Found problems: 1148
[b]i.)[/b] Let $ABC$ be a triangle with $AB = 12$ and $AC = 16.$ Suppose $M$ is the midpoint of side $BC$ and points $E$ and $F$ are chosen on sides $AC$ and $AB$, respectively, and suppose that lines $EF$ and $AM$ intersect at $G.$ If $AE = 2 \cdot AF$ then find the ratio
\[ \frac{EG}{GF} \]
[b]ii.)[/b] Let $E$ be a point external to a circle and suppose that two chords $EAB$ and $EDC$ meet at angle of $40^{\circ}.$ If $AB = BC = CD$ find the size of angle $ACD.$
$ a_{1} \equal{}1$, $ a_{n\plus{}1} \equal{}\frac{a_{n} }{\sqrt{1\plus{}4a_{n}^{2} } }$ for $ n\ge 1$. What is the least $ k$ such that $ a_{k} <10^{\minus{}2}$ ?
$\textbf{(A)}\ 2501 \qquad\textbf{(B)}\ 251 \qquad\textbf{(C)}\ 2499 \qquad\textbf{(D)}\ 249 \qquad\textbf{(E)}\ \text{None}$
The roots of $ (x^2\minus{}3x\plus{}2)(x)(x\minus{}4)\equal{}0$ are:
$\textbf{(A)}\ 4\qquad
\textbf{(B)}\ 0\text{ and }4 \qquad
\textbf{(C)}\ 1\text{ and }2 \qquad
\textbf{(D)}\ 0,1,2\text{ and }4\qquad
\textbf{(E)}\ 1,2\text{ and }4$
Prove that the equation $x^{2}+p|x| = qx - 1 $ has 4 distinct real solutions if and only if $p+|q|+2<0$
($p$ and $q$ are two real parameters).
Find all real solutions to the following system of equations:
\[\begin{cases} x+y+z+t=5\\xy+yz+zt+tx=4\\xyz+yzt+ztx+txy=3\\xyzt=-1\end{cases}\]
Let $a$ and $b$ be two positive rational numbers such that $\sqrt[3] {a} + \sqrt[3]{b}$ is also a rational number. Prove that $\sqrt[3]{a}$ and $\sqrt[3] {b}$ themselves are rational numbers.
For integers $x$, $y$, and $z$, we have $(x-y)(y-z)(z-x)=x+y+z$. Prove that $27|x+y+z$.
If $a$ , $b$ , $c$ , $d$ are integers and $A=2(a-2b+c)^4+2(b-2c+a)^4+2(c-2a+b)^4$ , $B=d(d+1)(d+2)(d+3)+1$ , then prove that $\left (\sqrt{A}+1 \right )^2 +B$ cannot be a perfect square.
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.
Let $A,B$ be two points on a give circle, and $M$ be the midpoint of one of the arcs $AB$ . Point $C$ is the orthogonal projection of $B$ onto the tangent $l$ to the circle at $A$. The tangent at $M$ to the circle meets $AC,BC$ at $A',B'$ respectively. Prove that if $\hat{BAC}<\frac{\pi}{8}$ then $S_{ABC}<2S_{A'B'C'}$.
Points $(\sqrt{\pi}, a)$ and $(\sqrt{\pi}, b)$ are distinct points on the graph of $y^2+x^4=2x^2y+1$. What is $|a-b|$?
$ \textbf{(A) }1\qquad\textbf{(B) }\dfrac{\pi}{2}\qquad\textbf{(C) }2\qquad\textbf{(D) }\sqrt{1+\pi}\qquad\textbf{(E) }1+\sqrt{\pi} $
On the figure, the quadrilateral $ ABCD$ is a rectangle, $ P$ lies on $ AD$ and $ Q$ on $ AB.$ The triangles $ PAQ, QBC,$ and $ PCD$ all have the same areas, and $ BQ \equal{} 2.$ How long is $ AQ$?
[img]http://i250.photobucket.com/albums/gg265/geometry101/NielsHenrikAbel1995Number2.jpg[/img]
A. 7/2
B. $ \sqrt{7}$
C. $ 2 \sqrt{3}$
D. $ 1 \plus{} \sqrt{5}$
E. Not uniquely determined
Let $k$ be a fixed integer greater than 1, and let ${m=4k^2-5}$. Show that there exist positive integers $a$ and $b$ such that the sequence $(x_n)$ defined by \[x_0=a,\quad x_1=b,\quad x_{n+2}=x_{n+1}+x_n\quad\text{for}\quad n=0,1,2,\dots,\] has all of its terms relatively prime to $m$.
[i]Proposed by Jaroslaw Wroblewski, Poland[/i]
Let $a_{i}$ and $b_{i}$, where $i \in \{1,2, \dots, 2005 \}$, be real numbers such that the inequality $(a_{i}x-b_{i})^{2} \ge \sum_{j=1, j \not= i}^{2005} (a_{j}x-b_{j})$ holds for all $x \in \mathbb{R}$ and all $i \in \{1,2, \dots, 2005 \}$. Find the maximum possible number of positive numbers amongst $a_{i}$ and $b_{i}$, $i \in \{1,2, \dots, 2005 \}$.
Given are distinct natural numbers $a$, $b$, and $c$. Prove that
\[ \gcd(ab+1, ac+1, bc+1)\le \frac{a+b+c}{3}\]
Let $ABCD$ be a rectangle with: $AB=a$, $BC=b$. Inside the rectangle we have to exteriorly tangents circles such that one is tangent to the sides $AB$ and $AD$,the other is tangent to the sides $CB$ and $CD$.
1. Find the distance between the centers of the circles(using $a$ and $b$).
2. When the radiums of both circles change the tangency point between both of them changes, and describes a locus. Find that locus.
Suppose real numbers $A,B,C$ such that for all real numbers $x,y,z$ the following inequality holds:
\[A(x-y)(x-z) + B(y-z)(y-x) + C(z-x)(z-y) \geq 0.\]
Find the necessary and sufficient condition $A,B,C$ must satisfy (expressed by means of an equality or an inequality).
Let $\mathcal K$ be a field with $2^{n}$ elements, $n \in \mathbb N^\ast$, and $f$ be the polynomial $X^{4}+X+1$. Prove that:
(a) if $n$ is even, then $f$ is reducible in $\mathcal K[X]$;
(b) if $n$ is odd, then $f$ is irreducible in $\mathcal K[X]$.
[hide="Remark."]I saw the official solution and it wasn't that difficult, but I just couldn't solve this bloody problem.[/hide]
Calculate $ \int_1^4 \frac{\ln x}{(1+x)(4+x)} dx . $
[i]Ovidiu Țâțan[/i]
Given that $P(x)$ is the least degree polynomial with rational coefficients such that
\[P(\sqrt{2} + \sqrt{3}) = \sqrt{2},\] find $P(10)$.
The quadratic equation $x^2+mx+n=0$ has roots that are twice those of $x^2+px+m=0$, and none of $m$, $n$, and $p$ is zero. What is the value of $\frac{n}{p}$?
$\text{(A)} \ 1 \qquad \text{(B)} \ 2 \qquad \text{(C)} \ 4 \qquad \text{(D)} \ 8\qquad \text{(E)} \ 16$
Let $ x, y, z$ be distinct non-negative real numbers. Prove that
\[ \frac{1}{(x\minus{}y)^2} \plus{} \frac{1}{(y\minus{}z)^2} \plus{} \frac{1}{(z\minus{}x)^2} \geq \frac{4}{xy \plus{} yz \plus{} zx}.\]
When does the equality hold?
Suppose that $a$ is an irrational number.
(a) If there is a real number $b$ such that both $(a+b)$ and $ab$ are rational numbers, show that $a$ is a quadratic surd. ($a$ is a quadratic surd if it is of the form $r+\sqrt{s}$ or $r-\sqrt{s}$ for some rationals $r$ and $s$, where $s$ is not the square of a rational number).
(b) Show that there are two real numbers $b_1$ and $b_2$ such that
i) $a+b_1$ is rational but $ab_1$ is irrational.
ii) $a+b_2$ is irrational but $ab_2$ is rational.
(Hint: Consider the two cases, where $a$ is a quadratic surd and $a$ is not a quadratic surd, separately).
For which value of $A$, does the equation $3m^2n = n^3 + A$ have a solution in natural numbers?
$
\textbf{(A)}\ 301
\qquad\textbf{(B)}\ 403
\qquad\textbf{(C)}\ 415
\qquad\textbf{(D)}\ 427
\qquad\textbf{(E)}\ 481
$
Find the last three digits of the product of the positive roots of \[ \sqrt{1995}x^{\log_{1995}x}=x^2. \]