Found problems: 85335
Let $E$ be the midpoint of the side $[BC]$ of $\triangle ABC$ with $|AB|=7$, $|BC|=6$, and $|AC|=5$. The line, which passes through $E$ and is perpendicular to the angle bisector of $\angle A$, intersects $AB$ at $D$. What is $|AD|$?
$
\textbf{(A)}\ 5
\qquad\textbf{(B)}\ 6
\qquad\textbf{(C)}\ \frac 92
\qquad\textbf{(D)}\ 3\sqrt 2
\qquad\textbf{(E)}\ \text{None of above}
$
Which of the following statements is false?
$ \textbf{(A)}\ \text{All equilateral triangles are congruent to each other.}$
$ \textbf{(B)}\ \text{All equilateral triangles are convex.}$
$ \textbf{(C)}\ \text{All equilateral triangles are equilangular.}$
$ \textbf{(D)}\ \text{All equilateral triangles are regular polygons.}$
$ \textbf{(E)}\ \text{All equilateral triangles are similar to each other.}$
Let $A$ be a finite set made up of prime numbers. Determine if there exists an infinite set $B$ that satisfies the following conditions:
$(i)$ the prime factors of any element of $B$ are in $A$;
$(ii)$ no term of $B$ divides another element of this set.
Let $ABC$ be a non-degenerate triangle inscribed in a circle, such that $AB$ is the diameter of the circle. Let the angle bisectors of the angles at $A$ and $B$ meet at $P.$ Determine the maximum possible value of $\angle APB,$ in degrees.
Evaluate $ \displaystyle I \equal{} \int_0^{\frac\pi4}\left(\sin^62x \plus{} \cos^62x\right)\cdot \ln(1 \plus{} \tan x)\text{d}x$.
Is there a solution of the Cauchy problem
\[x(x^2+y^2)\frac{\partial u}{\partial x}+y^3\frac{\partial u}{\partial y}=0,\;u|_{y=0}=1\]
in a neighbourhood of the point $(x_0,0)$ of the $x$-axis? Is it unique?
Find all integers $n$ for which $\log_{2n-2} (n^2 + 2)$ is a rational number.
Find all primes $p>5$ for which there exists an integer $a$ and an integer $r$ satisfying $1\leq r\leq p-1$ with the following property: the sequence $1,\,a,\,a^2,\,\ldots,\,a^{p-5}$ can be rearranged to form a sequence $b_0,\,b_1,\,b_2,\,\ldots,\,b_{p-5}$ such that $b_n-b_{n-1}-r$ is divisible by $p$ for $1\leq n\leq p-5$.
In a village with at least one inhabitant, there are several associations. Each inhabitant is a member of at least $ k$ associations, and any two associations have at most one common member.
Prove that at least $ k$ associations have the same number of members.
Let $ABCD$ be a convex quadrilateral such that $\angle ABC = \angle ADC =135^o$ and $$AC^2 BD^2=2AB\cdot BC \cdot CD\cdot DA.$$ Prove that the diagonals of $ABCD$ are perpendicular.
Mr. Lopez has a choice of two routes to get to work. Route A is $6$ miles long, and his average speed along this route is $30$ miles per hour. Route B is $5$ miles long, and his average speed along this route is $40$ miles per hour, except for a $\frac{1}{2}$-mile stretch in a school zone where his average speed is $20$ miles per hour. By how many minutes is Route B quicker than Route A?
$\textbf{(A)}\ 2 \frac{3}{4} \qquad\textbf{(B)}\ 3 \frac{3}{4} \qquad\textbf{(C)}\ 4 \frac{1}{2} \qquad\textbf{(D)}\
5 \frac{1}{2} \qquad\textbf{(E)}\ 6 \frac{3}{4}$
Let $f$ be the quadratic function with leading coefficient $1$ whose graph is tangent to that of the lines $y=-5x+6$ and $y=x-1$. The sum of the coefficients of $f$ is $\tfrac pq$, where $p$ and $q$ are positive relatively prime integers. Find $100p + q$.
[i]Proposed by David Altizio[/i]
In the plane, a set of points $ M $ is given with the following properties:
1. The points of the set $ M $ do not lie on one straight line,
2. If the points $ A, B, C$, and $D$ are vertices of a parallelogram and $ A, B, C \in M $, then $ D \in M $,
3. If $ A, B \in M $, then $ AB \geq 1 $.
Prove that there exist two families of parallel lines such that $ M $ is the set of all intersection points of the lines of the first family with the lines of the second family.
Let $ABC$ be an acute triangle such that $\angle BAC = 45^o$. Let $D$ a point on $AB$ such that $CD \perp AB$. Let $P$ be an internal point of the segment $CD$. Prove that $AP\perp BC$ if and only if $|AP| = |BC|$.
Let ABCD be a parallelogram, and P a point such that
$2 PDA=ABP$ and
$2 PAD=PCD$
Show that $AB=BP=CP$
The largest integer $n$ for which $n^{200} < 5^{300}$ is
$\textbf{(A) }8\qquad
\textbf{(B) }9\qquad
\textbf{(C) }10\qquad
\textbf{(D) }11\qquad
\textbf{(E) }12$
Let $n$ be a positive integer. Given is a subset $A$ of $\{0,1,...,5^n\}$ with $4n+2$ elements. Prove that there exist three elements $a<b<c$ from $A$ such that $c+2a>3b$.
[i]Proposed by Dominik Burek and Tomasz Ciesla, Poland[/i]
Find the maximum value of:
$E(a,b)=\frac{a+b}{(4a^2+3)(4b^2+3)}$
For $a,b$ real numbers.
Let $C_1,C_2$ be two circles such that the center of $C_1$ is on the circumference of $C_2$. Let $C_1,C_2$ intersect each other at points $M,N$. Let $A,B$ be two points on the circumference of $C_1$ such that $AB$ is the diameter of it. Let lines $AM,BN$ meet $C_2$ for the second time at $A',B'$, respectively. Prove that $A'B'=r_1$ where $r_1$ is the radius of $C_1$.
The base of a triangle is twice as long as a side of a square and their areas are the same. Then the ratio of the altitude of the triangle to the side of the square is:
$ \textbf{(A)}\ \frac{1}{4} \qquad
\textbf{(B)}\ \frac{1}{2} \qquad
\textbf{(C)}\ 1 \qquad
\textbf{(D)}\ 2 \qquad
\textbf{(E)}\ 4$
Segments $AB, BC$ and $CD$ of the broken line $ABCD$ are equal and are tangent to a circle with centre at the point $O$. Prove that the point of contact of this circle with $BC$, the point $O$ and the intersection point of $AC$ and $BD$ are collinear.
On two parallel lines, the distinct points $A_1,A_2,A_3,\ldots $ respectively $B_1,B_2,B_3,\ldots $ are marked in such a way that $|A_iA_{i+1}|=1$ and $|B_iB_{i+1}|=2$ for $i=1,2,\ldots $. Provided that $A_1A_2B_1=\alpha$, find the infinite sum $\angle A_1B_1A_2+\angle A_2B_2A_3+\angle A_3B_3A_4+\ldots $
Find all positive integers $k$ that can be expressed as the sum of $50$ fractions such that the numerators are the $50$ natural numbers from $1$ to $50$ and the denominators are positive integers, that is, $k = \dfrac{1}{a_1} + \dfrac{2}{a_2} + \ldots + \dfrac{50}{a_{50}}$ with a$_1 , a_2 , \ldots , a_n$ positive integers.
Two semicircles are tangent to middle circle, and both semicircles and middle circle are tangent to the horizontal line as shown. If $ PQ \equal{} QR \equal{} RS \equal{} 24,$ then find the length of radius $ r$.
[img]http://i250.photobucket.com/albums/gg265/geometry101/NielsHenrikAbel1999Number3.jpg[/img]
Find all integers $n$ such that $n^4 + 8n + 11$ is a product of two or more consecutive integers.