler's totient function of $ n$ and $ \sigma (n)$ is the sum of the positive divisors of $ n$. Prove that $ n$ has at least three distinct prime factors.
2018 District Olympiad, 1
Let $\mathcal{F}$ be the set of continuous functions $f : [0, 1]\to\mathbb{R}$ satisfying $\max_{0\le x\le 1} |f(x)| = 1$ and let $I : \mathcal{F} \to \mathbb{R}$,
\[I(f) = \int_0^1 f(x)\, \text{d}x - f(0) + f(1).\]
a) Show that $I(f) < 3$, for any $f \in \mathcal{F}$.
b) Determine $\sup\{I(f) \mid f \in \mathcal{F}\}$.
1969 Canada National Olympiad, 8
Let $f$ be a function with the following properties:
1) $f(n)$ is defined for every positive integer $n$;
2) $f(n)$ is an integer;
3) $f(2)=2$;
4) $f(mn)=f(m)f(n)$ for all $m$ and $n$;
5) $f(m)>f(n)$ whenever $m>n$.
Prove that $f(n)=n$.
2007 IMC, 1
Let $ f : \mathbb{R}\to \mathbb{R}$ be a continuous function. Suppose that for any $ c > 0$, the graph of $ f$ can be moved to the graph of $ cf$ using only a translation or a rotation. Does this imply that $ f(x) = ax+b$ for some real numbers $ a$ and $ b$?
2014 JHMMC 7 Contest, 3
Let $a\# b$ be defined as $ab-a-3$. For example, $4\#5=20-4-3=13$ Compute $(2\#0)\#(1\#4)$.
2007 QEDMO 5th, 6
Find all functions $ f: \mathbb{R}\to\mathbb{R}$ that satisfy the equation:
$ f\left(\left(f\left(x\right)\right)^2 \plus{} f\left(y\right)\right) \equal{} xf\left(x\right) \plus{} y$
for any two real numbers $ x$ and $ y$.
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 $\angle{APD}$.
2012 Serbia Team Selection Test, 1
Let $P(x)$ be a polynomial of degree $2012$ with real coefficients satisfying the condition \[P(a)^3 + P(b)^3 + P(c)^3 \geq 3P(a)P(b)P(c),\] for all real numbers $a,b,c$ such that $a+b+c=0$. Is it possible for $P(x)$ to have exactly $2012$ distinct real roots?
1984 AMC 12/AHSME, 11
A calculator has a key which replaces the displayed entry with its square, and another key which replaces the displayed entry with its reciprocal. Let $y$ be the final result if one starts with an entry $x \neq 0$ and alternately squares and reciprocates $n$ times each. Assuming the calculator is completely accurate (e.g., no roundoff or overflow), then $y$ equals
A. $x^{((-2)^n)}$
B. $x^{2n}$
C. $x^{-2n}$
D. $x^{-(2^n)}$
E. $x^{((-1)^n 2n)}$
PEN G Problems, 27
Let $1<a_{1}<a_{2}<\cdots$ be a sequence of positive integers. Show that \[\frac{2^{a_{1}}}{{a_{1}}!}+\frac{2^{a_{2}}}{{a_{2}}!}+\frac{2^{a_{3}}}{{a_{3}}!}+\cdots\] is irrational.
2003 China Team Selection Test, 3
Let $x_0+\sqrt{2003}y_0$ be the minimum positive integer root of Pell function $x^2-2003y^2=1$. Find all the positive integer solutions $(x,y)$ of the equation, such that $x_0$ is divisible by any prime factor of $x$.
1985 Miklós Schweitzer, 9
Let $D=\{ z\in \mathbb C\colon |z|<1\}$ and $D=\{ w\in \mathbb C \colon |w|=1\}$. Prove that if for a function $f\colon D\times B\rightarrow\mathbb C$ the equality
$$f\left( \frac{az+b}{\overline{b}z+\overline{a}}, \frac{aw+b}{\overline{b}w+\overline a} \right)=f(z,w)+f\left(\frac{b}{\overline a}, \frac{aw+b}{\overline b w+\overline a} \right)$$
holds for all $z\in D, w\in B$ and $a, b\in \mathbb C,|a|^2=|b|^2+1$, then there is a function $L\colon (0, \infty )\rightarrow \mathbb C$ satisfying
$$L(pq)=L(p)+L(q)\,\,\,\text{for all}\,\,\, p,q > 0$$
such that $f$ can be represented as
$$f(z,w)=L\left( \frac{1-|z|^2}{|w-z|^2}\right)\,\,\,\text{for all}\,\,\, z\in D, w\in B$$.
[Gy. Maksa]
2010 Germany Team Selection Test, 3
Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that
\[f(x)f(y) = (x+y+1)^2 \cdot f \left( \frac{xy-1}{x+y+1} \right)\] $\forall x,y \in \mathbb{R}$ with $x+y+1 \neq 0$ and $f(x) > 1$ $\forall x > 0.$
2009 Putnam, A3
Let $ d_n$ be the determinant of the $ n\times n$ matrix whose entries, from left to right and then from top to bottom, are $ \cos 1,\cos 2,\dots,\cos n^2.$ (For example, $ d_3 \equal{} \begin{vmatrix}\cos 1 & \cos2 & \cos3 \\
\cos4 & \cos5 & \cos 6 \\
\cos7 & \cos8 & \cos 9\end{vmatrix}.$ The argument of $ \cos$ is always in radians, not degrees.)
Evaluate $ \lim_{n\to\infty}d_n.$
2012 Singapore MO Open, 2
Find all functions $f:\mathbb{R}\to\mathbb{R}$ so that $(x+y)(f(x)-f(y))=(x-y)f(x+y)$ for all $x,y$ that belongs to $\mathbb{R}$.
2009 Tournament Of Towns, 4
Denote by $[n]!$ the product $ 1 \cdot 11 \cdot 111\cdot ... \cdot \underbrace{111...1}_{\text{n ones}}$.($n$ factors in total). Prove that $[n + m]!$ is divisible by $ [n]! \times [m]!$
[i](8 points)[/i]
2007 Harvard-MIT Mathematics Tournament, 19
Define $x\star y=\frac{\sqrt{x^2+3xy+y^2-2x-2y+4}}{xy+4}$. Compute \[((\cdots ((2007\star 2006)\star 2005)\star\cdots )\star 1).\]
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 $f(b) \ge f(c)$.
2006 Iran Team Selection Test, 4
Let $x_1,x_2,\ldots,x_n$ be real numbers. Prove that
\[ \sum_{i,j=1}^n |x_i+x_j|\geq n\sum_{i=1}^n |x_i| \]
1962 Miklós Schweitzer, 5
Let $ f$ be a finite real function of one variable. Let $ \overline{D}f$ and $ \underline{D}f$ be its upper and lower derivatives, respectively, that is, \[ \overline{D}f\equal{}\limsup_{{h,k\rightarrow 0}_{{h,k \geq 0}_{h\plus{}k>0}}} \frac{f(x\plus{}h)\minus{}f(x\minus{}k)}{h\plus{}k}\] ,
\[ \underline{D}f\equal{}\liminf_{{h,k\rightarrow 0}_{{h,k \geq 0}_{h\plus{}k>0}}} \frac{f(x\plus{}h)\minus{}f(x\minus{}k)}{h\plus{}k}.\] Show that $ \overline{D}f$ and $ \underline{D}f$ are Borel-measurable functions. [A. Csaszar]
1998 USAMO, 3
Let $a_0,a_1,\cdots ,a_n$ be numbers from the interval $(0,\pi/2)$ such that \[ \tan (a_0-\frac{\pi}{4})+ \tan (a_1-\frac{\pi}{4})+\cdots +\tan (a_n-\frac{\pi}{4})\geq n-1. \] Prove that \[ \tan a_0\tan a_1 \cdots \tan a_n\geq n^{n+1}. \]