Found problems: 85335
Let $ABCD$ be a quadrilateral with $\angle BAD = \angle ABC = 90^{\circ}$, and suppose $AB=BC=1$, $AD=2$. The circumcircle of $ABC$ meets $\overline{AD}$ and $\overline{BD}$ at point $E$ and $F$, respectively. If lines $AF$ and $CD$ meet at $K$, compute $EK$.
Let $ABCD$ be a trapezoid with $\angle A = \angle B = 90^o$ and a point $E$ lies on the segment $CD$. Denote $(\omega)$ as incircle of triangle $ABE$ and it is tangent to $AB,AE,BE$ respectively at $P, F,K$. Suppose that $KF$ cuts $BC,AD$ at $M,N$ and $PM,PN$ cut $(\omega)$ at $H, T$. Prove that $PH = PT$.
For each positive integer $ m > 1$, let $ P(m)$ denote the greatest prime factor of $ m$. For how many positive integers $ n$ is it true that both $ P(n) \equal{} \sqrt{n}$ and $ P(n \plus{} 48) \equal{} \sqrt{n \plus{} 48}$?
$ \textbf{(A)}\ 0\qquad
\textbf{(B)}\ 1\qquad
\textbf{(C)}\ 3\qquad
\textbf{(D)}\ 4\qquad
\textbf{(E)}\ 5$
In a lattice, connected the elements $ a \wedge b$ and $ a \vee b$ by an edge whenever $ a$ and $ b$ are incomparable. Prove that in the obtained graph every connected component is a sublattice.
[i]M. Ajtai[/i]
Every point with integer coordinates in the plane is the center of a disk with radius $1/1000$.
(1) Prove that there exists an equilateral triangle whose vertices lie in different discs.
(2) Prove that every equilateral triangle with vertices in different discs has side-length greater than $96$.
[i]Radu Gologan, Romania[/i]
[hide="Remark"]
The "> 96" in [b](b)[/b] can be strengthened to "> 124". By the way, part [b](a)[/b] of this problem is the place where I used [url=http://mathlinks.ro/viewtopic.php?t=5537]the well-known "Dedekind" theorem[/url].
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Let $\operatorname{rad}(k)$ denote the product of prime divisors of a natural number $k$ (define $\operatorname{rad}(1)=1$). A sequence $(a_n)$ is defined by setting $a_1$ arbitrarily, and $a_{n+1}=a_n+\operatorname{rad}(a_n)$ for $n\ge1$. Prove that the sequence $(a_n)$ contains arithmetic progressions of arbitrary length.
In a triangle $ ABC,$ choose any points $ K \in BC, L \in AC, M \in AB, N \in LM, R \in MK$ and $ F \in KL.$ If $ E_1, E_2, E_3, E_4, E_5, E_6$ and $ E$ denote the areas of the triangles $ AMR, CKR, BKF, ALF, BNM, CLN$ and $ ABC$ respectively, show that
\[ E \geq 8 \cdot \sqrt [6]{E_1 E_2 E_3 E_4 E_5 E_6}.
\]
Let $y(x)$ be a continuously differentiable real-valued function of a real variable $x$. Show that if $y'(x)^2 +y(x)^3 \to 0$ as $x\to \infty,$ then $y(x)$ and $y'(x) \to 0$ as $x \to \infty.$
Find all values of parameters $a,b$ for which the polynomial
$$x^4+(2a+1)x^3+(a-1)^2x^2+bx+4$$can be written as a product of two monic quadratic polynomials $\Phi(x)$ and $\Psi(x)$, such that the equation $\Psi(x)=0$ has two distinct roots $\alpha,\beta$ which satisfy $\Phi(\alpha)=\beta$ and $\Phi(\beta)=\alpha$.
For a positive real number $r$, let $G(r)$ be the minimum value of $|r - \sqrt{m^2+2n^2}|$ for all integers $m$ and $n$. Prove or disprove the assertion that $\lim_{r\to \infty}G(r)$ exists and equals $0.$
Given an equilateral triangle $ABC$ of side $a$ in a plane, let $M$ be a point on the circumcircle of the triangle. Prove that the sum $s = MA^4 +MB^4 +MC^4$ is independent of the position of the point $M$ on the circle, and determine that constant value as a function of $a$.
The polynomial $P$ of degree $n$ satisfies $P(k) = \frac{k}{k +1}$ for $k = 0,1,2,...,n$. Find $P(n+1)$.
Let triangle $ABC$ be an acute triangle with $AB\neq AC$. The bisector of $BC$ intersects the lines $AB$ and $AC$ at points $F$ and $E$, respectively. The circumcircle of triangle $AEF$ has center $P$ and intersects the circumcircle of triangle $ABC$ at point $D$ with $D$ different to $A$.
Prove that the line $PD$ is tangent to the circumcircle of triangle $ABC$.
The incircle $(I)$ of a triangle $ABC$ touches $BC,CA,AB$ at $D,E, F$. Let $ID, IE, IF$ intersect $EF, FD,DE$ at $X,Y,Z$, respectively. The lines $\ell_a, \ell_b, \ell_c$ through $A,B,C$ respectively and are perpendicular to $YZ,ZX,XY$ .
Prove that $\ell_a, \ell_b, \ell_c$ are concurrent at a point that is on the line segment joining $I$ and the centroid of triangle $ABC$ .
Nguyễn Minh Hà
Three different collinear points are given. What is the number of isosceles triangles such that these points are their circumcenter, incenter and excenter (in some order)?
Real numbers $a$ and $b$ satisfy $a^3+b^3-6ab=-11$. Prove that $-\frac{7}{3}<a+b<-2$.
[i]Proposed by Serbia[/i]
Let $ ABCD$ be a cyclic quadrilateral and let $ E$ and $ F$ be the feet of perpendiculars from the intersection of diagonals $ AC$ and $ BD$ to $ AB$ and $ CD$, respectively. Prove that $ EF$ is perpendicular to the line through the midpoints of $ AD$ and $ BC$.
Find all natural $n{}$ such that for every natural $a{}$ that is mutually prime with $n{}$, the number $a^n - 1$ is divisible by $2n^2$.
Diameter $ \overline{AB}$ of a circle with center $ O$ is $ 10$ units. $ C$ is a point $ 4$ units from $ A$, and on $ \overline{AB}$. $ D$ is a point $ 4$ units from $ B$, and on $ \overline{AB}$. $ P$ is any point on the circle. Then the broken-line path from $ C$ to $ P$ to $ D$:
$ \textbf{(A)}\ \text{has the same length for all positions of }{P}\qquad\\
\textbf{(B)}\ \text{exceeds }{10}\text{ units for all positions of }{P}\qquad \\
\textbf{(C)}\ \text{cannot exceed }{10}\text{ units}\qquad \\
\textbf{(D)}\ \text{is shortest when }{\triangle CPD}\text{ is a right triangle}\qquad \\
\textbf{(E)}\ \text{is longest when }{P}\text{ is equidistant from }{C}\text{ and }{D}.$
Two circles are tangent to each other internally at a point $T$. Let the chord $AB$ of the larger circle be tangent to the smaller circle at a point $P$. Prove that the line TP bisects $\angle ATB$.
In the rectangular coordinate system every point with integer coordinates is called laticeal point. Let $P_n(n, n + 5)$ be a laticeal point and denote by $f(n)$ the number of laticeal points on the open segment $(OP_n)$, where the point $0(0,0)$ is the coordinates system origine. Calculate the number $f(1) +f(2) + f(3) + ...+ f(2002) + f(2003)$.
Let $a,b,$ and $c$ be real numbers such that $a^2(b+1)=1, b^2(c+a)=2,$ and $c^2(a+b)=5.$ Given that there are three possible values for $abc,$ compute the minimum possible value of $abc.$
Let $\omega_1,\omega_2$ be two circles with centers $O_1$ and $O_2$, respectively. These two circles intersect each other at points $A$ and $B$. Line $O_1B$ intersects $\omega_2$ for the second time at point $C$, and line $O_2A$ intersects $\omega_1$ for the second time at point $D$ . Let $X$ be the second intersection of $AC$ and $\omega_1$. Also $Y$ is the second intersection point of $BD$ and $\omega_2$. Prove that $CX = DY$ .
Proposed by Alireza Dadgarnia
Let be two complex numbers $ a,b. $ Show that the following affirmations are equivalent:
$ \text{(i)} $ there are four numbers $ x_1,x_2,x_3,x_4\in\mathbb{C} $ such that $ \big| x_1 \big| =\big| x_3 \big|, \big| x_2 \big| =\big| x_4 \big|, $ and
$$ x_{j_1}^2-ax_{j_1}+b=0=x_{j_2}^2-bx_{j_2}+a,\quad\forall j_1\in\{ 1,2\} ,\quad\forall j_2\in\{ 3,4\} . $$
$ \text{(ii)} a^3=b^3 $ or $ b=\overline{a} $ (the conjugate of a).
In cyclic quadrilateral $ABCD$ with $AB = AD = 49$ and $AC = 73$, let $I$ and $J$ denote the incenters of triangles $ABD$ and $CBD$. If diagonal $\overline{BD}$ bisects $\overline{IJ}$, find the length of $IJ$.