Found problems: 85335
From a point $S$, which lies outside the circle $\Omega$, tangent lines $SA$ and $SB$ to that circle are drawn. On the chord $AB$ an arbitrary point $K$ is chosen. $SK$ intersects $\Omega$ at points $P$ and $Q$, and chords $RT$ and $UW$ pass through $K$ such that $W, Q$ and $T$ lie in the same half-plane with respect to $AB$. Lines $WR$ and $TU$ intersect chord $AB$ at $C$ and $D$, and $M$ is the midpoint of $PQ$.
Prove that $\angle AMC = \angle BMD$
Prove or disprove: given any quadrilateral inscribed in a convex polygon, we can find a rhombus inscribed in the polygon with side not less than the shortest side of the quadrilateral.
Find (with proof) all natural numbers $n$ such that, for some natural numbers $a$ and $b$, $a\ne b$, the digits in the decimal representations of the two numbers $n^a+1$ and $n^b+1$ are in reverse order.
The circle $x^2 + y^2 = 25$ intersects the abscissa in points $A$ and $B$. Let $P$ be a point that lies on the line $x = 11$, $C$ is the intersection point of this line with the $Ox$ axis, and the point $Q$ is the intersection point of $AP$ with the given circle. It turned out that the area of the triangle $AQB$ is four times smaller than the area of the triangle $APC$. Find the coordinates of $Q$.
Show that the straight lines passing through the feet of the altitudes of an acute-angled triangle form a triangle in which the altitudes of the original triangle are angle bisectors.
Let $ABCDS$ be a pyramid with four faces and with $ABCD$ as a base, and let a plane $\alpha$ through the vertex $A$ meet its edges $SB$ and $SD$ at points $M$ and $N$, respectively. Prove that if the intersection of the plane $\alpha$ with the pyramid $ABCDS$ is a parallelogram, then $SM \cdot SN > BM \cdot DN$.
155 birds $ P_1, \ldots, P_{155}$ are sitting down on the boundary of a circle $ C.$ Two birds $ P_i, P_j$ are mutually visible if the angle at centre $ m(\cdot)$ of their positions $ m(P_iP_j) \leq 10^{\circ}.$ Find the smallest number of mutually visible pairs of birds, i.e. minimal set of pairs $ \{x,y\}$ of mutually visible pairs of birds with $ x,y \in \{P_1, \ldots, P_{155}\}.$ One assumes that a position (point) on $ C$ can be occupied simultaneously by several birds, e.g. all possible birds.
In the blackboard there are drawn $25$ points as shown in the figure.
Gastón must choose $4$ points that are vertices of a square.
In how many different ways can he make this choice?$$\begin{matrix}\bullet & \bullet & \bullet & \bullet & \bullet \\\bullet & \bullet & \bullet & \bullet & \bullet \\\bullet & \bullet & \bullet & \bullet & \bullet \\\bullet & \bullet & \bullet & \bullet & \bullet \\\bullet & \bullet & \bullet & \bullet & \bullet \end{matrix}$$
Let $a,b,c\ge0$. Show that $(a^2+2bc)^{2012}+(b^2+2ca)^{2012}+(c^2+2ab)^{2012}\le (a^2+b^2+c^2)^{2012}+2(ab+bc+ca)^{2012}$.
[i]Calvin Deng.[/i]
Given a finite set $K_0$ of points (in the plane or space). The sequence of sets $K_1, K_2, ... , K_n, ...$ is constructed according to the rule: [i]we take all the points of $K_i$, add all the symmetric points with respect to all its points, and, thus obtain $K_{i+1}$.[/i]
a) Let $K_0$ consist of two points $A$ and $B$ with the distance $1$ unit between them. For what $n$ the set $K_n$ contains the point that is $1000$ units far from $A$?
b) Let $K_0$ consist of three points that are the vertices of the equilateral triangle with the unit square. Find the area of minimal convex polygon containing $K_n. K_0$ below is the set of the unit volume tetrahedron vertices.
c) How many faces contain the minimal convex polyhedron containing $K_1$?
d) What is the volume of the above mentioned polyhedron?
e) What is the volume of the minimal convex polyhedron containing $K_n$?
An object moves $ 8$ cm in a straight line from $ A$ to $ B$, turns at an angle $ \alpha$, measured in radians and chosen at random from the interval $ (0,\pi)$, and moves $ 5$ cm in a straight line to $ C$. What is the probability that $ AC<7$?
$ \textbf{(A)}\ \frac{1}{6} \qquad
\textbf{(B)}\ \frac{1}{5} \qquad
\textbf{(C)}\ \frac{1}{4} \qquad
\textbf{(D)}\ \frac{1}{3} \qquad
\textbf{(E)}\ \frac{1}{2}$
A sextic funtion $ y \equal{} ax^6 \plus{} bx^5 \plus{} cx^4 \plus{} dx^3 \plus{} ex^2 \plus{} fx \plus{} g\ (a\neq 0)$ touches the line $ y \equal{} px \plus{} q$ at $ x \equal{} \alpha ,\ \beta ,\ \gamma \ (\alpha < \beta < \gamma ).$
Find the area of the region bounded by these graphs in terms of $ a,\ \alpha ,\ \beta ,\gamma .$
created by kunny
[b]7.[/b] Let $a_0$ and $a_1$ be arbitrary real numbers, and let
$a_{n+1}=a_n + \frac{2}{n+1}a_{n-1}$ $(n= 1, 2, \dots)$
Show that the sequence $\left (\frac{a_n}{n^2} \right )_{n=1}^{\infty}$ is convergent and find its limit. [b](S. 10)[/b]
A point $P$ is given inside a triangle $ABC$. Let $D,E,F$ be the midpoints of $AP,BP,CP$, and let $L,M,N$ be the intersection points of $ BF$ and $CE, AF$ and $CD, AE$ and $BD$, respectively.
(a) Prove that the area of hexagon $DNELFM$ is equal to one third of the area of triangle $ABC$.
(b) Prove that $DL,EM$, and $FN$ are concurrent.
Let $x$ be the answer to this question. Find the value of $2017 - 2016x$.
[i]Proposed by Michael Tang[/i]
Let $n\ge 2$ be a given integer. Show that the number of strings of length $n$ consisting of $0'$s and $1'$s such that there are equal number of $00$ and $11$ blocks in each string is equal to
\[2\binom{n-2}{\left \lfloor \frac{n-2}{2}\right \rfloor}\]
Let $AD$ be the internal angle bisector of triangle $ABC$. The incircles of triangles
$ABC$ and $ACD$ touch each other externally. Prove that $\angle ABC > 120^{\circ}$. (Recall that the incircle of a triangle is a circle inside the triangle that is tangent to its three sides.)
[i]Proposed by Volodymyr Brayman (Ukraine)[/i]
Let $P$ and $Q$ be interior points in $\Delta ABC$ such that $PQ$ doesn't contain any vertices of $\Delta ABC$.
Let $A_1$, $B_1$, and $C_1$ be the points of intersection of $BC$, $CA$, and $AB$ with $AQ$, $BQ$, and $CQ$, respectively.
Let $K$, $L$, and $M$ be the intersections of $AP$, $BP$, and $CP$ with $B_1C_1$, $C_1A_1$, and $A_1B_1$, respectively.
Prove that $A_1K$, $B_1L$, and $C_1M$ are concurrent.
[b](a)[/b] [Problem for class 11]
Let r be the inradius and $r_a$, $r_b$, $r_c$ the exradii of a triangle ABC. Prove that $\frac{1}{r}=\frac{1}{r_a}+\frac{1}{r_b}+\frac{1}{r_c}$.
[b](b)[/b] [Problem for classes 12/13]
Let r be the radius of the insphere and let $r_a$, $r_b$, $r_c$, $r_d$ the radii of the four exspheres of a tetrahedron ABCD. (An [i]exsphere[/i] of a tetrahedron is a sphere touching one sideface and the extensions of the three other sidefaces.)
Prove that $\frac{2}{r}=\frac{1}{r_a}+\frac{1}{r_b}+\frac{1}{r_c}+\frac{1}{r_d}$.
I am really sorry for posting these, but else, Orl will probably post them. This time, we really did not have any challenging problem on the DeMO. But at least, the problems were simple enough that I solved all of them. ;)
Darij
In the triangle $ABC$ it is known that $AC = 21$ cm, $BC = 28$ cm and $\angle C = 90^o$. On the hypotenuse $AB$, we construct a square $ABMN$ with center $O$ such that the segment $CO$ intersects the hypotenuse $AB$ at the point $K$. Find the lengths of the segments $AK$ and $KB$.
Three distinct points $A$, $B$, and $C$ are fixed on a line in this order. Let $\Gamma$ be a circle passing through $A$ and $C$ whose center does not lie on the line $AC$. Denote by $P$ the intersection of the tangents to $\Gamma$ at $A$ and $C$. Suppose $\Gamma$ meets the segment $PB$ at $Q$. Prove that the intersection of the bisector of $\angle AQC$ and the line $AC$ does not depend on the choice of $\Gamma$.
Andy rolls a fair 4-sided dice, numbered 1 to 4, until he rolls a number that is less than his last roll. If the expected number of times that Andy will roll the dice can be expressed as a reduced fraction $\frac{p}{q}$, find $p + q$.
Let $T = TNYWR$. The solutions in $z$ to the equation \[\left(z + \frac Tz\right)^2 = 1\] form the vertices of a quadrilateral in the complex plane. Compute the area of this quadrilateral.
The trapeze $ABCD$ with bases $AB$ and $CD$ is inscribed in a circle $\Gamma$. Let $X$ be a variable point of the arc $\overarc{AB}$ that does not contain either $C$ or $D$. Let $Y$ be the point of intersection of $AB$ and $DX$, and let $Z$ be the point of the segment $CX$ such that $\frac{XZ}{XC}=\frac{AY}{AB}$. Prove that the measure of the angle $\angle AZX$ does not depend on the choice of $X$.
Define a class of graphs $G_k$ for each positive integer k as follows. A graph G = ( V , E ) is an element of $G_k$ if and only if there exists an edge coloring $\psi: E\to [ k ] = \{1,2, ..., k\}$ such that for all vertex coloring $\phi: V\to [ k ]$ there exist an edge e = { x , y } such that $\phi ( x ) = \phi( y ) = \psi( e )$. Prove that there exist $c_1< c_2$ positive constants with the following two properties:
(i) each graph in $G_k$ has at least $c_1 k^2$ vertices;
(ii) there is a graph in $G_k$ which has at most $c_2 k^2$ vertices.
Let the quadratic $P(x)=x^2+5x+1$. Two distinct real numbers $a,b$ satisfy
\[P(a+b)=ab\]
\[P(ab)=a+b\]
Find the sum of all possible values of $a^2$.
[i]Proposed by Harry Kim[/i]