Found problems: 1581
On the real number line, paint red all points that correspond to integers of the form $81x+100y$, where $x$ and $y$ are positive integers. Paint the remaining integer point blue. Find a point $P$ on the line such that, for every integer point $T$, the reflection of $T$ with respect to $P$ is an integer point of a different colour than $T$.
Suppose $ABCD$ is a parallelogram. Consider circles $w_1$ and $w_2$ such that $w_1$ is tangent to segments $AB$ and $AD$ and $w_2$ is tangent to segments $BC$ and $CD$. Suppose that there exists a circle which is tangent to lines $AD$ and $DC$ and externally tangent to $w_1$ and $w_2$. Prove that there exists a circle which is tangent to lines $AB$ and $BC$ and also externally tangent to circles $w_1$ and $w_2$.
[i]Proposed by Ali Khezeli[/i]
Circles $\alpha$ and $\beta$ of the same radius intersect in two points, one of which is $P$. Denote by $A$ and $B$, respectively, the points diametrically opposite to $P$ on each of $\alpha$ and $\beta$ . A third circle of the same radius passes through $P$ and intersects $\alpha$ and $\beta$ in the points $X$ and $Y$ , respectively. Show that the line $XY$ is parallel to the line $AB$.
Let $n$ be a positive integer, $n \ge 2$, and put $\theta=\frac{2\pi}{n}$. Define points $P_k=(k,0)$ in the [i]xy[/i]-plane, for $k=1,2,\dots,n$. Let $R_k$ be the map that rotates the plane counterclockwise by the angle $\theta$ about the point $P_k$. Let $R$ denote the map obtained by applying in order, $R_1$, then $R_2$, ..., then $R_n$. For an arbitrary point $(x,y)$, find and simplify the coordinates of $R(x,y)$.
Given is a circular bus route with $n\geqslant2$ bus stops. The route can be frequented in both directions. The way between two stops is called [i]section[/i] and one of the bus stops is called [i]Zürich[/i]. A bus shall start at Zürich, pass through all the bus stops [b]at least once[/b] and drive along exactly $n+2$ sections before it returns to Zürich in the end. Assuming that the bus can chance directions at each bus stop, how many possible routes are there?
EDIT: Sorry, there was a mistake...corrected now, thanks mavropnevma! :oops:
Point $P$ is taken in the interior of the triangle $ABC$, so that
\[\angle PAB = \angle PCB = \dfrac {1} {4} (\angle A + \angle C).\]
Let $L$ be the foot of the angle bisector of $\angle B$. The line $PL$ meets the circumcircle of $\triangle APC$ at point $Q$. Prove that $QB$ is the angle bisector of $\angle AQC$.
[i]Proposed by S. Berlov[/i]
Assume $ABC$ is an isosceles triangle that $AB=AC$ Suppose $P$ is a point on extension of side $BC$. $X$ and $Y$ are points on $AB$ and $AC$ that:
\[PX || AC \ , \ PY ||AB \]
Also $T$ is midpoint of arc $BC$. Prove that $PT \perp XY$
Let $ ABC$ be a triangle with $ BC > AC > AB$. Let $ A',B',C'$ be feet of perpendiculars from $ A,B,C$ to $ BC,AC,AB$, such that $ AA' \equal{} BB' \equal{} CC' \equal{} x$. Prove that:
a) If $ ABC\sim A'B'C'$ then $ x \equal{} 2r$
b) Prove that if $ A',B'$ and $ C'$ are collinear, then $ x \equal{} R \plus{} d$ or $ x \equal{} R \minus{} d$.
(In this problem $ R$ is the radius of circumcircle, $ r$ is radius of incircle and $ d \equal{} OI$)
Consider the cyclic quadrilateral with side lengths $1$, $4$, $8$, $7$ in that order. What is its circumdiameter? Let the answer be of the form $a\sqrt b+c$, for $b$ squarefree. Find $a+b+c$.
Consider a triangle $ABC$ with circumcenter $O$ and let $A_1,B_1,C_1$ be the midpoints of the sides $BC,AC,AB,$ respectively.
Points $A_2,B_2,C_2$ are defined as $\overrightarrow{OA_2}=\lambda \cdot \overrightarrow{OA_1}, \ \overrightarrow{OB_2}=\lambda \cdot \overrightarrow{OB_1}, \ \overrightarrow{OC_2}=\lambda \cdot \overrightarrow{OC_1},$ where $\lambda >0.$
Prove that lines $AA_2,BB_2,CC_2$ are concurrent.
Let $ABC$ be a triangle and $D$, $E$, $F$ be the midpoints of arcs $BC$, $CA$, $AB$ on the circumcircle. Line $\ell_a$ passes through the feet of the perpendiculars from $A$ to $DB$ and $DC$. Line $m_a$ passes through the feet of the perpendiculars from $D$ to $AB$ and $AC$. Let $A_1$ denote the intersection of lines $\ell_a$ and $m_a$. Define points $B_1$ and $C_1$ similarly. Prove that triangle $DEF$ and $A_1B_1C_1$ are similar to each other.
Triangle $ABC$ has a right angle at $B$. Point $D$ lies on side $BC$ such that $3\angle BAD = \angle BAC$. Given $AC=2$ and $CD=1$, compute $BD$.
Let $P$ be a given point inside the triangle $ABC$. Suppose $L,M,N$ are the midpoints of $BC, CA, AB$ respectively and \[PL: PM: PN= BC: CA: AB.\] The extensions of $AP, BP, CP$ meet the circumcircle of $ABC$ at $D,E,F$ respectively. Prove that the circumcentres of $APF, APE, BPF, BPD, CPD, CPE$ are concyclic.
Restore a triangle by one of its vertices, the circumcenter and the Lemoine's point.
[i](The Lemoine's point is the intersection point of the reflections of the medians in the correspondent angle bisectors)[/i]
Let $ABC$ be a right angled triangle at $A$. Denote $D$ the foot of the altitude through $A$ and $O_1, O_2$ the incentres of triangles $ADB$ and $ADC$. The circle with centre $A$ and radius $AD$ cuts $AB$ in $K$ and $AC$ in $L$. Show that $O_1, O_2, K$ and $L$ are on a line.
I have $8$ unit cubes of different colors, which I want to glue together into a $2\times 2\times 2$ cube. How many distinct $2\times 2\times 2$ cubes can I make? Rotations of the same cube are not considered distinct, but reflections are.
Isosceles $ \triangle ABC$ has a right angle at $ C$. Point $ P$ is inside $ \triangle ABC$, such that $ PA \equal{} 11, PB \equal{} 7,$ and $ PC \equal{} 6$. Legs $ \overline{AC}$ and $ \overline{BC}$ have length $ s \equal{} \sqrt {a \plus{} b\sqrt {2}}$, where $ a$ and $ b$ are positive integers. What is $ a \plus{} b$?
[asy]pointpen = black;
pathpen = linewidth(0.7);
pen f = fontsize(10);
size(5cm);
pair B = (0,sqrt(85+42*sqrt(2)));
pair A = (B.y,0);
pair C = (0,0);
pair P = IP(arc(B,7,180,360),arc(C,6,0,90));
D(A--B--C--cycle);
D(P--A);
D(P--B);
D(P--C);
MP("A",D(A),plain.E,f);
MP("B",D(B),plain.N,f);
MP("C",D(C),plain.SW,f);
MP("P",D(P),plain.NE,f);[/asy]
$ \textbf{(A) } 85 \qquad \textbf{(B) } 91 \qquad \textbf{(C) } 108 \qquad \textbf{(D) } 121 \qquad \textbf{(E) } 127$
Let $ABC$ be a triangle and let $\omega$ be its incircle. Denote by $D_1$ and $E_1$ the points where $\omega$ is tangent to sides $BC$ and $AC$, respectively. Denote by $D_2$ and $E_2$ the points on sides $BC$ and $AC$, respectively, such that $CD_2=BD_1$ and $CE_2=AE_1$, and denote by $P$ the point of intersection of segments $AD_2$ and $BE_2$. Circle $\omega$ intersects segment $AD_2$ at two points, the closer of which to the vertex $A$ is denoted by $Q$. Prove that $AQ=D_2P$.
A rectangle that is inscribed in a larger rectangle (with one vertex on each side) is called [i]unstuck[/i] if it is possible to rotate (however slightly) the smaller rectangle about its center within the confines of the larger. Of all the rectangles that can be inscribed unstuck in a 6 by 8 rectangle, the smallest perimeter has the form $\sqrt{N}$, for a positive integer $N$. Find $N$.
Let $K$ be a closed convex polygonal region, and let $X$ be a point in the plane of $K$. Show that there exists a finite sequence of reflections in the sides of $K$, such that $K$ contains the image of $X$ after these reflections.
Let $ABC$ be a triangle and $X,Y,Z$ be the tangency points of its inscribed circle with the sides $BC, CA, AB$, respectively. Suppose that $C_1, C_2, C_3$ are circle with chords $YZ, ZX, XY$, respectively, such that $C_1$ and $C_2$ intersect on the line $CZ$ and that $C_1$ and $C_3$ intersect on the line $BY$. Suppose that $C_1$ intersects the chords $XY$ and $ZX$ at $J$ and $M$, respectively; that $C_2$ intersects the chords $YZ$ and $XY$ at $L$ and $I$, respectively; and that $C_3$ intersects the chords $YZ$ and $ZX$ at $K$ and $N$, respectively. Show that $I, J, K, L, M, N$ lie on the same circle.
Jonathan and Eric are standing one kilometer apart on a large, flat, empty field. Jonathan rotates an angle of $\theta = 120^o$ counterclockwise around Eric, then Eric moves half of the distance to Jonathan. They keep repeating the previous two movements in this order. After a very long time, their locations approach a point $P$ on the field. What is the distance, in kilometers, from Jonathan’s starting location to $P$?
$P$ is a point in the interior of acute triangle $ABC$. $D,E,F$ are the reflections of $P$ across $BC,CA,AB$ respectively. Rays $AP,BP,CP$ meet the circumcircle of $\triangle ABC$ at $L,M,N$ respectively. Prove that the circumcircles of $\triangle PDL,\triangle PEM,\triangle PFN$ meet at a point $T$ different from $P$.
Let $ABC$ be a triangle with incenter $I$ and circumcenter $O$ such that $|AC|>|BC|>|AB|$ and the incircle touches the sides $BC, CA, AB$ at $D, E, F$ respectively. Let the reflection of $A$ with respect to $F$ and $E$ be $F_1$ and $E_1$ respectively. The circle tangent to $BC$ at $D$ and passing through $F_1$ intersects $AB$ a second time at $F_2$ and the circle tangent to $BC$ at $D$ and passing through $E_1$ intersects $AC$ a second time at $E_2$. The midpoints of the segments $|OE|$ and $|IF|$ are $P$ and $Q$ respectively. Prove that \[|AB| + |AC| = 2|BC| \iff PQ\perp E_2F_2 \].
Let $C: y=x^2+ax+b$ be a parabola passing through the point $(1,\ -1)$. Find the minimum volume of the figure enclosed by $C$ and the $x$ axis by a rotation about the $x$ axis.
Proposed by kunny