Found problems: 1581
Let $\Gamma_1$ be a circle and $P$ a point outside of $\Gamma_1$. The tangents from $P$ to $\Gamma_1$ touch the circle at $A$ and $B$. Let $M$ be the midpoint of $PA$ and $\Gamma_2$ the circle through $P$, $A$ and $B$. Line $BM$ cuts $\Gamma_2$ at $C$, line $CA$ cuts $\Gamma_1$ at $D$, segment $DB$ cuts $\Gamma_2$ at $E$ and line $PE$ cuts $\Gamma_1$ at $F$, with $E$ in segment $PF$. Prove lines $AF$, $BP$, and $CE$ are concurrent.
$I$ is the incenter of triangle $ABC$. perpendicular from $I$ to $AI$ meet $AB$ and $AC$ at ${B}'$ and ${C}'$ respectively .
Suppose that ${B}''$ and ${C}''$ are points on half-line $BC$ and $CB$ such that $B{B}''=BA$ and $C{C}''=CA$.
Suppose that the second intersection of circumcircles of $A{B}'{B}''$ and $A{C}'{C}''$ is $T$.
Prove that the circumcenter of $AIT$ is on the $BC$.
Let $L$ be a subset in the coordinate plane defined by $L = \{(41x + 2y, 59x + 15y) | x, y \in \mathbb Z \}$, where $\mathbb Z$ is set of integers. Prove that for any parallelogram with center in the origin of coordinate and area $1990$, there exist at least two points of $L$ located in it.
The vertices A,B,C of an acute-angled triangle ABC lie on the sides B1C1, C1A1, A1B1 respectively of a triangle A1B1C1 similar to the triangle ABC (∠A = ∠A1, etc.). Prove that the orthocenters of triangles ABC and A1B1C1 are equidistant from the circumcenter of △ABC.
Let $ABCD$ be a quadrilateral inscribed in a circle $\omega$. The lines $AB$ and $CD$ meet at $P$, the lines $AD$ and $BC$ meet at $Q$, and the diagonals $AC$ and $BD$ meet at $R$. Let $M$ be the midpoint of the segment $PQ$, and let $K$ be the common point of the segment $MR$ and the circle $\omega$. Prove that the circumcircle of the triangle $KPQ$ and $\omega$ are tangent to one another.
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 $k$ and $a$ are positive constants. Denote by $V_1$ the volume of the solid generated by a rotation of the figure enclosed
by the curve $C: y=\frac{x}{x+k}\ (x\geq 0)$, the line $x=a$ and the $x$-axis around the $x$-axis, and denote by $V_2$ that of
the solid by a rotation of the figure enclosed by the curve $C$, the line $y=\frac{a}{a+k}$ and the $y$-axis around the $y$-axis.
Find the ratio $\frac{V_2}{V_1}.$
Let $O$ denote the circumcentre of an acute-angled triangle $ABC$. Let point $P$ on side $AB$ be such that $\angle BOP = \angle ABC$, and let point $Q$ on side $AC$ be such that $\angle COQ = \angle ACB$. Prove that the reflection of $BC$ in the line $PQ$ is tangent to the circumcircle of triangle $APQ$.
The sum of the squares of three positive numbers is $160$. One of the numbers is equal to the sum of the other two. The difference between the smaller two numbers is $4.$ What is the difference between the cubes of the smaller two numbers?
[i]Author: Ray Li[/i]
[hide="Clarification"]The problem should ask for the positive difference.[/hide]
Let $ S$ be the set of points in the Cartesian plane that satisfy
\[ \Big|\big|{|x| \minus{} 2}\big| \minus{} 1\Big| \plus{} \Big|\big|{|y| \minus{} 2}\big| \minus{} 1\Big| \equal{} 1.
\]
If a model of $ S$ were built from wire of negligible thickness, then the total length of wire required would be $ a\sqrt {b},$ where $ a$ and $ b$ are positive integers and $ b$ is not divisible by the square of any prime number. Find $ a \plus{} b.$
Let $ABCD$ be a cyclic quadrilateral. Points $K, L, M, N$ are chosen on $AB, BC, CD, DA$ such that $KLMN$ is a rhombus with $KL \parallel AC$ and $LM \parallel BD$. Let $\omega_A, \omega_B, \omega_C, \omega_D$ be the incircles of $\triangle ANK, \triangle BKL, \triangle CLM, \triangle DMN$.
Prove that the common internal tangents to $\omega_A$, and $\omega_C$ and the common internal tangents to $\omega_B$ and $\omega_D$ are concurrent.
By the time a party is over, $ 28$ handshakes have occurred. If everyone shook everyone else's hand once, how many people attended the party?
Let $P$ be a point inside a triangle $ABC$ such that $\angle PAC= \angle PCB$. Let the projections of $P$ onto $BC$, $CA$, and $AB$ be $X,Y,Z$ respectively. Let $O$ be the circumcenter of $\triangle XYZ$, $H$ be the foot of the altitude from $B$ to $AC$, $N$ be the midpoint of $AC$, and $T$ be the point such that $TYPO$ is a parallelogram. Show that $\triangle THN$ is similar to $\triangle PBC$.
[i]Proposed by Sammy Luo[/i]
The points $A_1,B_1,C_1$ lie on the sides $BC,CA$ and $AB$ of the triangle $ABC$ respectively. Suppose that $AB_1-AC_1=CA_1-CB_1=BC_1-BA_1$. Let $O_A,O_B$ and $O_C$ be the circumcentres of triangles $AB_1C_1,A_1BC_1$ and $A_1B_1C$ respectively. Prove that the incentre of triangle $O_AO_BO_C$ is the incentre of triangle $ABC$ too.
Given circle $O$ with radius $R$, the inscribed triangle $ABC$ is an acute scalene triangle, where $AB$ is the largest side. $AH_A, BH_B,CH_C$ are heights on $BC,CA,AB$. Let $D$ be the symmetric point of $H_A$ with respect to $H_BH_C$, $E$ be the symmetric point of $H_B$ with respect to $H_AH_C$. $P$ is the intersection of $AD,BE$, $H$ is the orthocentre of $\triangle ABC$. Prove: $OP\cdot OH$ is fixed, and find this value in terms of $R$.
(Edited)
[color=darkred]Quadrilateral $ ABCD$ is inscribed in the circle of diameter $ BD$. Point $ A_1$ is reflection of point $ A$ wrt $ BD$ and $ B_1$ is reflection of $ B$ wrt $ AC$. Denote $ \{P\}\equal{}CA_1 \cap BD$ and $ \{Q\}\equal{}DB_1\cap AC$. Prove that $ AC\perp PQ$.[/color]
Let $f : \mathbb R \to \mathbb R$ be a function such that
\[f(x+y)=f(x) \cdot f(y) \qquad \forall x,y \in \mathbb R\]
Suppose that $f(0) \neq 0$ and $f(0)$ exists and it is finite $(f(0) \neq \infty)$. Prove that $f$ has derivative in each point $x \in \mathbb R.$
Let $\triangle ABC$ be an acute triangle with $AB>AC$, let $I$ be the center of the incircle. Let $M,N$ be the midpoint of $AC$ and $AB$ respectively. $D,E$ are on $AC$ and $AB$ respectively such that $BD\parallel IM$ and $CE\parallel IN$. A line through $I$ parallel to $DE$ intersects $BC$ in $P$. Let $Q$ be the projection of $P$ on line $AI$. Prove that $Q$ is on the circumcircle of $\triangle ABC$.
In acute triangle $ABC$, $AB > AC$.
$D$ and $E$ are the midpoints of $AB$, $AC$ respectively.
The circumcircle of $ADE$ intersects the circumcircle of $BCE$ again at $P$.
The circumcircle of $ADE$ intersects the circumcircle $BCD$ again at $Q$.
Prove that $AP = AQ$.
Suppose circles $\mathit{W}_1$ and $\mathit{W}2$, with centres $\mathit{O}_1$ and $\mathit{O}_2$ respectively, intersect at points $\mathit{M}$ and $\mathit{N}$. Let the tangent on $\mathit{W}_2$ at point $\mathit{N}$ intersect $\mathit{W}_1$ for the second time at $\mathit{B}_1$. Similarly, let the tangent on $\mathit{W}_1$ at point $\mathit{N}$ intersect $\mathit{W}_2$ for the second time at $\mathit{B}_2$. Let $\mathit{A}_1$ be a point on $\mathit{W}_1$ which is on arc $\mathit{B}_1\mathit{N}$ not containing $\mathit{M}$ and suppose line $\mathit{A}_1\mathit{N}$ intersects $\mathit{W}_2$ at point $\mathit{A}_2$. Denote the incentres of triangles $\mathit{B}_1\mathit{A}_1\mathit{N}$ and $\mathit{B}_2\mathit{A}_2\mathit{N}$ by $\mathit{I}_1$ and $\mathit{I}_2$, respectively.*
[asy]
/* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki, go to User:Azjps/geogebra */
import graph; size(10.1cm);
real labelscalefactor = 0.5; /* changes label-to-point distance */
pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */
pen dotstyle = black; /* point style */
real xmin = -0.9748626324969808, xmax = 13.38440254515721, ymin = 0.5680051903627492, ymax = 10.99430986899034; /* image dimensions */
pair O_2 = (7.682929606970993,6.084708172218866), O_1 = (2.180000000000002,6.760000000000007), M = (4.560858774883258,8.585242858926296), B_2 = (10.07334553576748,9.291873850408265), A_2 = (11.49301008867042,4.866805580476367), B_1 = (2.113311869970955,9.759258690628950), A_1 = (0.2203184186713625,4.488514120712773);
/* draw figures */
draw(circle(O_2, 4.000000000000000));
draw(circle(O_1, 3.000000000000000));
draw((4.048892687647541,4.413249028538064)--B_2);
draw(B_2--A_2);
draw(A_2--(4.048892687647541,4.413249028538064));
draw((4.048892687647541,4.413249028538064)--B_1);
draw(B_1--A_1);
draw(A_1--(4.048892687647541,4.413249028538064));
/* dots and labels */
dot(O_2,dotstyle);
label("$O_2$", (7.788512439159622,6.243082420501817), NE * labelscalefactor);
dot(O_1,dotstyle);
label("$O_1$", (2.298205165350667,6.929370829727937), NE * labelscalefactor);
dot(M,dotstyle);
label("$M$", (4.383466101076183,8.935444641311980), NE * labelscalefactor);
dot((4.048892687647541,4.413249028538064),dotstyle);
label("$N$", (3.855551940133015,3.761885864068922), NE * labelscalefactor);
dot(B_2,dotstyle);
label("$B_2$", (10.19052187145104,9.463358802255147), NE * labelscalefactor);
dot(A_2,dotstyle);
label("$A_2$", (11.80066006232771,4.659339937672310), NE * labelscalefactor);
dot(B_1,dotstyle);
label("$B_1$", (1.981456668784765,10.09685579538695), NE * labelscalefactor);
dot(A_1,dotstyle);
label("$A_1$", (0.08096568938935705,3.973051528446190), NE * labelscalefactor);
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);
/* end of picture */[/asy]
Show that \[\angle\mathit{I}_1\mathit{MI}_2=\angle\mathit{O}_1\mathit{MO}_2.\]
*[size=80]Given a triangle ABC, the incentre of the triangle is defined to be the intersection of the angle bisectors of A, B, and C. To avoid cluttering, the incentre is omitted in the provided diagram. Note also that the diagram serves only as an aid and is not necessarily drawn to scale.[/size]
The plane is divided by three infinite sets of parallel lines into equilateral triangles of equal area. Let $M$ be the set of their vertices, and $A$ and $B$ be two vertices of such an equilateral triangle. One may rotate the plane through $120^o$ around any vertex of the set $M$. Is it possible to move the point $A$ to the point $B$ by a number of such rotations
(N Vasiliev, Moscow)
Let $ ABC$ be a non-obtuse triangle with $ CH$ and $ CM$ are the altitude and median, respectively. The angle bisector of $ \angle BAC$ intersects $ CH$ and $ CM$ at $ P$ and $ Q$, respectively. Assume that \[ \angle ABP\equal{}\angle PBQ\equal{}\angle QBC,\]
(a) prove that $ ABC$ is a right-angled triangle, and
(b) calculate $ \dfrac{BP}{CH}$.
[i]Soewono, Bandung[/i]
Aaron takes a square sheet of paper, with one corner labeled $A$. Point $P$ is chosen at random inside of the square and Aaron folds the paper so that points $A$ and $P$ coincide. He cuts the sheet along the crease and discards the piece containing $A$. Let $p$ be the probability that the remaining piece is a pentagon. Find the integer nearest to $100p$.
[i]Proposed by Aaron Lin[/i]
In a group of $n$ people, each one knows exactly three others. They are seated around a table. We say that the seating is $perfect$ if everyone knows the two sitting by their sides. Show that, if there is a perfect seating $S$ for the group, then there is always another perfect seating which cannot be obtained from $S$ by rotation or reflection.
Let $ABC$ be a triangle in which $\angle A = 60^\circ$. Let $BE$ and $CF$ be the bisectors of $\angle B$ and $\angle C$ with $E$ on $AC$ and $F$ on $AB$. Let $M$ be the reflection of $A$ in line $EF$. Prove that $M$ lies on $BC$.