Found problems: 85335
Let $ABC$ be a triangle with height $AH$. $P$ lies on the circle over 3 midpoint of $AB,BC,CA$ ($P \notin BC$). Prove that the line connect 2 center of $(PBH)$ and $(PCH)$ go through a fixed point.
(where $(XYZ)$ be a circumscribed circle of triangle $XYZ$)
Let $a \geq b \geq 1 \geq c \geq 0$ be real numbers such that $a+b+c=3$. Show that
$$3 \left( \frac{a}{b}+\frac{b}{a} \right ) \geq 4c^2+\frac{a^2}{b}+\frac{b^2}{a}$$
Over a period of $k$ consecutive days, a total of $2014$ babies were born in a certain city, with at least one baby being born each day. Show that:
(a) If $1014 < k \le 2014$, there must be a period of consecutive days during which exactly $100$ babies were born.
(b) By contrast, if $k = 1014$, such a period might not exist.
Compute the base 10 value of $14641_{99}$
Let $A$ be a point on a circle with center $O$ and $B$ be the midpoint of $[OA]$. Let $C$ and $D$ be points on the circle such that they lie on the same side of the line $OA$ and $\widehat{CBO} = \widehat{DBA}$. Show that the reflection of the midpoint of $[CD]$ over $B$ lies on the circle.
Let $m$ and $n$ be positive integers such that $m - n = 189$ and such that the least common multiple of $m$ and $n$ is equal to $133866$. Find $m$ and $n$.
Each citizen in a town knows at least $ 30$% of the remaining citizens. A citizen votes in elections if he/she knows at least one candidate. Prove that it is possible to schedule elections with two candidates for the mayor of the city so that at least $ 50$% of the citizen can vote.
Determine the least possible value of $f(1998),$ where $f:\Bbb{N}\to \Bbb{N}$ is a function such that for all $m,n\in {\Bbb N}$,
\[f\left( n^{2}f(m)\right) =m\left( f(n)\right) ^{2}. \]
New license plates consist of two letters, three digits, and two letters (from the English alphabet of$ 26$ letters). What is the largest possible number of such license plates if it is required that every two of them differ at no less than two positions?
Which of the following functions are differentiable at $x=0$?
$\textbf{(A)}~f(x)=\begin{cases}\tan^{-1}\left(\frac1{|x|}\right)&\text{if }x\ne0\\\frac\pi2&\text{if }x=0\end{cases}$
$\textbf{(B)}~f(x)=|x|^{1/2}x$
$\textbf{(C)}~f(x)=\begin{cases}x^2\left|\cos\frac{\pi}x\right|&\text{if }x\ne0\\0&\text{if }x=0\end{cases}$
$\textbf{(D)}~\text{None of the above}$
Let $\mathcal{P}$ be a convex polygon with $50$ vertices. A set $\mathcal{F}$ of diagonals of $\mathcal{P}$ is said to be [i]$minimally friendly$ [/i] if any diagonal $d \in \mathcal{F}$ intersects at most one other diagonal in $\mathcal{F}$ at a point interior to $\mathcal{P}.$ Find the largest possible number of elements in a $\text{minimally friendly}$ set $\mathcal{F}$.
Suppose that every two opposite edges of a tetrahedron are orthogonal. Show that the midpoints of the six edges lie on a sphere.
In a convex quadrilateral $ABCD$, the diagonal $BD$ bisects neither the angle $ABC$ nor the angle $CDA$. The point $P$ lies inside $ABCD$ and satisfies \[\angle PBC=\angle DBA\quad\text{and}\quad \angle PDC=\angle BDA.\] Prove that $ABCD$ is a cyclic quadrilateral if and only if $AP=CP$.
Suppose that $ \frac{2}{3}$ of $ 10$ bananas are worth as much as $ 8$ oranges. How many oranges are worth as much is $ \frac{1}{2}$ of $ 5$ bananas?
$ \textbf{(A)}\ 2 \qquad
\textbf{(B)}\ \frac{5}{2} \qquad
\textbf{(C)}\ 3 \qquad
\textbf{(D)}\ \frac{7}{2} \qquad
\textbf{(E)}\ 4$
Determine all integers $x,y,z$ such that \[x^2+y^2=3z^2.\]
Find the max and minimum without using dervivate:
$\sqrt{x} +4 \cdot \sqrt{\frac{1}{2} - x}$
The number $5^{867}$ is between $2^{2013}$ and $2^{2014}$. How many pairs of integers $(m,n)$ are there such that $1\leq m\leq 2012$ and \[5^n<2^m<2^{m+2}<5^{n+1}?\]
$\textbf{(A) }278\qquad
\textbf{(B) }279\qquad
\textbf{(C) }280\qquad
\textbf{(D) }281\qquad
\textbf{(E) }282\qquad$
Let a, b, c, d, and e be positive integers. Are there any solutions to $a^2+b^3+c^5+d^7=e^{11}$?
Calculate $\displaystyle \sum_{n=1}^\infty \ln \left(1+\frac{1}{n}\right) \ln\left( 1+\frac{1}{2n}\right)\ln\left( 1+\frac{1}{2n+1}\right)$.
Consider a polynomial $P(x) = \prod^9_{j=1}(x+d_j),$ where $d_1, d_2, \ldots d_9$ are nine distinct integers. Prove that there exists an integer $N,$ such that for all integers $x \geq N$ the number $P(x)$ is divisible by a prime number greater than 20.
[i]Proposed by Luxembourg[/i]
$\frac{2}{10}+\frac{4}{100}+\frac{6}{1000} =$
$\text{(A)}\ .012 \qquad \text{(B)}\ .0246 \qquad \text{(C)}\ .12 \qquad \text{(D)}\ .246 \qquad \text{(E)}\ 246$
A gumball machine contains $9$ red, $7$ white, and $8$ blue gumballs. The least number of gumballs a person must buy to be sure of getting four gumballs of the same color is
${\text{(A)}\ 8 \qquad \text{(B)}\ 9 \qquad \text(C)}\ 10 \qquad \text{(D)}\ 12 \qquad \text{(E)}\ 18$
In a Cartesian coordinate plane, call a rectangle $standard$ if all of its sides are parallel to the $x$- and $y$- axes, and call a set of points $nice$ if no two of them have the same $x$- or $y$- coordinate. First, Bert chooses a nice set $B$ of $2016$ points in the coordinate plane. To mess with Bert, Ernie then chooses a set $E$ of $n$ points in the coordinate plane such that $B\cup E$ is a nice set with $2016+n$ points. Bert returns and then miraculously notices that there does not exist a standard rectangle that contains at least two points in $B$ and no points in $E$ in its interior. For a given nice set $B$ that Bert chooses, define $f(B)$ as the smallest positive integer $n$ such that Ernie can find a nice set $E$ of size $n$ with the aforementioned properties. Help Bert determine the minimum and maximum possible values of $f(B)$.
[i]Yannick Yao[/i]
The positive integers $a$ and $b$ are such that the numbers $15a+16b$ and $16a-15b$ are both squares of positive integers. What is the least possible value that can be taken on by the smaller of these two squares?
Let $A$ be a set of positive integers such that
a) if $a\in A$, the all the positive divisors of $a$ are also in $A$;
b) if $a,b\in A$, with $1<a<b$, then $1+ab \in A$.
Prove that if $A$ has at least 3 elements, then $A$ is the set of all positive integers.