Found problems: 85335
Let $a$, $b$, $c$ be positive integers. Prove that for infinitely many positive odd integers $n$, there exists an integer $m > n$ such that $a^n + b^n + c^n$ divides $a^m + b^m + c^m$.
A digital watch displays hours and minutes with $ \text c{AM}$ and $ \text c{PM}$. What is the largest possible sum of the digits in the display?
$ \textbf{(A) } 17\qquad \textbf{(B) } 19\qquad \textbf{(C) } 21\qquad \textbf{(D) } 22\qquad \textbf{(E) } 23$
Two circles $\gamma_1, \gamma_2$ are given, with centers at points $O_1, O_2$ respectively. Select a point $K$ on circle $\gamma_2$ and construct two circles, one $\gamma_3$ that touches circle $\gamma_2$ at point $K$ and circle $\gamma_1$ at a point $A$, and the other $\gamma_4$ that touches circle $\gamma_2$ at point $K$ and circle $\gamma_1$ at a point $B$. Prove that, regardless of the choice of point K on circle $\gamma_2$, all lines $AB$ pass through a fixed point of the plane.
$a,b$ are two skew lines, the angle they form is $\theta$. Length of their common perpendicular $AA'$ is $d$($A'\in a,A\in b)$. $E\in a,F\in b,|A'E|=m,|AF|=n$. Calculate $|EF|$.
We say a triple of real numbers $ (a_1,a_2,a_3)$ is [b]better[/b] than another triple $ (b_1,b_2,b_3)$ when exactly two out of the three following inequalities hold: $ a_1 > b_1$, $ a_2 > b_2$, $ a_3 > b_3$. We call a triple of real numbers [b]special[/b] when they are nonnegative and their sum is $ 1$.
For which natural numbers $ n$ does there exist a collection $ S$ of special triples, with $ |S| \equal{} n$, such that any special triple is bettered by at least one element of $ S$?
If $x, y, z$ are positive real numbers such that $x + y + z = 9xyz.$ Prove that:
$$\frac {x} {\sqrt {x^2+2yz+2}}+\frac {y} {\sqrt {y^2+2zx+2}}+\frac {z} {\sqrt {z^2+2xy+2}}\ge 1.$$
Consider when equality applies.
Prove that if two triangles are inscribed in the same circle, then their incircles are not strictly contained one into each other.
Let $ABC$ be an acute triangle with $AB<AC<BC$ and let $D$ be a point on it's extension of $BC$ towards $C$. Circle $c_1$, with center $A$ and radius $AD$, intersects lines $AC,AB$ and $CB$ at points $E,F$, and $G$ respectively. Circumscribed circle $c_2$ of triangle $AFG$ intersects again lines $FE,BC,GE$ and $DF$ at points $J,H,H' $ and $J'$ respectively. Circumscribed circle $c_3$ of triangle $ADE$ intersects again lines $FE,BC,GE$ and $DF$ at points $I,K,K' $ and $I' $ respectively. Prove that the quadrilaterals $HIJK$ and $H'I'J'K '$ are cyclic and the centers of their circumscribed circles coincide.
by Evangelos Psychas, Greece
Vijay chooses three distinct integers $a,b,c$ from the set $\{1,2,3,4,5,6,7,8,9,10,11\}.$ If $k$ is the minimum value taken on by the polynomial $a(x - b)(x - c)$ over all real numbers $x,$ and $l$ is the minimum value taken on by the polynomial $a(x-b)(x+c)$ over all real numbers $x,$ compute the maximum possible value of $k -l.$
Let $ABC$ be an isosceles triangle with orthocenter $H.$ Let $M$ and $N$ be the midpoints of sides $\overline{AB}$ and $\overline{AC},$ respectively. The circumcircle of triangle $MHN$ intersects line $BC$ at two points $X$ and $Y.$ Given $XY=AB=AC=2,$ compute $BC^2.$
A Haiku is a Japanese poem of seventeen syllables, in three lines of five, seven, and five.
Using the four words
“Hi”, “hey”, “hello”, and “haiku”,
How many haikus
Can somebody make?
(Repetition is allowed,
Order does matter.)
[i]Proposed by Jeff Lin[/i]
Consider two polygons $ P$ and $ Q$. We want to cut $ P$ into some smaller polygons and put them together in such a way to obtain $ Q$. We can translate the pieces but we can not rotate them or reflect them. We call $ P,Q$ equivalent if and only if we can obtain $ Q$ from $ P$(which is obviously an equivalence relation).
[img]http://i3.tinypic.com/4lrb43k.png[/img]
a) Let $ P,Q$ be two rectangles with the same area(their sides are not necessarily parallel). Prove that $ P$ and $ Q$ are equivalent.
b) Prove that if two triangles are not translation of each other, they are not equivalent.
c) Find a necessary and sufficient condition for polygons $ P,Q$ to be equivalent.
Two vertices of the rectangle are located on side $BC$ of triangle $ABC$, and the other two are on sides $AB$ and $AC$. It is known that the midpoint of the altitude of this triangle, drawn on the side $BC$, lies on one of the diagonals of the rectangle, and the side of the rectangle located on $BC$ is three times less than $BC$. In what ratio does the altitude of the triangle divide the side $BC$ ?
A triangle $ABC$ and a point $P$ inside it are given. $A', B', C'$ are the projections of $P$ onto the straight lines ot the sides $BC,CA,AB$. Prove that the center of the circle circumscribed around the triangle $A'B'C'$ lies inside the triangle $ABC$.
The volume of a pyramid whose base is an equilateral triangle of side length 6 and whose other edges are each of length $ \sqrt{15}$ is
$ \textbf{(A)}\ 9 \qquad
\textbf{(B)}\ 9/2 \qquad
\textbf{(C)}\ 27/2 \qquad
\textbf{(D)}\ \frac{9\sqrt3}{2} \qquad
\textbf{(E)}\ \text{none of these}$
Higher Secondary P4
If the fraction $\dfrac{a}{b}$ is greater than $\dfrac{31}{17}$ in the least amount while $b<17$, find $\dfrac{a}{b}$.
Let $s(n)$ the sum of digits of $n$. Find the greatest 3-digits number $m$ such that $3s(m)=s(3m)$.
Rectangle $ABCD$ has sides $AB = 3$, $BC = 2$. Point $ P$ lies on side $AB$ is such that the bisector of the angle $CDP$ passes through the midpoint $M$ of $BC$. Find $BP$.
In the figure, the rectangle is formed by $4$ smaller equal rectangles.
If we count the total number of rectangles in the figure we find $10$.
How many rectangles in total will there be in a rectangle that is formed by $n$ smaller equal rectangles?
Solve $p^n+144=m^2$ where $m,n\in \mathbb{N}$ and $p$ is a prime number.
Call a positive number $n$ [i]fine[/i], if there exists a prime number $p$ such that $p|n$ and $p^2\nmid n$. Prove that at least 99% of numbers $1, 2, 3, \ldots, 10^{12}$ are fine numbers.
Let $ n$ be a positive integer. Prove that $ x^ne^{1\minus{}x}\leq n!$ for $ x\geq 0$,
Fix positive integers $k,n$. A candy vending machine has many different colours of candy, where there are $2n$ candies of each colour. A couple of kids each buys from the vending machine $2$ candies of different colours. Given that for any $k+1$ kids there are two kids who have at least one colour of candy in common, find the maximum number of kids.
Number of solutions of the equation $3^x+4^x=8^x$ in reals is
[list=1]
[*] 0
[*] 1
[*] 2
[*] $\infty$
[/list]
Given a sheet of tin $6\times 6$. It is allowed to bend it and to cut it but in such a way that it doesn’t fall to pieces. How to make a cube with edge $2$, divided by partitions into unit cubes?