Found problems: 25757
2015 CCA Math Bonanza, L4.4
Sierpinski's triangle is formed by taking a triangle, and drawing an upside down triangle inside each upright triangle that appears. A snake sees the fractal, but decides that the triangles need circles inside them. Therefore, she draws a circle inscribed in every upside down triangle she sees (assume that the snake can do an infinite amount of work). If the original triangle had side length $1$, what is the total area of all the individual circles?
[i]2015 CCA Math Bonanza Lightning Round #4.4[/i]
2009 Indonesia TST, 1
Given an $ n\times n$ chessboard.
a) Find the number of rectangles on the chessboard.
b) Assume there exists an $ r\times r$ square (label $ B$) with $ r<n$ which is located on the upper left corner of the board. Define "inner border" of $ A$ as the border of $ A$ which is not the border of the chessboard. How many rectangles in $ B$ that touch exactly one inner border of $ B$?
2015 JHMT, 5
Points $ABCDEF$ are evenly spaced on a unit circle and line segments $AD$, $DF$, $FB$, $BE$, $EC$, $CA$ are drawn. The line segments intersect each other at seven points inside the circle. Denote these intersections $p_1$, $p_2$, $...$,$p_7$, where $p_7$ is the center of the circle. What is the area of the $12$-sided shape $A_{p_1}B_{p_2}C_{p_3}D_{p_4}E_{p_5}F_{p_6}$?
[img]https://cdn.artofproblemsolving.com/attachments/9/2/645b3c31b0bb7f81d0c0b86e13b27ec5b32864.png[/img]
2016 Brazil Team Selection Test, 1
We say that a triangle $ABC$ is great if the following holds: for any point $D$ on the side $BC$, if $P$ and $Q$ are the feet of the perpendiculars from $D$ to the lines $AB$ and $AC$, respectively, then the reflection of $D$ in the line $PQ$ lies on the circumcircle of the triangle $ABC$. Prove that triangle $ABC$ is great if and only if $\angle A = 90^{\circ}$ and $AB = AC$.
[i]Senior Problems Committee of the Australian Mathematical Olympiad Committee[/i]
2020 HMIC, 3
Let $P_1P_2P_3P_4$ be a tetrahedron in $\mathbb{R}^3$ and let $O$ be a point equidistant from each of its vertices. Suppose there exists a point $H$ such that for each $i$, the line $P_iH$ is perpendicular to the plane through the other three vertices. Line $P_1H$ intersects the plane through $P_2, P_3, P_4$ at $A$, and contains a point $B\neq P_1$ such that $OP_1=OB$. Show that $HB=3HA$.
[i]Michael Ren[/i]
2002 Regional Competition For Advanced Students, 3
In the convex $ABCDEF$ (has all interior angles less than $180^o$) with the perimeter $s$ the triangles $ACE$ and $BDF$ have perimeters $u$ and $v$ respectively.
a) Show the inequalities $\frac{1}{2} \le \frac{s}{u+v}\le 1$
b) Check whether $1$ is replaced by a smaller number or $1/2$ by a larger number can the inequality remains valid for all convex hexagons.
1970 IMO Longlists, 8
Consider a regular $2n$-gon and the $n$ diagonals of it that pass through its center. Let $P$ be a point of the inscribed circle and let $a_1, a_2, \ldots , a_n$ be the angles in which the diagonals mentioned are visible from the point $P$. Prove that
\[\sum_{i=1}^n \tan^2 a_i = 2n \frac{\cos^2 \frac{\pi}{2n}}{\sin^4 \frac{\pi}{2n}}.\]
2011 JHMT, 5
Let $ABCD$ be a unit square. Point $E$ is on $BC$, point $F$ is on $DC$, $\vartriangle AEF$ is equilateral, and $GHIJ$ is a square in $\vartriangle AEF$ such that $GH$ is on $EF$. Compute the area of square $GHIJ$.
Denmark (Mohr) - geometry, 2014.3
The points $C$ and $D$ lie on a halfline from the midpoint $M$ of a segment $AB$, so that $|AC| = |BD|$. Prove that the angles $u = \angle ACM$ and $v = \angle BDM$ are equal.
[img]https://1.bp.blogspot.com/-tQEJ1VBCa8U/XzT7IhwlZHI/AAAAAAAAMVI/xpRdlj5Rl64VUt_tCRsQ1UxIsv_SGrMlACLcBGAsYHQ/s0/2014%2BMohr%2Bp3.png[/img]
2006 Oral Moscow Geometry Olympiad, 6
Given triangle $ABC$ and points $P$. Let $A_1,B_1,C_1$ be the second points of intersection of straight lines $AP, BP, CP$ with the circumscribed circle of $ABC$. Let points $A_2, B_2, C_2$ be symmetric to $A_1,B_1,C_1$ wrt $BC,CA,AB$, respectively. Prove that the triangles $A_1B_1C_1$ and $A_2B_2C_2$ are similar.
(A. Zaslavsky)
2014 BMT Spring, P2
Let $ABC$ be a fixed scalene triangle. Suppose that $X, Y$ are variable points on segments $AB$, $AC$, respectively such that $BX = CY$ . Prove that the circumcircle of $\vartriangle AXY$ passes through a fixed point other than $A$.
2010 Today's Calculation Of Integral, 559
In $ xyz$ space, consider two points $ P(1,\ 0,\ 1),\ Q(\minus{}1,\ 1,\ 0).$ Let $ S$ be the surface generated by rotation the line segment $ PQ$ about $ x$ axis. Answer the following questions.
(1) Find the volume of the solid bounded by the surface $ S$ and two planes $ x\equal{}1$ and $ x\equal{}\minus{}1$.
(2) Find the cross-section of the solid in (1) by the plane $ y\equal{}0$ to sketch the figure on the palne $ y\equal{}0$.
(3) Evaluate the definite integral $ \int_0^1 \sqrt{t^2\plus{}1}\ dt$ by substitution $ t\equal{}\frac{e^s\minus{}e^{\minus{}s}}{2}$.
Then use this to find the area of (2).
2019 Junior Balkan Team Selection Tests - Moldova, 11
Let $I$ be the center of inscribed circle of right triangle $\Delta ABC$ with $\angle A = 90$ and point $M$ is the midpoint of $(BC)$.The bisector of $\angle BAC$ intersects the circumcircle of $\Delta ABC $ in point $W$.Point $U$ is situated on the line $AB$ such that the lines $AB$ and $WU$ are perpendiculars.Point $P$ is situated on the line $WU$ such that the lines $PI$ and $WU$ are perpendiculars.Prove that the line $MP$ bisects the segment $CI$.
2006 MOP Homework, 4
1.14. Let P and Q be interior points of triangle ABC such that
\ACP = \BCQ and \CAP = \BAQ. Denote by D;E and
F the feet of the perpendiculars from P to the lines BC, CA
and AB, respectively. Prove that if \DEF = 90, then Q is the
orthocenter of triangle BDF.
2012 Greece Junior Math Olympiad, 1
Let $ABC$ be an acute angled triangle (with $AB<AC<BC$) inscribed in circle $c(O,R)$ (with center $O$ and radius $R$). Circle $c_1(A,AB)$ (with center $A$ and radius $AB$) intersects side $BC$ at point $D$ and the circumcircle $c(O,R)$ at point $E$. Prove that side $AC$ bisects angle $\angle DAE$.
Durer Math Competition CD 1st Round - geometry, 2018.D2
In an isosceles triangle, we drew one of the angle bisectors. At least one of the resulting two smaller ones triangles is similar to the original. What can be the leg of the original triangle if the length of its base is $1$ unit?
2013 Olympic Revenge, 2
Let $ABC$ to be an acute triangle. Also, let $K$ and $L$ to be the two intersections of the perpendicular from $B$ with respect to side $AC$ with the circle of diameter $AC$, with $K$ closer to $B$ than $L$. Analogously, $X$ and $Y$ are the two intersections of the perpendicular from $C$ with respect to side $AB$ with the circle of diamter $AB$, with $X$ closer to $C$ than $Y$. Prove that the intersection of $XL$ and $KY$ lies on $BC$.
1990 AMC 12/AHSME, 4
Let $ABCD$ be a parallelogram with $\angle ABC=120^\circ$, $AB=16$ and $BC=10$. Extend $\overline{CD}$ through $D$ to $E$ so that $DE=4$. If $\overline{BE}$ intersects $\overline{AD}$ at $F$, then $FD$ is closest to
$\textbf{(A) }1\qquad
\textbf{(B) }2\qquad
\textbf{(C) }3\qquad
\textbf{(D) }4\qquad
\textbf{(E) }5$
[asy]
size(200);
defaultpen(linewidth(0.8));
pair A=origin,B=(16,0),C=(26,10*sqrt(3)),D=(10,10*sqrt(3)),E=(0,10*sqrt(3));
draw(A--B--C--E--B--A--D);
label("$A$",A,S);
label("$B$",B,S);
label("$C$",C,N);
label("$D$",D,N);
label("$E$",E,N);
label("$F$",extension(A,D,B,E),W);
label("$4$",(D+E)/2,N);
label("$16$",(8,0),S);
label("$10$",(B+C)/2,SE);
[/asy]
2024 Kyiv City MO Round 2, Problem 3
Let $AH_A, BH_B, CH_C$ be the altitudes of the triangle $ABC$. Points $A_1$ and $C_1$ are the projections of the point $H_B$ onto the sides $AB$ and $BC$, respectively. $B_1$ is the projection of $B$ onto $H_AH_C$. Prove that the diameter of the circumscribed circle of $\triangle A_1B_1C_1$ is equal to $BH_B$.
[i]Proposed by Anton Trygub[/i]
2021 Brazil Team Selection Test, 3
Let $ABC$ be an acute triangle with $AC>CB$ and let $M$ be the midpoint of side $AB$. Denote by $Q$ the midpoint of the big arc $AB$ which cointais $C$ and by $B_1$ the point inside $AC$ such that $BC=CB_1$. $B_1Q$ touches $BC$ in $E$ and $K$ is the intersection of $(BB_1M)$ and $(ABC)$. Prove that $KC$ bissects $B_1E$.
OIFMAT II 2012, 3
In the interior of an equilateral triangle $ ABC $ a point $ P $ is chosen such that $ PA ^2 = PB ^2 + PC ^2 $. Find the measure of $ \angle BPC $.
2013 Romanian Master of Mathematics, 6
A token is placed at each vertex of a regular $2n$-gon. A [i]move[/i] consists in choosing an edge of the $2n$-gon and swapping the two tokens placed at the endpoints of that edge. After a finite number of moves have been performed, it turns out that every two tokens have been swapped exactly once. Prove that some edge has never been chosen.
1990 IMO Longlists, 60
Unit cubes are made into beads by drilling a hole through them along a diagonal. The beads are put on a string in such a way that they can move freely in space under the restriction that the vertices of two neighboring cubes are touching. Let $ A$ be the beginning vertex and $ B$ be the end vertex. Let there be $ p \times q \times r$ cubes on the string $ (p, q, r \geq 1).$
[i](a)[/i] Determine for which values of $ p, q,$ and $ r$ it is possible to build a block with dimensions $ p, q,$ and $ r.$ Give reasons for your answers.
[i](b)[/i] The same question as (a) with the extra condition that $ A \equal{} B.$
2024 District Olympiad, P4
Let $H{}$ be the orthocenter of the triangle $ABC{}$ and $X{}$ be the midpoint of the side $BC.$ The perpendicular at $H{}$ to $HX{}$ intersects the sides $(AB)$ and $(AC)$ at $Y{}$ and $Z{}$ respectively. Let $O{}$ be the circumcenter of $ABC{}$ and $O'$ be the circumcenter of $BHC.$ [list=a]
[*]Prove that $HY=HZ.$
[*]Prove that $\overrightarrow{AY}+\overrightarrow{AZ}=2\overrightarrow{OO'}.$
[/list]
2023 CMIMC Geometry, 6
Let $ABCD$ be a regular tetrahedron. Suppose points $X$, $Y$, and $Z$ lie on rays $AB$, $AC$, and $AD$ respectively such that $XY=YZ=7$ and $XZ=5$. Moreover, the lengths $AX$, $AY$, and $AZ$ are all distinct. Find the volume of tetrahedron $AXYZ$.
[i]Proposed by Connor Gordon[/i]