Found problems: 4776
2002 Switzerland Team Selection Test, 5
Find all $f: R\rightarrow R$ such that
(i) The set $\{\frac{f(x)}{x}| x\in R-\{0\}\}$ is finite
(ii) $f(x-1-f(x)) = f(x)-1-x$ for all $x$
2005 Today's Calculation Of Integral, 35
Determine the value of $a,b$ for which $\int_0^1 (\sqrt{1-x}-ax-b)^2 dx$ is minimized.
2022-IMOC, A4
Let the set of all bijective functions taking positive integers to positive integers be $\mathcal B.$ Find all functions $\mathbf F:\mathcal B\to \mathbb R$ such that $$(\mathbf F(p)+\mathbf F(q))^2=\mathbf F(p \circ p)+\mathbf F(p\circ q)+\mathbf F(q\circ p)+\mathbf F(q\circ q)$$ for all $p,q \in \mathcal B.$
[i]Proposed by ckliao914[/i]
1990 Federal Competition For Advanced Students, P2, 4
For each nonzero integer $ n$ find all functions $ f: \mathbb{R} \minus{} \{\minus{}3,0 \} \rightarrow \mathbb{R}$ satisfying:
$ f(x\plus{}3)\plus{}f \left( \minus{}\frac{9}{x} \right)\equal{}\frac{(1\minus{}n)(x^2\plus{}3x\minus{}9)}{9n(x\plus{}3)}\plus{}\frac{2}{n}$ for all $ x \not\equal{} 0,\minus{}3.$
Furthermore, for each fixed $ n$ find all integers $ x$ for which $ f(x)$ is an integer.
2001 Bundeswettbewerb Mathematik, 2
For a sequence $ a_i \in \mathbb{R}, i \in \{0, 1, 2, \ldots\}$ we have $ a_0 \equal{} 1$ and \[ a_{n\plus{}1} \equal{} a_n \plus{} \sqrt{a_{n\plus{}1} \plus{} a_n} \quad \forall n \in \mathbb{N}.\] Prove that this sequence is unique and find an explicit formula for this recursively defined sequence.
2018 Poland - Second Round, 1
Determine all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ which satisfy conditions:
$f(x) + f(y) \ge xy$ for all real $x, y$ and
for each real $x$ exists real $y$, such that $f(x) + f(y) = xy$.
2018 Ramnicean Hope, 3
Consider two positive real numbers $ a,b $ and the function $ f:(0,\infty )\longrightarrow\left( \sqrt{ab} ,\frac{a+b}{2} \right) $ defined as $ f(x)=-x+\sqrt{x^2+(a+b)x+ab}. $ Prove that it's bijective.
[i]D.M. Bătineți-Giurgiu[/i] and [i]Neculai Stanciu[/i]
2007 Today's Calculation Of Integral, 184
(1) For real numbers $x,\ a$ such that $0<x<a,$ prove the following inequality.
\[\frac{2x}{a}<\int_{a-x}^{a+x}\frac{1}{t}\ dt<x\left(\frac{1}{a+x}+\frac{1}{a-x}\right). \]
(2) Use the result of $(1)$ to prove that $0.68<\ln 2<0.71.$
2008 Harvard-MIT Mathematics Tournament, 7
Compute $ \sum_{n \equal{} 1}^\infty\sum_{k \equal{} 1}^{n \minus{} 1}\frac {k}{2^{n \plus{} k}}$.
2001 CentroAmerican, 2
Let $ a,b$ and $ c$ real numbers such that the equation $ ax^2\plus{}bx\plus{}c\equal{}0$ has two distinct real solutions $ p_1,p_2$ and the equation $ cx^2\plus{}bx\plus{}a\equal{}0$ has two distinct real solutions $ q_1,q_2$. We know that the numbers $ p_1,q_1,p_2,q_2$ in that order, form an arithmetic progression. Show that $ a\plus{}c\equal{}0$.
2007 Junior Balkan MO, 1
Let $a$ be positive real number such that $a^{3}=6(a+1)$. Prove that the equation $x^{2}+ax+a^{2}-6=0$ has no real solution.
2005 Today's Calculation Of Integral, 72
Let $f(x)$ be a continuous function satisfying $f(x)=1+k\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} f(t)\sin (x-t)dt\ (k:constant\ number)$
Find the value of $k$ for which $\int_0^{\pi} f(x)dx$ is maximized.
2002 Greece National Olympiad, 3
In a triangle $ABC$ we have $\angle C>10^0$ and $\angle B=\angle C+10^0.$We consider point $E$ on side $AB$ such that $\angle ACE=10^0,$ and point $D$ on side $AC$ such that $\angle DBA=15^0.$ Let $Z\neq A$ be a point of interection of the circumcircles of the triangles $ABD$ and $AEC.$Prove that $\angle ZBA>\angle ZCA.$
2012 Brazil National Olympiad, 6
Find all surjective functions $f\colon (0,+\infty) \to (0,+\infty)$ such that $2x f(f(x)) = f(x)(x+f(f(x)))$ for all $x>0$.
2007 China Team Selection Test, 1
Find all functions $ f: \mathbb{Q}^{\plus{}} \mapsto \mathbb{Q}^{\plus{}}$ such that:
\[ f(x) \plus{} f(y) \plus{} 2xy f(xy) \equal{} \frac {f(xy)}{f(x\plus{}y)}.\]
2014 Harvard-MIT Mathematics Tournament, 9
Given $a$, $b$, and $c$ are complex numbers satisfying
\[ a^2+ab+b^2=1+i \]
\[ b^2+bc+c^2=-2 \]
\[ c^2+ca+a^2=1, \]
compute $(ab+bc+ca)^2$. (Here, $i=\sqrt{-1}$)
2010 ELMO Shortlist, 1
For a permutation $\pi$ of $\{1,2,3,\ldots,n\}$, let $\text{Inv}(\pi)$ be the number of pairs $(i,j)$ with $1 \leq i < j \leq n$ and $\pi(i) > \pi(j)$.
[list=1]
[*] Given $n$, what is $\sum \text{Inv}(\pi)$ where the sum ranges over all permutations $\pi$ of $\{1,2,3,\ldots,n\}$?
[*] Given $n$, what is $\sum \left(\text{Inv}(\pi)\right)^2$ where the sum ranges over all permutations $\pi$ of $\{1,2,3,\ldots,n\}$?[/list]
[i]Brian Hamrick.[/i]
2014 Contests, 2
Does there exist a function $f: \mathbb R \to \mathbb R $ satisfying the following conditions:
(i) for each real $y$ there is a real $x$ such that $f(x)=y$ , and
(ii) $f(f(x)) = (x - 1)f(x) + 2$ for all real $x$ ?
[i]Proposed by Igor I. Voronovich, Belarus[/i]
2011 Iran MO (3rd Round), 7
Suppose that $f:P(\mathbb N)\longrightarrow \mathbb N$ and $A$ is a subset of $\mathbb N$. We call $f$ $A$-predicting if the set $\{x\in \mathbb N|x\notin A, f(A\cup x)\neq x \}$ is finite. Prove that there exists a function that for every subset $A$ of natural numbers, it's $A$-predicting.
[i]proposed by Sepehr Ghazi-Nezami[/i]
2004 Italy TST, 3
Find all functions $f:\mathbb{N}\rightarrow \mathbb{N}$ such that for all $m,n\in\mathbb{N}$,
\[(2^m+1)f(n)f(2^mn)=2^mf(n)^2+f(2^mn)^2+(2^m-1)^2n. \]
1984 IMO Longlists, 38
Determine all continuous functions $f: \mathbb R \to \mathbb R$ such that
\[f(x + y)f(x - y) = (f(x)f(y))^2, \quad \forall(x, y) \in\mathbb{R}^2.\]
2010 AIME Problems, 4
Jackie and Phil have two fair coins and a third coin that comes up heads with probability $ \frac47$. Jackie flips the three coins, and then Phil flips the three coins. Let $ \frac{m}{n}$ be the probability that Jackie gets the same number of heads as Phil, where $ m$ and $ n$ are relatively prime positive integers. Find $ m \plus{} n$.
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 $|x|\leq1$.
2012 Today's Calculation Of Integral, 788
For a function $f(x)=\ln (1+\sqrt{1-x^2})-\sqrt{1-x^2}-\ln x\ (0<x<1)$, answer the following questions:
(1) Find $f'(x)$.
(2) Sketch the graph of $y=f(x)$.
(3) Let $P$ be a mobile point on the curve $y=f(x)$ and $Q$ be a point which is on the tangent at $P$ on the curve $y=f(x)$ and such that $PQ=1$. Note that the $x$-coordinate of $Q$ is les than that of $P$. Find the locus of $Q$.
2014 Kazakhstan National Olympiad, 2
$\mathbb{Q}$ is set of all rational numbers. Find all functions $f:\mathbb{Q}\times\mathbb{Q}\rightarrow\mathbb{Q}$ such that for all $x$, $y$, $z$ $\in\mathbb{Q}$ satisfy
$f(x,y)+f(y,z)+f(z,x)=f(0,x+y+z)$