Found problems: 4776
2009 Putnam, A2
Functions $ f,g,h$ are differentiable on some open interval around $ 0$ and satisfy the equations and initial conditions
\begin{align*}f'&=2f^2gh+\frac1{gh},\ f(0)=1,\\
g'&=fg^2h+\frac4{fh},\ g(0)=1,\\
h'&=3fgh^2+\frac1{fg},\ h(0)=1.\end{align*}
Find an explicit formula for $ f(x),$ valid in some open interval around $ 0.$
2014 Tuymaada Olympiad, 4
Positive numbers $a,\ b,\ c$ satisfy $\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}=3$. Prove the inequality
\[\dfrac{1}{\sqrt{a^3+1}}+\dfrac{1}{\sqrt{b^3+1}}+\dfrac{1}{\sqrt{c^3+1}}\le \dfrac{3}{\sqrt{2}}. \]
[i](N. Alexandrov)[/i]
2002 National High School Mathematics League, 1
The increasing interval of $f(x)=\log_{\frac{1}{2}}(x^2-2x-3)$ is
$\text{(A)}(-\infty,-1)\qquad\text{(B)}(-\infty,1)\qquad\text{(C)}(1,+\infty)\qquad\text{(D)}(3,+\infty)$
1993 Romania Team Selection Test, 4
Let $Y$ be the family of all subsets of $X = \{1,2,...,n\}$ ($n > 1$) and let $f : Y \to X$ be an arbitrary mapping. Prove that there exist distinct subsets $A,B$ of $X$ such that $f(A) = f(B) = max A\triangle B$, where $A\triangle B = (A-B)\cup(B-A)$.
2007 ITest, 41
The sequence of digits \[123456789101112131415161718192021\ldots\] is obtained by writing the positive integers in order. If the $10^n$th digit in this sequence occurs in the part of the sequence in which the $m$-digit numbers are placed, define $f(n)$ to be $m$. For example, $f(2) = 2$ because the $100^{\text{th}}$ digit enters the sequence in the placement of the two-digit integer $55$. Find the value of $f(2007)$.
2008 Brazil Team Selection Test, 3
Let $ n$ be a positive integer, and let $ x$ and $ y$ be a positive real number such that $ x^n \plus{} y^n \equal{} 1.$ Prove that
\[ \left(\sum^n_{k \equal{} 1} \frac {1 \plus{} x^{2k}}{1 \plus{} x^{4k}} \right) \cdot \left( \sum^n_{k \equal{} 1} \frac {1 \plus{} y^{2k}}{1 \plus{} y^{4k}} \right) < \frac {1}{(1 \minus{} x) \cdot (1 \minus{} y)}.
\]
[i]Author: Juhan Aru, Estonia[/i]
2014 Nordic, 1
Find all functions ${ f : N \rightarrow N}$ (where ${N}$ is the set of the natural numbers and is assumed to contain ${0}$), such that ${f(x^2) - f(y^2) = f(x + y)f(x - y)}$ for all ${x, y \in N}$ with ${x \ge y}$.
2011 ELMO Shortlist, 2
Find all functions $f:\mathbb{R}^+\to\mathbb{R}^+$ such that whenever $a>b>c>d>0$ and $ad=bc$,
\[f(a+d)+f(b-c)=f(a-d)+f(b+c).\]
[i]Calvin Deng.[/i]
2006 IberoAmerican, 3
Consider a regular $n$-gon with $n$ odd. Given two adjacent vertices $A_{1}$ and $A_{2},$ define the sequence $(A_{k})$ of vertices of the $n$-gon as follows: For $k\ge 3,\, A_{k}$ is the vertex lying on the perpendicular bisector of $A_{k-2}A_{k-1}.$ Find all $n$ for which each vertex of the $n$-gon occurs in this sequence.
2012 Indonesia TST, 1
Find all functions $f : \mathbb{R} \rightarrow \mathbb{R}$ such that
\[f(x+y) + f(x)f(y) = f(xy) + (y+1)f(x) + (x+1)f(y)\]
for all $x,y \in \mathbb{R}$.
2020 Korea National Olympiad, 1
Determine all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that
$$x^2f(x)+yf(y^2)=f(x+y)f(x^2-xy+y^2)$$
for all $x,y\in\mathbb{R}$.
2004 Romania National Olympiad, 3
Let $f : (a,b) \to \mathbb R$ be a function with the property that for all $x \in (a,b)$ there is a non-degenerated interval $[ a_x,b_x ]$ with $a < a_x \leq x \leq b_x < b$ such that $f$ is constant on $\left[ a_x,b_x \right]$.
(a) Prove that $\textrm{Im} \, f$ is finite or numerable.
(b) Find all continuous functions which have the property mentioned in the hypothesis.
2002 China Team Selection Test, 2
For any two rational numbers $ p$ and $ q$ in the interval $ (0,1)$ and function $ f$, there is always $ \displaystyle f \left( \frac{p\plus{}q}{2} \right) \leq \frac{f(p) \plus{} f(q)}{2}$. Then prove that for any rational numbers $ \lambda, x_1, x_2 \in (0,1)$, there is always:
\[ f( \lambda x_1 \plus{} (1\minus{}\lambda) x_2 ) \leq \lambda f(x_i) \plus{} (1\minus{}\lambda) f(x_2)\]
2016 Fall CHMMC, 14
For a unit circle $O$, arrange points $A,B,C,D$ and $E$ in that order evenly along $O$'s circumference. For each of those points, draw the arc centered at that point inside O from the point to its left to the point to its right. Denote the outermost intersections of these arcs as $A', B', C', D'$ and $E'$, where the prime of any point is opposite the point. The length of $AC'$ can be written as an expression $f(x)$, where $f$ is a trigonometric function. Find this expression.
2000 China Team Selection Test, 2
[b]a.)[/b] Let $a,b$ be real numbers. Define sequence $x_k$ and $y_k$ such that
\[x_0 = 1, y_0 = 0, x_{k+1} = a \cdot x_k - b \cdot y_l, \quad y_{k+1} = x_k - a \cdot y_k \text{ for } k = 0,1,2, \ldots \]
Prove that
\[x_k = \sum^{[k/2]}_{l=0} (-1)^l \cdot a^{k - 2 \cdot l} \cdot \left(a^2 + b \right)^l \cdot \lambda_{k,l}\]
where $\lambda_{k,l} = \sum^{[k/2]}_{m=l} \binom{k}{2 \cdot m} \cdot \binom{m}{l}$
[b]b.)[/b] Let $u_k = \sum^{[k/2]}_{l=0} \lambda_{k,l} $. For positive integer $m,$ denote the remainder of $u_k$ divided by $2^m$ as $z_{m,k}$. Prove that $z_{m,k},$ $k = 0,1,2, \ldots$ is a periodic function, and find the smallest period.
1997 Greece National Olympiad, 1
Let $P$ be a point inside or on the boundary of a square $ABCD$. Find the minimum and maximum values of $f(P ) = \angle ABP + \angle BCP + \angle CDP + \angle DAP$.
1985 Vietnam National Olympiad, 2
Find all functions $ f \colon \mathbb{Z} \mapsto \mathbb{R}$ which satisfy:
i) $ f(x)f(y) \equal{} f(x \plus{} y) \plus{} f(x \minus{} y)$ for all integers $ x$, $ y$
ii) $ f(0) \neq 0$
iii) $ f(1) \equal{} \frac {5}{2}$
2010 Postal Coaching, 4
Five distinct points $A, B, C, D$ and $E$ lie in this order on a circle of radius $r$ and satisfy $AC = BD = CE = r$. Prove that the orthocentres of the triangles $ACD, BCD$ and $BCE$ are the vertices of a right-angled triangle.
1994 China Team Selection Test, 1
Given $5n$ real numbers $r_i, s_i, t_i, u_i, v_i \geq 1 (1 \leq i \leq n)$, let $R = \frac {1}{n} \sum_{i=1}^{n} r_i$, $S = \frac {1}{n} \sum_{i=1}^{n} s_i$, $T = \frac {1}{n} \sum_{i=1}^{n} t_i$, $U = \frac {1}{n} \sum_{i=1}^{n} u_i$, $V = \frac {1}{n} \sum_{i=1}^{n} v_i$. Prove that $\prod_{i=1}^{n}\frac {r_i s_i
t_i u_i v_i + 1}{r_i s_i t_i u_i v_i - 1} \geq \left(\frac {RSTUV +1}{RSTUV - 1}\right)^n$.
1989 IMO Longlists, 81
A real-valued function $ f$ on $ \mathbb{Q}$ satisfies the following conditions for arbitrary $ \alpha, \beta \in \mathbb{Q}:$
[b](i)[/b] $ f(0) \equal{} 0,$
[b](ii)[/b] $ f(\alpha) > 0 \text{ if } \alpha \neq 0,$
[b](iii)[/b] $ f(\alpha \cdot \beta) \equal{} f(\alpha)f(\beta),$
[b](iv)[/b] $ f(\alpha \plus{} \beta) \leq f(\alpha) \plus{} f(\beta),$
[b](v)[/b] $ f(m) \leq 1989$ $ \forall m \in \mathbb{Z}.$
Prove that \[ f(\alpha \plus{} \beta) \equal{} \max\{f(\alpha), f(\beta)\} \text{ if } f(\alpha) \neq f(\beta).\]
2022 Estonia Team Selection Test, 1
Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ that satisfy the following condition for any real numbers $x{}$ and $y$ $$f(x)+f(x+y) \leq f(xy)+f(y).$$
2011 Kosovo National Mathematical Olympiad, 2
It is given the function $f: \left(\mathbb{R} - \{0\}\right) \to \mathbb{R}$ such that $f(x)=x+\frac{1}{x}$.
Is this function injective ? Justify your answer.
2010 Laurențiu Panaitopol, Tulcea, 2
Let be a real number $ c $ and a differentiable function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ such that
$$ f(c)\neq \frac{1}{b-a}\int_a^b f(x)dx, $$
for any real numbers $ a\neq b. $
Prove that $ f'(c)=0. $
[i]Florin Rotaru[/i]
2005 Today's Calculation Of Integral, 79
Find the area of the domain expressed by the following system inequalities.
\[x\geq 0,\ y\geq 0,\ x^{\frac{1}{p}}+y^{\frac{1}{p}} \leq 1\ (p=1,2,\cdots)\]
2022 CCA Math Bonanza, TB4
Let $f(x)$ be a function such that $f(1) = 1234$, $f(2)=1800$, and $f(x) = f(x-1) + 2f(x-2)-1$ for all integers $x$. Evaluate the number of divisors of
\[\sum_{i=1}^{2022}f(i)\]
[i]2022 CCA Math Bonanza Tiebreaker Round #4[/i]