Found problems: 4776
2008 IMS, 3
Let $ A,B$ be different points on a parabola. Prove that we can find $ P_1,P_2,\dots,P_{n}$ between $ A,B$ on the parabola such that area of the convex polygon $ AP_1P_2\dots P_nB$ is maximum. In this case prove that the ratio of $ S(AP_1P_2\dots P_nB)$ to the sector between $ A$ and $ B$ doesn't depend on $ A$ and $ B$, and only depends on $ n$.
2019 Kosovo National Mathematical Olympiad, 4
Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that:
$$f(xy+f(x))=xf(y)$$
for all $x,y\in\mathbb{R}$.
1969 IMO Shortlist, 59
$(SWE 2)$ For each $\lambda (0 < \lambda < 1$ and $\lambda = \frac{1}{n}$ for all $n = 1, 2, 3, \cdots)$, construct a continuous function $f$ such that there do not exist $x, y$ with $0 < \lambda < y = x + \lambda \le 1$ for which $f(x) = f(y).$
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:
\[ f( \lambda x_1 \plus{} (1\minus{}\lambda) x_2 ) \leq \lambda f(x_i) \plus{} (1\minus{}\lambda) f(x_2)\]
2002 Olympic Revenge, 1
Show that there is no function \(f:\mathbb{N}^* \rightarrow \mathbb{N}^*\) such that \(f^n(n)=n+1\) for all \(n\) (when \(f^n\) is the \(n\)th iteration of \(f\))
2001 India IMO Training Camp, 2
Find all functions $f \colon \mathbb{R_{+}}\to \mathbb{R_{+}}$ satisfying : \[f ( f (x)-x) = 2x\] for all $x > 0$.
1998 All-Russian Olympiad, 8
A figure $\Phi$ composed of unit squares has the following property: if the squares of an $m \times n$ rectangle ($m,n$ are fixed) are filled with numbers whose sum is positive, the figure $\Phi$ can be placed within the rectangle (possibly after being rotated) so that the sum of the covered numbers is also positive. Prove that a number of such figures can be put on the $m\times n$ rectangle so that each square is covered by the same number of figures.
2003 Germany Team Selection Test, 1
At a chess tournament the winner gets 1 point and the defeated one 0 points. A tie makes both obtaining $\frac{1}{2}$ points. 14 players, none of them equally aged, participated in a competition where everybody played against all the other players. After the competition a ranking was carried out. Of the two players with the same number of points the younger received the better ranking. After the competition Jan realizes that the best three players together got as many points as the last 9 players obtained points together. And Joerg noted that the number of ties was maximal. Determine the number of ties.
2007 Hungary-Israel Binational, 1
You have to organize a fair procedure to randomly select someone from $ n$ people so that every one of them would be chosen with the probability $ \frac{1}{n}$. You are allowed to choose two real numbers $ 0<p_1<1$ and $ 0<p_2<1$ and order two coins which satisfy the following requirement: the probability of tossing "heads" on the first coin $ p_1$ and the probability of tossing "heads" on the second coin is $ p_2$. Before starting the procedure, you are supposed to announce an upper bound on the total number of times that the two coins are going to be flipped altogether. Describe a procedure that achieves this goal under the given conditions.
2003 China Western Mathematical Olympiad, 4
$ 1650$ students are arranged in $ 22$ rows and $ 75$ columns. It is known that in any two columns, the number of pairs of students in the same row and of the same sex is not greater than $ 11$. Prove that the number of boys is not greater than $ 928$.
1998 French Mathematical Olympiad, Problem 3
Let $k\ge2$ be an integer. The function $f:\mathbb N\to\mathbb N$ is defined by
$$f(n)=n+\left\lfloor\sqrt[k]{n+\sqrt[k]n}\right\rfloor.$$Determine the set of values taken by the function $f$.
2013 Today's Calculation Of Integral, 890
A function $f_n(x)\ (n=1,\ 2,\ \cdots)$ is defined by $f_1(x)=x$ and
\[f_n(x)=x+\frac{e}{2}\int_0^1 f_{n-1}(t)e^{x-t}dt\ (n=2,\ 3,\ \cdots)\].
Find $f_n(x)$.
2009 Today's Calculation Of Integral, 510
(1) Evaluate $ \int_0^{\frac{\pi}{2}} (x\cos x\plus{}\sin ^ 2 x)\sin x\ dx$.
(2) For $ f(x)\equal{}\int_0^x e^t\sin (x\minus{}t)\ dt$, find $ f''(x)\plus{}f(x)$.
2012 Online Math Open Problems, 22
Let $c_1,c_2,\ldots,c_{6030}$ be 6030 real numbers. Suppose that for any 6030 real numbers $a_1,a_2,\ldots,a_{6030}$, there exist 6030 real numbers $\{b_1,b_2,\ldots,b_{6030}\}$ such that \[a_n = \sum_{k=1}^{n} b_{\gcd(k,n)}\] and \[b_n = \sum_{d\mid n} c_d a_{n/d}\] for $n=1,2,\ldots,6030$. Find $c_{6030}$.
[i]Victor Wang.[/i]
2004 Harvard-MIT Mathematics Tournament, 8
If $x$ and $y$ are real numbers with $(x+y)^4=x-y$, what is the maximum possible value of $y$?
1976 Miklós Schweitzer, 4
Let $ \mathbb{Z}$ be the ring of rational integers. Construct an integral domain $ I$ satisfying the following conditions:
a)$ \mathbb{Z} \varsubsetneqq I$;
b) no element of $ I \minus{} \mathbb{Z}$ (only in $ I$) is algebraic over $ \mathbb{Z}$ (that is, not a root of a polynomial with coefficients in $ \mathbb{Z}$);
c) $ I$ only has trivial endomorphisms.
[i]E. Fried[/i]
2023 European Mathematical Cup, 4
Let $f\colon\mathbb{N}\rightarrow\mathbb{N}$ be a function such that for all positive integers $x$ and $y$, the number $f(x)+y$ is a perfect square if and only if $x+f(y)$ is a perfect square. Prove that $f$ is injective.
[i]Remark.[/i] A function $f\colon\mathbb{N}\rightarrow\mathbb{N}$ is injective if for all pairs $(x,y)$ of distinct positive integers, $f(x)\neq f(y)$ holds.
[i]Ivan Novak[/i]
1980 IMO Shortlist, 8
Three points $A,B,C$ are such that $B \in ]AC[$. On the side of $AC$ we draw the three semicircles with diameters $[AB], [BC]$ and $[AC]$. The common interior tangent at $B$ to the first two semi-circles meets the third circle in $E$. Let $U$ and $V$ be the points of contact of the common exterior tangent to the first two semi-circles. Denote the area of the triangle $ABC$ as $S(ABC)$. Evaluate the ratio $R=\frac{S(EUV)}{S(EAC)}$ as a function of $r_1 = \frac{AB}{2}$ and $r_2 = \frac{BC}{2}$.
2015 Indonesia MO Shortlist, A6
Let functions $f, g: \mathbb{R}^+ \to \mathbb{R}^+$ satisfy the following:
\[ f(g(x)y + f(x)) = (y+2015)f(x) \]
for every $x,y \in \mathbb{R}^+$.
(a) Prove that $g(x) = \frac{f(x)}{2015}$ for every $x \in \mathbb{R}^+. $
(b) State an example of function that satisfy the equation above and $f(x), g(x) \ge 1$ for every $x \in \mathbb{R}^+$.
1993 Poland - First Round, 2
The sequence of functions $f_0,f_1,f_2,...$ is given by the conditions:
$f_0(x) = |x|$ for all $x \in R$
$f_{n+1}(x) = |f_n(x)-2|$ for $n=0,1,2,...$ and all $x \in R$.
For each positive integer $n$, solve the equation $f_n(x)=1$.
2017 District Olympiad, 2
[b]a)[/b] Prove that there exist two functions $ f,g:\mathbb{R}\longrightarrow\mathbb{R} $ having the properties:
$ \text{(i)}\quad f\circ g=g\circ f $
$\text{(ii)}\quad f\circ f=g\circ g $
$ \text{(iii)}\quad f(x)\neq g(x), \quad \forall x\in\mathbb{R} $
[b]b)[/b] Show that if there are two functions $ f_1,g_1:\mathbb{R}\longrightarrow\mathbb{R} $ with the properties $ \text{(i)} $ and $ \text{(iii)} $ from above, then $ \left( f_1\circ f_1\right)(x) \neq \left( g_1\circ g_1 \right)(x) , $ for all real numbers $ x. $
2007 IberoAmerican, 1
Given an integer $ m$, define the sequence $ \left\{a_{n}\right\}$ as follows:
\[ a_{1}\equal{}\frac{m}{2},\ a_{n\plus{}1}\equal{}a_{n}\left\lceil a_{n}\right\rceil,\textnormal{ if }n\geq 1\]
Find all values of $ m$ for which $ a_{2007}$ is the first integer appearing in the sequence.
Note: For a real number $ x$, $ \left\lceil x\right\rceil$ is defined as the smallest integer greater or equal to $ x$. For example, $ \left\lceil\pi\right\rceil\equal{}4$, $ \left\lceil 2007\right\rceil\equal{}2007$.
2004 239 Open Mathematical Olympiad, 1
Given non-constant linear functions $p(x), q(x), r(x)$. Prove that at least one of three trinomials $pq+r, pr+q, qr+p$ has a real root.
[b]proposed by S. Berlov[/b]
2010 Iran Team Selection Test, 1
Let $f:\mathbb N\rightarrow\mathbb N$ be a non-decreasing function and let $n$ be an arbitrary natural number. Suppose that there are prime numbers $p_1,p_2,\dots,p_n$ and natural numbers $s_1,s_2,\dots,s_n$ such that for each $1\leq i\leq n$ the set $\{f(p_ir+s_i)|r=1,2,\dots\}$ is an infinite arithmetic progression. Prove that there is a natural number $a$ such that
\[f(a+1), f(a+2), \dots, f(a+n)\]
form an arithmetic progression.
2014 AIME Problems, 15
For any integer $k\ge1$, let $p(k)$ be the smallest prime which does not divide $k$. Define the integer function $X(k)$ to be the product of all primes less than $p(k)$ if $p(k)>2$, and $X(k)=1$ if $p(k)=2$. Let $\{x_n\}$ be the sequence defined by $x_0=1$, and $x_{n+1}X(x_n)=x_np(x_n)$ for $n\ge0$. Find the smallest positive integer, $t$ such that $x_t=2090$.