Found problems: 85335
2000 VJIMC, Problem 3
Prove that if m,n are nonnegative integers and 0<=x<=1 then
$(1-x^n)^m + (1-(1-x)^m)^n \ge 1$
1973 AMC 12/AHSME, 23
There are two cards; one is red on both sides and the other is red on one side and blue on the other. The cards have the same probability (1/2) of being chosen, and one is chosen and placed on the table. If the upper side of the card on the table is red, then the probability that the under-side is also red is
$ \textbf{(A)}\ \frac14 \qquad
\textbf{(B)}\ \frac13 \qquad
\textbf{(C)}\ \frac12 \qquad
\textbf{(D)}\ \frac23 \qquad
\textbf{(E)}\ \frac34$
2010 Romania Team Selection Test, 2
Let $\ell$ be a line, and let $\gamma$ and $\gamma'$ be two circles. The line $\ell$ meets $\gamma$ at points $A$ and $B$, and $\gamma'$ at points $A'$ and $B'$. The tangents to $\gamma$ at $A$ and $B$ meet at point $C$, and the tangents to $\gamma'$ at $A'$ and $B'$ meet at point $C'$. The lines $\ell$ and $CC'$ meet at point $P$. Let $\lambda$ be a variable line through $P$ and let $X$ be one of the points where $\lambda$ meets $\gamma$, and $X'$ be one of the points where $\lambda$ meets $\gamma'$. Prove that the point of intersection of the lines $CX$ and $C'X'$ lies on a fixed circle.
[i]Gazeta Matematica[/i]
1992 Putnam, A4
Let $ f$ be an infinitely differentiable real-valued function defined on the real numbers. If $ f(1/n)\equal{}\frac{n^{2}}{n^{2}\plus{}1}, n\equal{}1,2,3,...,$ Compute the values of the derivatives of $ f^{k}(0), k\equal{}0,1,2,3,...$
2018 Turkey Junior National Olympiad, 2
We are placing rooks on a $n \cdot n$ chess table that providing this condition:
Every two rooks will threaten an empty square at least.
What is the most number of rooks?
2013 IberoAmerican, 2
Let $X$ and $Y$ be the diameter's extremes of a circunference $\Gamma$ and $N$ be the midpoint of one of the arcs $XY$ of $\Gamma$. Let $A$ and $B$ be two points on the segment $XY$. The lines $NA$ and $NB$ cuts $\Gamma$ again in $C$ and $D$, respectively. The tangents to $\Gamma$ at $C$ and at $D$ meets in $P$. Let $M$ the the intersection point between $XY$ and $NP$. Prove that $M$ is the midpoint of the segment $AB$.
2023 SG Originals, Q1
Let $n$ be a positive integer. A sequence $a_1$, $a_2$,$ ...$ , $a_n$ is called [i]good [/i] if the following conditions hold:
$\bullet$ For each $i \in \{1, 2, ..., n\}$, $1 \le a_i \le n$
$\bullet$ For all positive integers $i, j$ with $1 \le i \le j \le n$, the expression $a_i + a_{i+1} + ...+ a_j$ is not divisible by $ n + 1$.
Find the number of good sequences (in terms of $n$).
2004 Moldova Team Selection Test, 5
Let $n\in\mathbb{N}$, the set $A=\{(x_1,x_2...,x_n)|x_i\in\mathbb{R}_{+}, i=1,2,...,n\}$ and the function $$f:A\rightarrow\mathbb{R}, f(x_1,...,x_n)=\frac{1}{x_1}+\frac{1}{2x_2}+\ldots+\frac{1}{(n-1)x_{n-1}}+\frac{1}{nx_n}.$$
Prove that $f(\textstyle\binom{n}{1},\binom{n}{2},...,\binom{n}{n-1},\binom{n}{n})=f(2^{n-1},2^{n-2},...,2,1).$
1998 Tournament Of Towns, 2
On the plane are $n$ paper disks of radius $1$ whose boundaries all pass through a certain point, which lies inside the region covered by the disks. Find the perimeter of this region.
(P Kozhevnikov)
2014 Junior Balkan Team Selection Tests - Romania, 3
Consider six points in the interior of a square of side length $3$.
Prove that among the six points, there are two whose distance is less than $2$.
1983 All Soviet Union Mathematical Olympiad, 356
The sequences $a_n$ and $b_n$ members are the last digits of $[\sqrt{10}^n]$ and $[\sqrt{2}^n]$ respectively (here $[ ...]$ denotes the whole part of a number). Are those sequences periodical?
2012 Balkan MO, 4
Let $\mathbb{Z}^+$ be the set of positive integers. Find all functions $f:\mathbb{Z}^+ \rightarrow\mathbb{Z}^+$ such that the following conditions both hold:
(i) $f(n!)=f(n)!$ for every positive integer $n$,
(ii) $m-n$ divides $f(m)-f(n)$ whenever $m$ and $n$ are different positive integers.
2000 Rioplatense Mathematical Olympiad, Level 3, 3
Let $n>1$ be an integer. For each numbers $(x_1, x_2,\dots, x_n)$ with $x_1^2+x_2^2+x_3^2+\dots +x_n^2=1$, denote
$m=\min\{|x_i-x_j|, 0<i<j<n+1\}$
Find the maximum value of $m$.
2002 AIME Problems, 6
The solutions to the system of equations
\begin{eqnarray*} \log_{225}{x}+\log_{64}{y} &=& 4\\ \log_x{225}-\log_y{64} &=& 1 \end{eqnarray*}
are $(x_1,y_1)$ and $(x_2, y_2).$ Find $\log_{30}{(x_1y_1x_2y_2)}.$
2002 IMC, 3
Let $n$ be a positive integer and let $a_k = \dfrac{1}{\binom{n}{k}}, b_k = 2^{k-n},\ (k=1..n)$.
Show that $\sum_{k=1}^n \dfrac{a_k-b_k}{k} = 0$.
2013 JBMO Shortlist, 3
Let $ABC$ be an acute-angled triangle with $AB<AC$ and let $O$ be the centre of its circumcircle $\omega$. Let $D$ be a point on the line segment $BC$ such that $\angle BAD = \angle CAO$. Let $E$ be the second point of intersection of $\omega$ and the line $AD$. If $M$, $N$ and $P$ are the midpoints of the line segments $BE$, $OD$ and $AC$, respectively, show that the points $M$, $N$ and $P$ are collinear.
2012 Princeton University Math Competition, A5
Call a positive integer $x$ a leader if there exists a positive integer $n$ such that the decimal representation of $x^n$ starts ([u]not ends[/u]) with $2012$. For example, $586$ is a leader since $586^3 =201230056$. How many leaders are there in the set $\{1, 2, 3, ..., 2012\}$?
2019 Benelux, 1
[list=a]
[*]Let $a,b,c,d$ be real numbers with $0\leqslant a,b,c,d\leqslant 1$. Prove that
$$ab(a-b)+bc(b-c)+cd(c-d)+da(d-a)\leqslant \frac{8}{27}.$$[/*]
[*]Find all quadruples $(a,b,c,d)$ of real numbers with $0\leqslant a,b,c,d\leqslant 1$ for which equality holds in the above inequality.
[/list]
2012 China National Olympiad, 1
Let $f(x)=(x + a)(x + b)$ where $a,b>0$. For any reals $x_1,x_2,\ldots ,x_n\geqslant 0$ satisfying $x_1+x_2+\ldots +x_n =1$, find the maximum of $F=\sum\limits_{1 \leqslant i < j \leqslant n} {\min \left\{ {f({x_i}),f({x_j})} \right\}} $.
2000 Belarus Team Selection Test, 4.1
Find all functions $f ,g,h : R\to R$ such that $f(x+y^3)+g(x^3+y) = h(xy)$ for all $x,y \in R$
2024 Rioplatense Mathematical Olympiad, 2
Let $ABC$ be a triangle with $AB < AC$, incentre $I$, and circumcircle $\omega$. Let $D$ be the intersection of the external bisector of angle $\widehat{ BAC}$ with line $BC$. Let $E$ be the midpoint of the arc $BC$ of $\omega$ that does not contain $A$. Let $M$ be the midpoint of $DI$, and $X$ the intersection of $EM$ with $\omega$. Prove that $IX$ and $EM$ are perpendicular.
2004 China Western Mathematical Olympiad, 3
Let $\ell$ be the perimeter of an acute-angled triangle $ABC$ which is not an equilateral triangle. Let $P$ be a variable points inside the triangle $ABC$, and let $D,E,F$ be the projections of $P$ on the sides $BC,CA,AB$ respectively. Prove
that \[ 2(AF+BD+CE ) = \ell \] if and only if $P$ is collinear with the incenter and the circumcenter of the triangle $ABC$.
2015 Kazakhstan National Olympiad, 4
$P_k(n) $ is the product of all positive divisors of $n$ that are divisible by $k$ (the empty product is equal to $1$). Show that $P_1(n)P_2(n)\cdots P_n(n)$ is a perfect square, for any positive integer $n$.
1999 Poland - Second Round, 1
Let $f : (0,1) \to R$ be a function such that $f(1/n) = (-1)^n$ for all n ∈ N.
Prove that there are no increasing functions $g,h : (0,1) \to R$ such that $f = g - h$.
1959 AMC 12/AHSME, 4
If $78$ is divided into three parts which are proportional to $1, \frac13, \frac16$, the middle part is:
$ \textbf{(A)}\ 9\frac13 \qquad\textbf{(B)}\ 13\qquad\textbf{(C)}\ 17\frac13 \qquad\textbf{(D)}\ 18\frac13\qquad\textbf{(E)}\ 26 $