Found problems: 1581
In an acute angled triangle $ABC$, $\angle A = 30^{\circ}$, $H$ is the orthocenter, and $M$ is the midpoint of $BC$. On the line $HM$, take a point $T$ such that $HM = MT$. Show that $AT = 2 BC$.
In $\triangle ABC$, $AB>AC$. In the circumcircle $(O)$ of $\triangle ABC$, $M$ is the midpoint of arc $BAC$. The incircle $(I)$ of $\triangle ABC$ touches $BC$ at $D$, the line through $D$ parallel to $AI$ intersects $(I)$ again at $P$. Prove that $AP$ and $IM$ intersect at a point on $(O)$.
Find the volume $V$ of the solid formed by a rotation of the region enclosed by the curve $y=2^{x}-1$ and two lines $x=0,\ y=1$ around the $y$ axis.
Two circles $ C_1,C_2$ with different radii are given in the plane, they touch each other externally at $ T$. Consider any points $ A\in C_1$ and $ B\in C_2$, both different from $ T$, such that $ \angle ATB \equal{} 90^{\circ}$.
(a) Show that all such lines $ AB$ are concurrent.
(b) Find the locus of midpoints of all such segments $ AB$.
Let $ ABC$ be a triangle, line $ l$ cuts its sides $ BC,CA,AB$ at $ D,E,F$, respectively. Denote by $ O_{1},O_{2},O_{3}$ the circumcenters of triangle $ AEF,BFD,CDE$, respectively. Prove that the orthocenter of triangle $ O_{1}O_{2}O_{3}$ lies on line $ l$.
Let $ABC$ be an acute angled triangle satisfying the conditions $AB>BC$ and $AC>BC$. Denote by $O$ and $H$ the circumcentre and orthocentre, respectively, of the triangle $ABC.$ Suppose that the circumcircle of the triangle $AHC$ intersects the line $AB$ at $M$ different from $A$, and the circumcircle of the triangle $AHB$ intersects the line $AC$ at $N$ different from $A.$ Prove that the circumcentre of the triangle $MNH$ lies on the line $OH$.
In $\triangle ABC$ we have $BC>CA>AB$. The nine point circle is tangent to the incircle, $A$-excircle, $B$-excircle and $C$-excircle at the points $T,T_A,T_B,T_C$ respectively. Prove that the segments $TT_B$ and lines $T_AT_C$ intersect each other.
On a board, there are $n$ equations in the form $*x^2+*x+*$. Two people play a game where they take turns. During a turn, you are aloud to change a star into a number not equal to zero. After $3n$ moves, there will be $n$ quadratic equations. The first player is trying to make more of the equations not have real roots, while the second player is trying to do the opposite. What is the maximum number of equations that the first player can create without real roots no matter how the second player acts?
Points $X$ and $Y$ inside the rhombus $ABCD$ are such that $Y$ is inside the convex quadrilateral $BXDC$ and $2\angle XBY = 2\angle XDY = \angle ABC$. Prove that the lines $AX$ and $CY$ are parallel.
[i]S. Berlov[/i]
Let $a,b$ be integers with $0<a<b$. A set $\{x,y,z\}$ of non-negative integers is [i]olympic[/i] if $x<y<z$ and if $\{z-y,y-x\}=\{a,b\}$. Show that the set of all non-negative integers is the union of pairwise disjoint olympic sets.
Alice and Bob are in a hardware store. The store sells coloured sleeves that fit over keys to distinguish them. The following conversation takes place:
[color=#0000FF]Alice:[/color] Are you going to cover your keys?
[color=#FF0000]Bob:[/color] I would like to, but there are only $7$ colours and I have $8$ keys.
[color=#0000FF]Alice:[/color] Yes, but you could always distinguish a key by noticing that the red key next to the green key was different from the red key next to the blue key.
[color=#FF0000]Bob:[/color] You must be careful what you mean by "[i]next to[/i]" or "[i]three keys over from[/i]" since you can turn the key ring over and the keys are arranged in a circle.
[color=#0000FF]Alice:[/color] Even so, you don't need $8$ colours.
[b]Problem:[/b] What is the smallest number of colours needed to distinguish $n$ keys if all the keys are to be covered.
$A,B,C,D,E$ are points on a circle $\omega$, satisfying $AB=BD$, $BC=CE$. $AC$ meets $BE$ at $P$. $Q$ is on $DE$ such that $BE//AQ$. Suppose $\odot(APQ)$ intersects $\omega$ again at $T$. $A'$ is the reflection of $A$ wrt $BC$. Prove that $A'BPT$ lies on the same circle.
Let $ABC$ be an acute triangle. Let $\omega$ be a circle whose centre $L$ lies on the side $BC$. Suppose that $\omega$ is tangent to $AB$ at $B'$ and $AC$ at $C'$. Suppose also that the circumcentre $O$ of triangle $ABC$ lies on the shorter arc $B'C'$ of $\omega$. Prove that the circumcircle of $ABC$ and $\omega$ meet at two points.
[i]Proposed by Härmel Nestra, Estonia[/i]
Let $ABC$ be a triangle and $D$ be a point on $BC$ such that $AB+BD=AC+CD$. The line $AD$ intersects the incircle of triangle $ABC$ at $X$ and $Y$ where $X$ is closer to $A$ than $Y$ i. Suppose $BC$ is tangent to the incircle at $E$, prove that:
1) $EY$ is perpendicular to $AD$;
2) $XD=2IM$ where $I$ is the incentre and $M$ is the midpoint of $BC$.
Two three-letter strings, $aaa$ and $bbb$, are transmitted electronically. Each string is sent letter by letter. Due to faulty equipment, each of the six letters has a 1/3 chance of being received incorrectly, as an $a$ when it should have been a $b$, or as a $b$ when it should be an $a$. However, whether a given letter is received correctly or incorrectly is independent of the reception of any other letter. Let $S_a$ be the three-letter string received when $aaa$ is transmitted and let $S_b$ be the three-letter string received when $bbb$ is transmitted. Let $p$ be the probability that $S_a$ comes before $S_b$ in alphabetical order. When $p$ is written as a fraction in lowest terms, what is its numerator?
In triangle ABC, let $P$ and $Q$ be two interior points such that $\angle ABP = \angle QBC$ and $\angle ACP = \angle QCB$. Point $D$ lies on segment $BC$. Prove that $\angle APB + \angle DPC = 180^\circ$ if and only if $\angle AQC + \angle DQB = 180^\circ$.
A convex 2012-gon $A_1A_2A_3 \dots A_{2012}$ has the property that for every integer $1 \le i \le 1006$, $\overline{A_iA_{i+1006}}$ partitions the polygon into two congruent regions. Show that for every pair of integers $1 \le j < k \le 1006$, quadrilateral $A_jA_kA_{j+1006}A_{k+1006}$ is a parallelogram.
[i]Proposed by Lewis Chen[/i]
$ABC$ is a non-isosceles triangle.
$T_A$ is the tangency point of incircle of $ABC$ in the side $BC$ (define $T_B$,$T_C$ analogously).
$I_A$ is the ex-center relative to the side BC (define $I_B$,$I_C$ analogously).
$X_A$ is the mid-point of $I_BI_C$ (define $X_B$,$X_C$ analogously).
Show that $X_AT_A$,$X_BT_B$,$X_CT_C$ meet in a common point, colinear with the incenter and circumcenter of $ABC$.
Let $ABC$ be a triangle, and $B_1,C_1$ be its excenters opposite $B,C$. $B_2,C_2$ are reflections of $B_1,C_1$ across midpoints of $AC,AB$. Let $D$ be the extouch at $BC$. Show that $AD$ is perpendicular to $B_2C_2$
Let in-circle of $ABC$ touch $AB$, $BC$, $AC$ in $C_1$, $A_1$, $B_1$ respectively.
Let $H$- intersection point of altitudes in $A_1B_1C_1$, $I$ and $O$-be in-center and circumcenter of $ABC$ respectively.
Prove, that $I, O, H$ lies on one line.
A tetrahedron is called a [i]MEMO-tetrahedron[/i] if all six sidelengths are different positive integers where one of them is $ 2$ and one of them is $ 3$. Let $ l(T)$ be the sum of the sidelengths of the tetrahedron $ T$.
(a) Find all positive integers $ n$ so that there exists a MEMO-Tetrahedron $ T$ with $ l(T)\equal{}n$.
(b) How many pairwise non-congruent MEMO-tetrahedrons $ T$ satisfying $ l(T)\equal{}2007$ exist? Two tetrahedrons are said to be non-congruent if one cannot be obtained from the other by a composition of reflections in planes, translations and rotations. (It is not neccessary to prove that the tetrahedrons are not degenerate, i.e. that they have a positive volume).
Let us consider an infinite grid plane as shown below. We start with 4 points $A$, $B$, $C$, $D$, that form a square.
We perform the following operation: We pick two points $X$ and $Y$ from the currant points. $X$ is reflected about $Y$ to get $X'$. We remove $X$ and add $X'$ to get a new set of 4 points and treat it as our currant points.
For example in the figure suppose we choose $A$ and $B$ (we can choose any other pair too). Then reflect $A$ about $B$ to get $A'$. We remove $A$ and add $A'$. Thus $A'$, $B$, $C$, $D$ is our new 4 points. We may again choose $D$ and $A'$ from the currant points. Reflect $D$ about $A'$ to obtain $D'$ and hence $A'$, $B$, $C$, $D'$ are now new set of points. Then similar operation is performed on this new 4 points and so on.
Starting with $A$, $B$, $C$, $D$ can you get a bigger square by some sequence of such operations?
Let $ABC$ be a triangle, and $O$ the center of its circumcircle.
Let a line through the point $O$ intersect the lines $AB$ and $AC$ at the points $M$ and $N$, respectively. Denote by $S$ and $R$ the midpoints of the segments $BN$ and $CM$, respectively.
Prove that $\measuredangle ROS=\measuredangle BAC$.
The point $(-1, -2)$ is rotated $270^{\circ}$ counterclockwise about the point $(3, 1)$. What are the coordinates of its new position?
$\textbf{(A)}\ (-3, -4) \qquad \textbf{(B)}\ (0,5) \qquad \textbf{(C)}\ (2,-1) \qquad \textbf{(D)}\ (4,3) \qquad \textbf{(E)}\ (6,-3)$
Let $E$ be a point inside the rhombus $ABCD$ such that $|AE|=|EB|$, $m(\widehat{EAB})=12^\circ$, and $m(\widehat{DAE})=72^\circ$. What is $m(\widehat{CDE})$ in degrees?
$
\textbf{(A)}\ 64
\qquad\textbf{(B)}\ 66
\qquad\textbf{(C)}\ 68
\qquad\textbf{(D)}\ 70
\qquad\textbf{(E)}\ 72
$