Found problems: 1581
In the triangle $ABC$, the excircle opposite to the vertex $A$ with centre $I$ touches the side BC at D. (The circle also touches the sides of $AB$, $AC$ extended.) Let $M$ be the midpoint of $BC$ and $N$ the midpoint of $AD$. Prove that $I,M,N$ are collinear.
Let $ABCD$ be an isosceles trapezoid with $AB \parallel CD$. The inscribed circle $\omega$ of triangle $BCD$ meets $CD$ at $E$. Let $F$ be a point on the (internal) angle bisector of $\angle DAC$ such that $EF \perp CD$. Let the circumscribed circle of triangle $ACF$ meet line $CD$ at $C$ and $G$. Prove that the triangle $AFG$ is isosceles.
Let $ABC$ be a triangle, and $\omega_{a}$, $\omega_{b}$, $\omega_{c}$ be circles inside $ABC$, that are tangent (externally) one to each other, such that $\omega_{a}$ is tangent to $AB$ and $AC$, $\omega_{b}$ is tangent to $BA$ and $BC$, and $\omega_{c}$ is tangent to $CA$ and $CB$. Let $D$ be the common point of $\omega_{b}$ and $\omega_{c}$, $E$ the common point of $\omega_{c}$ and $\omega_{a}$, and $F$ the common point of $\omega_{a}$ and $\omega_{b}$. Show that the lines $AD$, $BE$ and $CF$ have a common point.
Given that points $ D,E$ lie on the sidelines $ AB,BC$ of triangle $ ABC$, respectively, point $ P$ is in interior of triangle $ ABC$ such that $ PE \equal{} PC$ and $ \bigtriangleup DEP\sim \bigtriangleup PCA.$ Prove that $ BP$ is tangent of the circumcircle of triangle $ PAD.$
A trapezoid with bases $AD$ and $BC$ is circumscribed about a circle, $E$ is the intersection point of the diagonals. Prove that $\angle AED$ is not acute.
Let $ABC$ be a triangle and $M$ be the midpoint of $BC$, and let $AM$ meet the circumcircle of $ABC$ again at $R$. A line passing through $R$ and parallel to $BC$ meet the circumcircle of $ABC$ again at $S$. Let $U$ be the foot from $R$ to $BC$, and $T$ be the reflection of $U$ in $R$. $D$ lies in $BC$ such that $AD$ is an altitude. $N$ is the midpoint of $AD$. Finally let $AS$ and $MN$ meets at $K$. Prove that $AT$ bisector $MK$.
On the number line, consider the point $x$ that corresponds to the value $10$. Consider $24$ distinct integer points $y_1$, $y_2$, $\ldots$, $y_{24}$ on the number line such that for all $k$ such that $1\leq k\leq 12$, we have that $y_{2k-1}$ is the reflection of $y_{2k}$ across $x$. Find the minimum possible value of \[\textstyle\sum_{n=1}^{24}(|y_n-1|+|y_n+1|).\]
Two equal-radii circles with centres $ O_1$ and $ O_2$ intersect each other at $ P$ and $ Q$, $ O$ is the midpoint of the common chord $ PQ$. Two lines $ AB$ and $ CD$ are drawn through $ P$ ( $ AB$ and $ CD$ are not coincide with $ PQ$ ) such that $ A$ and $ C$ lie on circle $ O_1$ and $ B$ and $ D$ lie on circle $ O_2$. $ M$ and $ N$ are the mipoints of segments $ AD$ and $ BC$ respectively. Knowing that $ O_1$ and $ O_2$ are not in the common part of the two circles, and $ M$, $ N$ are not coincide with $ O$.
Prove that $ M$, $ N$, $ O$ are collinear.
A square plate has side length $L$ and negligible thickness. It is laid down horizontally on a table and is then rotating about the axis $\overline{MN}$ where $M$ and $N$ are the midpoints of two adjacent sides of the square. The moment of inertia of the plate about this axis is $kmL^2$, where $m$ is the mass of the plate and $k$ is a real constant. Find $k$.
[color=red]Diagram will be added to this post very soon. If you want to look at it temporarily, see the PDF.[/color]
[i]Problem proposed by Ahaan Rungta[/i]
Let the vertices $A$ and $C$ of a right triangle $ABC$ be on the arc with center $B$ and radius $20$. A semicircle with diameter $[AB]$ is drawn to the inner region of the arc. The tangent from $C$ to the semicircle touches the semicircle at $D$ other than $B$. Let $CD$ intersect the arc at $F$. What is $|FD|$?
$
\textbf{(A)}\ 1
\qquad\textbf{(B)}\ \frac 52
\qquad\textbf{(C)}\ 3
\qquad\textbf{(D)}\ 4
\qquad\textbf{(E)}\ 5
$
A diagonal of a 100-gon is called good if it divides the 100-gon into two polygons each with an odd number of sides. A 100-gon was split into triangles with non-intersecting diagonals, exactly 49 of which are good. The triangles are colored into two colors such that no two triangles that border each other are colored with the same color. Prove that there is the same number of triangles colored with one color as with the other.
Fresh translation; slightly reworded.
The circles $ C_1$ and $ C_2$ with centers $ O_1$ and $ O_2$ respectively are externally tangent. Their common tangent not intersecting the segment $ O_1O_2$ touches $ C_1$ at $ A$ and $ C_2$ at $ B$. Let $ C$ be the reflection of $ A$ in $ O_1O_2$ and $ P$ be the intersection of $ AC$ and $ O_1O_2$. Line $ BP$ meets $ C_2$ again at $ L$. Prove that line $ CL$ is tangent to the circle $ C_2$.
Lines tangent to circle $O$ in points $A$ and $B$, intersect in point $P$. Point $Z$ is the center of $O$. On the minor arc $AB$, point $C$ is chosen not on the midpoint of the arc. Lines $AC$ and $PB$ intersect at point $D$. Lines $BC$ and $AP$ intersect at point $E$. Prove that the circumcentres of triangles $ACE$, $BCD$, and $PCZ$ are collinear.
Let $K_a,K_b,K_c$ with centres $O_a,O_b,O_c$ be the excircles of a triangle $ABC$, touching the interiors of the sides $BC,CA,AB$ at points $T_a,T_b,T_c$ respectively.
Prove that the lines $O_aT_a,O_bT_b,O_cT_c$ are concurrent in a point $P$ for which $PO_a=PO_b=PO_c=2R$ holds, where $R$ denotes the circumradius of $ABC$. Also prove that the circumcentre $O$ of $ABC$ is the midpoint of the segment $PI$, where $I$ is the incentre of $ABC$.
Let $ABCD$ be a quadrilateral inscribed in the unit circle such that $\angle BAD$ is $30$ degrees. Let $m$ denote the minimum value of $CP + PQ + CQ$, where $P$ and $Q$ may be any points lying along rays $AB$ and $AD$, respectively. Determine the maximum value of $m$.
Two transformations are said to [i]commute[/i] if applying the first followed by the second gives the same result as applying the second followed by the first. Consider these four transformations of the coordinate plane:
- A translation $2$ units to the right
- A $90^\circ$- rotation counterclockwise about the origin.
- A reflection across the $x$-axis, and
- A dilation centered at the origin with scale factor $2$.
Of the $6$ pairs of distinct transformations from this list, how many commute?
$
\textbf{(A) }1 \qquad
\textbf{(B) }2 \qquad
\textbf{(C) }3 \qquad
\textbf{(D) }4 \qquad
\textbf{(E) }5 \qquad
$
Let a semicircle is given with diameter $AB$ and centre $O$ and let $C$ be a arbitrary point on the segment $OB$. Point $D$ on the semicircle is such that $CD$ is perpendicular to $AB$. A circle with centre $P$ is tangent to the arc $BD$ at $F$ and to the segment $CD$ and $AB$ at $E$ and $G$ respectively. Prove that the triangle $ADG$ is isosceles.
Let $ P$ be an interior point of an acute angled triangle $ ABC$. The lines $ AP,BP,CP$ meet $ BC,CA,AB$ at points $ D,E,F$ respectively. Given that triangle $ \triangle DEF$ and $ \triangle ABC$ are similar, prove that $ P$ is the centroid of $ \triangle ABC$.
Let $ABC$ be a triangle with orthocenter $H$ and circumcenter $O$. Let $P$ be a point in the plane such that $AP \perp BC$. Let $Q$ and $R$ be the reflections of $P$ in the lines $CA$ and $AB$, respectively. Let $Y$ be the orthogonal projection of $R$ onto $CA$. Let $Z$ be the orthogonal projection of $Q$ onto $AB$. Assume that $H \neq O$ and $Y \neq Z$. Prove that $YZ \perp HO$.
[asy]
import olympiad;
unitsize(30);
pair A,B,C,H,O,P,Q,R,Y,Z,Q2,R2,P2;
A = (-14.8, -6.6);
B = (-10.9, 0.3);
C = (-3.1, -7.1);
O = circumcenter(A,B,C);
H = orthocenter(A,B,C);
P = 1.2 * H - 0.2 * A;
Q = reflect(A, C) * P;
R = reflect(A, B) * P;
Y = foot(R, C, A);
Z = foot(Q, A, B);
P2 = foot(A, B, C);
Q2 = foot(P, C, A);
R2 = foot(P, A, B);
draw(B--(1.6*A-0.6*B));
draw(B--C--A);
draw(P--R, blue);
draw(R--Y, red);
draw(P--Q, blue);
draw(Q--Z, red);
draw(A--P2, blue);
draw(O--H, darkgreen+linewidth(1.2));
draw((1.4*Z-0.4*Y)--(4.6*Y-3.6*Z), red+linewidth(1.2));
draw(rightanglemark(R,Y,A,10), red);
draw(rightanglemark(Q,Z,B,10), red);
draw(rightanglemark(C,Q2,P,10), blue);
draw(rightanglemark(A,R2,P,10), blue);
draw(rightanglemark(B,P2,H,10), blue);
label("$\textcolor{blue}{H}$",H,NW);
label("$\textcolor{blue}{P}$",P,N);
label("$A$",A,W);
label("$B$",B,N);
label("$C$",C,S);
label("$O$",O,S);
label("$\textcolor{blue}{Q}$",Q,E);
label("$\textcolor{blue}{R}$",R,W);
label("$\textcolor{red}{Y}$",Y,S);
label("$\textcolor{red}{Z}$",Z,NW);
dot(A, filltype=FillDraw(black));
dot(B, filltype=FillDraw(black));
dot(C, filltype=FillDraw(black));
dot(H, filltype=FillDraw(blue));
dot(P, filltype=FillDraw(blue));
dot(Q, filltype=FillDraw(blue));
dot(R, filltype=FillDraw(blue));
dot(Y, filltype=FillDraw(red));
dot(Z, filltype=FillDraw(red));
dot(O, filltype=FillDraw(black));
[/asy]
Two segment $AB,CD$ of the same length are given in plane such that lines $AB,CD$ are not parallel. Consider a point $S$ with the following property: the image of segment $AB$ under point reflection with respect to $S$ is identical to the mirror-image of segment $CD$ with respect to some axis. Find the locus of all such points $S.$
Let $U$ be the incenter of a triangle $ABC$ and $O_{1}, O_{2}, O_{3}$ be the circumcenters of the triangles $BCU, CAU, ABU$ , respectively. Prove that the circumcircles of the triangles $ABC$ and $O_{1}O_{2}O_{3}$ have the same center.
In a scalene triangle $ABC$, $D$ is the foot of the altitude through $A$, $E$ is the intersection of $AC$ with the bisector of $\angle ABC$ and $F$ is a point on $AB$. Let $O$ the circumcenter of $ABC$ and $X=AD\cap BE$, $Y=BE\cap CF$, $Z=CF \cap AD$. If $XYZ$ is an equilateral triangle, prove that one of the triangles $OXY$, $OYZ$, $OZX$ must be equilateral.
Consider the following transformation of the Cartesian plane: choose a lattice point and rotate the plane $90^\circ$ counterclockwise about that lattice point. Is it possible, through a sequence of such transformations, to take the triangle with vertices $(0,0)$, $(1,0)$ and $(0,1)$ to the triangle with vertices $(0,0)$, $(1,0)$ and $(1,1)$?
Consider a triangle $ABC$. Let $S$ be a circumference in the interior of the triangle that is tangent to the sides $BC$, $CA$, $AB$ at the points $D$, $E$, $F$ respectively. In the exterior of the triangle we draw three circumferences $S_A$, $S_B$, $S_C$. The circumference $S_A$ is tangent to $BC$ at $L$ and to the prolongation of the lines $AB$, $AC$ at the points $M$, $N$ respectively. The circumference $S_B$ is tangent to $AC$ at $E$ and to the prolongation of the line $BC$ at $P$. The circumference $S_C$ is tangent to $AB$ at $F$ and to the prolongation of the line $BC$ at $Q$. Show that the lines $EP$, $FQ$ and $AL$ meet at a point of the circumference $S$.
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 $N$.