Found problems: 3585
Suppose $\omega$ is a circle centered at $O$ with radius $8$. Let $AC$ and $BD$ be perpendicular chords of $\omega$. Let $P$ be a point inside quadrilateral $ABCD$ such that the circumcircles of triangles $ABP$ and $CDP$ are tangent, and the circumcircles of triangles $ADP$ and $BCP$ are tangent. If $AC = 2\sqrt{61}$ and $BD = 6\sqrt7$,then $OP$ can be expressed as $\sqrt{a}-\sqrt{b}$ for positive integers $a$ and $b$. Compute $100a + b$.
A triangle $ABC$ with $AB>AC$ is given, $AD$ is the A-angle bisector with point $D$ on $BC$ and point $I$ is the incenter of triangle $ABC$. Point M is the midpoint of segment $AD$ and point $F$ is the second intersection of $MB$ with the circumcirle of triangle $BIC$. Prove that $AF\bot FC$.
Let $ABC$ a triangle with $AB<AC$ and let $I$ it´s incenter. Let $\Gamma$ the circumcircle of $\triangle BIC$. $AI$ intersect $\Gamma$ again in $P$. Let $Q$ a point in side $AC$ such that $AB=AQ$ and let $R$ a point in $AB$ with $B$ between $A$ and $R$ such that $AR=AC$. Prove that $IQPR$ is cyclic.
Let $ABC$ be a triangle with circumcenter $O$. Let $P$ be a point inside $ABC$, so let the points $D, E, F$ be on $BC, AC, AB$ respectively so that the Miquel point of $DEF$ with respect to $ABC$ is $P$. Let the reflections of $D, E, F$ over the midpoints of the sides that they lie on be $R, S, T$. Let the Miquel point of $RST$ with respect to the triangle $ABC$ be $Q$. Show that $OP = OQ$.
[i]Proposed by Yang Liu[/i]
The circle $\omega_1$ with diameter $[AB]$ and the circle $\omega_2$ with center $A$ intersects at points $C$ and $D$. Let $E$ be a point on the circle $\omega_2$, which is outside $\omega_1$ and at the same side as $C$ with respect to the line $AB$. Let the second point of intersection of the line $BE$ with $\omega_2$ be $F$. For a point $K$ on the circle $\omega_1$ which is on the same side as $A$ with respect to the diameter of $\omega_1$ passing through $C$ we have $2\cdot CK \cdot AC = CE \cdot AB$. Let the second point of intersection of the line $KF$ with $\omega_1$ be $L$. Show that the symmetric of the point $D$ with respect to the line $BE$ is on the circumcircle of the triangle $LFC$.
Let $D$ and $E$ be points on the sides $BC$ and $AC$ of the triangle $\triangle ABC$, respectively. The circumcircle of $\triangle ADC$ meets the circumcircle of $\triangle BCE$ for the second time at $F$. The line $FE$ meets the line $AD$ at $G$, while the line $FD$ meets the line $BE$ at $H$. Prove that the lines $CF$, $AH$ and $BG$ pass through the same point.
[i]Authored by Petar Filipovski[/i]
Given triangle $ABC$ with $A$-excenter $I_A$, the foot of the perpendicular from $I_A$ to $BC$ is $D$. Let the midpoint of segment $I_AD$ be $M$, $T$ lies on arc $BC$(not containing $A$) satisfying $\angle BAT=\angle DAC$, $I_AT$ intersects the circumcircle of $ABC$ at $S\neq T$. If $SM$ and $BC$ intersect at $X$, the perpendicular bisector of $AD$ intersects $AC,AB$ at $Y,Z$ respectively, prove that $AX,BY,CZ$ are concurrent.
Let $ABCC_1B_1A_1$ be a convex hexagon such that $AB=BC$, and suppose that the line segments $AA_1, BB_1$, and $CC_1$ have the same perpendicular bisector. Let the diagonals $AC_1$ and $A_1C$ meet at $D$, and denote by $\omega$ the circle $ABC$. Let $\omega$ intersect the circle $A_1BC_1$ again at $E \neq B$. Prove that the lines $BB_1$ and $DE$ intersect on $\omega$.
Quadrilateral $ABCD$ is inscribed in circle $O$ and has sides $AB = 3$, $BC = 2$, $CD = 6$, and $DA = 8$. Let $X$ and $Y$ be points on $\overline{BD}$ such that
\[\frac{DX}{BD} = \frac{1}{4} \quad \text{and} \quad \frac{BY}{BD} = \frac{11}{36}.\]
Let $E$ be the intersection of intersection of line $AX$ and the line through $Y$ parallel to $\overline{AD}$. Let $F$ be the intersection of line $CX$ and the line through $E$ parallel to $\overline{AC}$. Let $G$ be the point on circle $O$ other than $C$ that lies on line $CX$. What is $XF \cdot XG$?
$\textbf{(A) }17\qquad\textbf{(B) }\frac{59 - 5\sqrt{2}}{3}\qquad\textbf{(C) }\frac{91 - 12\sqrt{3}}{4}\qquad\textbf{(D) }\frac{67 - 10\sqrt{2}}{3}\qquad\textbf{(E) }18$
Lateral sidelines $AB$ and $CD$ of a trapezoid $ABCD$ ($AD >BC$) meet at point $P$. Let $Q$ be a point of segment $AD$ such that $BQ = CQ$. Prove that the line passing through the circumcenters of triangles $AQC$ and $BQD$ is perpendicular to $PQ$.
[b]p1.[/b] Let $f$ be a function such that $f(x + y) = f(x) + f(y)$ for all $x$ and $y$. Assume $f(5) = 9$. Compute $f(2015)$.
[b]p2.[/b] There are six cards, with the numbers $2, 2, 4, 4, 6, 6$ on them. If you pick three cards at random, what is the probability that you can make a triangles whose side lengths are the chosen numbers?
[b]p3. [/b]A train travels from Berkeley to San Francisco under a tunnel of length $10$ kilometers, and then returns to Berkeley using a bridge of length $7$ kilometers. If the train travels at $30$ km/hr underwater and 60 km/hr above water, what is the train’s average speed in km/hr on the round trip?
[b]p4.[/b] Given a string consisting of the characters A, C, G, U, its reverse complement is the string obtained by first reversing the string and then replacing A’s with U’s, C’s with G’s, G’s with C’s, and U’s with A’s. For example, the reverse complement of UAGCAC is GUGCUA. A string is a palindrome if it’s the same as its reverse. A string is called self-conjugate if it’s the same as its reverse complement. For example, UAGGAU is a palindrome and UAGCUA is self-conjugate. How many six letter strings with just the characters A, C, G (no U’s) are either palindromes or self-conjugate?
[b]p5.[/b] A scooter has $2$ wheels, a chair has $6$ wheels, and a spaceship has $11$ wheels. If there are $10$ of these objects, with a total of $50$ wheels, how many chairs are there?
[b]p6.[/b] How many proper subsets of $\{1, 2, 3, 4, 5, 6\}$ are there such that the sum of the elements in the subset equal twice a number in the subset?
[b]p7.[/b] A circle and square share the same center and area. The circle has radius $1$ and intersects the square on one side at points $A$ and $B$. What is the length of $\overline{AB}$ ?
[b]p8. [/b]Inside a circle, chords $AB$ and $CD$ intersect at $P$ in right angles. Given that $AP = 6$, $BP = 12$ and $CD = 15$, find the radius of the circle.
[b]p9.[/b] Steven makes nonstandard checkerboards that have $29$ squares on each side. The checkerboards have a black square in every corner and alternate red and black squares along every row and column. How many black squares are there on such a checkerboard?
[b]p10.[/b] John is organizing a race around a circular track and wants to put $3$ water stations at $9$ possible spots around the track. He doesn’t want any $2$ water stations to be next to each other because that would be inefficient. How many ways are possible?
[b]p11.[/b] In square $ABCD$, point $E$ is chosen such that $CDE$ is an equilateral triangle. Extend $CE$ and $DE$ to $F$ and $G$ on $AB$. Find the ratio of the area of $\vartriangle EFG$ to the area of $\vartriangle CDE$.
[b]p12.[/b] Let $S$ be the number of integers from $2$ to $8462$ (inclusive) which does not contain the digit $1,3,5,7,9$. What is $S$?
[b]p13.[/b] Let x, y be non zero solutions to $x^2 + xy + y^2 = 0$. Find $\frac{x^{2016} + (xy)^{1008} + y^{2016}}{(x + y)^{2016}}$ .
[b]p14.[/b] A chess contest is held among $10$ players in a single round (each of two players will have a match). The winner of each game earns $2$ points while loser earns none, and each of the two players will get $1$ point for a draw. After the contest, none of the $10$ players gets the same score, and the player of the second place gets a score that equals to $4/5$ of the sum of the last $5$ players. What is the score of the second-place player?
[b]p15.[/b] Consider the sequence of positive integers generated by the following formula
$a_1 = 3$, $a_{n+1} = a_n + a^2_n$ for $n = 2, 3, ...$
What is the tens digit of $a_{1007}$?
[b]p16.[/b] Let $(x, y, z)$ be integer solutions to the following system of equations
$x^2z + y^2z + 4xy = 48$
$x^2 + y^2 + xyz = 24$
Find $\sum x + y + z$ where the sum runs over all possible $(x, y, z)$.
[b]p17.[/b] Given that $x + y = a$ and $xy = b$ and $1 \le a, b \le 50$, what is the sum of all a such that $x^4 + y^4 - 2x^2y^2$ is a prime squared?
[b]p18.[/b] In $\vartriangle ABC$, $M$ is the midpoint of $\overline{AB}$, point $N$ is on side $\overline{BC}$. Line segments $\overline{AN}$ and $\overline{CM}$ intersect at $O$. If $AO = 12$, $CO = 6$, and $ON = 4$, what is the length of $OM$?
[b]p19.[/b] Consider the following linear system of equations.
$1 + a + b + c + d = 1$
$16 + 8a + 4b + 2c + d = 2$
$81 + 27a + 9b + 3c + d = 3$
$256 + 64a + 16b + 4c + d = 4$
Find $a - b + c - d$.
[b]p20.[/b] Consider flipping a fair coin $ 8$ times. How many sequences of coin flips are there such that the string HHH never occurs?
PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
Let $O$ be the circumcenter of an acute triangle $ABC$. Line $OA$ intersects the altitudes of $ABC$ through $B$ and $C$ at $P$ and $Q$, respectively. The altitudes meet at $H$. Prove that the circumcenter of triangle $PQH$ lies on a median of triangle $ABC$.
Points $P,Q,R,S$ lie on a circle and $\angle PSR$ is right. $H,K$ are the projections of $Q$ on lines $PR,PS$. Prove that $HK$ bisects segment $ QS$.
$ABCDE$ is convex pentagon. $\angle BCA=\angle BEA = \frac{\angle BDA}{2}, \angle BDC =\angle EDA$.
Prove, that $\angle DEB=\angle DAC$
Triangle $ABC$ satisfies $\angle ABC < \angle BCA < \angle CAB < 90^{\circ}$. $O$ is the circumcenter of triangle $ABC$, and $K$ is the reflection of $O$ in $BC$. $D,E$ is the foot of perpendicular line from $K$ to line $AB$, $AC$, respectively. Line $DE$ meets $BC$ at $P$, and a circle with diameter $AK$ meets the circumcircle of triangle $ABC$ at $Q(\neq A)$. If $PQ$ cuts the perpendicular bisector of $BC$ at $S$, then prove that $S$ lies on the circle with diameter $AK$.
$ABCD$ is a cyclic quadrilateral with circumcenter $O$. The lines $AC, BD$ intersect at $E$ and $AD, BC$ intersect at $F$. $O_1$ and $O_2$ are the circumcenters of $\triangle ABE$ and $\triangle CDE$, respectively. Assume that $(ABCD)$ and $(OO_1O_2)$ intersect at two points $P, Q$. Prove that $P, Q, F$ are collinear.
Let $ABC$ be an acute triangle and let $M$ be the midpoint of side $BC$. Let $D,E$ be the excircles of triangles $AMB,AMC$ respectively, towards $M$. Circumcirscribed circle of triangle $ABD$ intersects line $BC$ at points $B$ and $F$. Circumcirscribed circles of triangle $ACE$ intersects line $BC$ at points $C$ and $G$. Prove that $BF=CG$.
by Petru Braica, Romania
Let $\triangle ABC$ be an acute triangle with incenter $I$ and circumcenter $O$. The incircle touches sides $BC,CA,$ and $AB$ at $D,E,$ and $F$ respectively, and $A'$ is the reflection of $A$ over $O$. The circumcircles of $ABC$ and $A'EF$ meet at $G$, and the circumcircles of $AMG$ and $A'EF$ meet at a point $H\neq G$, where $M$ is the midpoint of $EF$. Prove that if $GH$ and $EF$ meet at $T$, then $DT\perp EF$.
[i]Proposed by Ankit Bisain[/i]
Let $I$ be the incentre of a non-equilateral triangle $ABC$, $I_A$ be the $A$-excentre, $I'_A$ be the reflection of $I_A$ in $BC$, and $l_A$ be the reflection of line $AI'_A$ in $AI$. Define points $I_B$, $I'_B$ and line $l_B$ analogously. Let $P$ be the intersection point of $l_A$ and $l_B$.
[list=a]
[*] Prove that $P$ lies on line $OI$ where $O$ is the circumcentre of triangle $ABC$.
[*] Let one of the tangents from $P$ to the incircle of triangle $ABC$ meet the circumcircle at points $X$ and $Y$. Show that $\angle XIY = 120^{\circ}$.
[/list]
Let $ABC$ be an acute-angled triangle with $AC>BC$ and circumcircle $\omega$. Suppose that $P$ is a point on $\omega$ such that $AP=AC$ and that $P$ is an interior point on the shorter arc $BC$ of $\omega$. Let $Q$ be the intersection point of the lines $AP$ and $BC$. Furthermore, suppose that $R$ is a point on $\omega$ such that $QA=QR$ and $R$ is an interior point of the shorter arc $AC$ of $\omega$. Finally, let $S$ be the point of intersection of the line $BC$ with the perpendicular bisector of the side $AB$.
Prove that the points $P, Q, R$ and $S$ are concyclic.
[i]Proposed by Patrik Bak, Slovakia[/i]
In the triangle $ABC$, $M$ is the midpoint of $AB$ and $B'$ is the foot of $B$-altitude. $CB'M$ intersects the line $BC$ for the second time at $D$. Circumcircles of $CB'M$ and $ABD$ intersect each other again at $K$. The parallel to $AB$ through $C$ intersects the $CB'M$ circle again at $L$. Prove that $KL$ cuts $CM$ in half.
Let $ABC$ be a triangle with $AB=AC$, and let $M$ be the midpoint of $BC$. Let $P$ be a point such that $PB<PC$ and $PA$ is parallel to $BC$. Let $X$ and $Y$ be points on the lines $PB$ and $PC$, respectively, so that $B$ lies on the segment $PX$, $C$ lies on the segment $PY$, and $\angle PXM=\angle PYM$. Prove that the quadrilateral $APXY$ is cyclic.
Let $ABC$ be a triangle such that $\angle C=2\angle B$ and $\omega$ be its circumcircle. a tangent from $A$ to $\omega$ intersect $BC$ at $E$. $\Omega$ is a circle passing throw $B$ that is tangent to $AC$ at $C$. Let $\Omega\cap AB=F$. $K$ is a point on $\Omega$ such that $EK$ is tangent to $\Omega$ ($A,K$ aren't in one side of $BC$). Let $M$ be the midpoint of arc $BC$ of $\omega$ (not containing $A$). Prove that $AFMK$ is a cyclic quadrilateral.
[asy]
import graph; size(15.424606256655986cm);
real labelscalefactor = 0.5; /* changes label-to-point distance */
pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */ real xmin = -7.905629294221492, xmax = 11.618976962434495, ymin = -5.154837585051625, ymax = 4.0091473316396895; /* image dimensions */
pen uuuuuu = rgb(0.26666666666666666,0.26666666666666666,0.26666666666666666);
/* draw figures */
draw(circle((1.4210145017438194,0.18096629151696939), 2.581514123077079));
draw(circle((1.4210145017438194,-1.3302878964546825), 2.8984706754484924));
draw(circle((-0.7076932767793396,-0.4161825262831505), 2.9101722408015513), linetype("4 4") + red);
draw((3.996177869179178,0.)--(-3.839514259733819,0.));
draw((3.996177869179178,0.)--(0.07833180472267817,2.385828723227042));
draw((0.07833180472267817,2.385828723227042)--(-1.154148865691539,0.));
draw((-3.839514259733819,0.)--(-0.6807342461448075,-3.3262298939043657));
draw((0.07833180472267817,2.385828723227042)--(-3.839514259733819,0.));
/* dots and labels */
dot((3.996177869179178,0.),blue);
label("$B$", (4.040279615036859,0.10218054796102663), NE * labelscalefactor,blue);
dot((-1.154148865691539,0.),blue);
label("$C$", (-1.3803811057738653,-0.14328333373606214), NE * labelscalefactor,blue);
dot((1.4210145017438194,1.5681827789938092),linewidth(4.pt));
label("$F$", (1.4629088572174203,1.6465574703052102), NE * labelscalefactor);
dot((0.07833180472267817,2.385828723227042),linewidth(3.pt) + blue);
label("$A$", (-0.04055741817725232,2.5568193649319144), NE * labelscalefactor,blue);
dot((-3.839514259733819,0.),linewidth(3.pt));
label("$E$", (-4.049800819229713,-0.06146203983703255), NE * labelscalefactor);
dot((1.4210145017438194,-2.40054783156011),linewidth(4.pt) + uuuuuu);
label("$M$", (1.4117705485305265,-2.6490604593938434), NE * labelscalefactor,uuuuuu);
dot((-0.6807342461448075,-3.3262298939043657),linewidth(4.pt));
label("$K$", (-0.7871767250058992,-3.5490946922831688), NE * labelscalefactor);
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);
/* end of picture */[/asy]
Problem 4
Let $ABCD$ be a rectangle with diagonals $AC$ and $BD$. Let $E$ be the intersection of the bisector of $\angle CAD$ with segment $CD$, $F$ on $CD$ such that $E$ is midpoint of $DF$, and $G$ on $BC$ such that $BG = AC$ (with $C$ between $B$ and $G$). Prove that the circumference through $D$, $F$ and $G$ is tangent to $BG$.
Let $ABCD$ be a trapezoid with $AB\parallel CD$ and $ \Omega $ is a circle passing through $A,B,C,D$. Let $ \omega $ be the circle passing through $C,D$ and intersecting with $CA,CB$ at $A_1$, $B_1$ respectively. $A_2$ and $B_2$ are the points symmetric to $A_1$ and $B_1$ respectively, with respect to the midpoints of $CA$ and $CB$. Prove that the points $A,B,A_2,B_2$ are concyclic.
[i]I. Bogdanov[/i]