Found problems: 25757
2005 Alexandru Myller, 2
Let $ ABC $ be a triangle with $ \angle BAC <90^{\circ } . $ In the exterior of $ ABC, $ choose the points $ D,E $ such that $ DA=DB,EA=EC $ and $ \angle ADB =\angle AEC =2\angle BAC . $ Show that the symmetric of $ A $ with respect to the midpoint of the segment $ DE $ is the circumcircle of $ ABC. $
2012 Peru IMO TST, 2
Let $a, b, c$ be the lengths of the sides of a triangle, and $h_a, h_b, h_c$ the lengths of the heights corresponding to the sides $a, b, c,$ respectively. If $t \geq \frac{1} {2}$ is a real number, show that there is a triangle with sidelengths $$ t\cdot a + h_a, \ t\cdot b + h_b , \ t\cdot c + h_c.$$
2004 India National Olympiad, 1
$ABCD$ is a convex quadrilateral. $K$, $L$, $M$, $N$ are the midpoints of the sides $AB$, $BC$, $CD$, $DA$. $BD$ bisects $KM$ at $Q$. $QA = QB = QC = QD$ , and$\frac{LK}{LM} = \frac{CD}{CB}$. Prove that $ABCD$ is a square
2001 Balkan MO, 4
A cube side 3 is divided into 27 unit cubes. The unit cubes are arbitrarily labeled 1 to 27 (each cube is given a different number). A move consists of swapping the cube labeled 27 with one of its 6 neighbours. Is it possible to find a finite sequence of moves at the end of which cube 27 is in its original position, but cube $n$ has moved to the position originally occupied by $27-n$ (for each $n = 1, 2, \ldots , 26$)?
2024 Turkey EGMO TST, 1
Let $ABC$ be a triangle and its circumcircle be $\omega$. Let $I$ be the incentre of the $ABC$. Let the line $BI$ meet $AC$ at $E$ and $\omega$ at $M$ for the second time. The line $CI$ meet $AB$ at $F$ and $\omega$ at $N$ for the second time. Let the circumcircles of $BFI$ and $CEI$ meet again at point $K$. Prove that the lines $BN$, $CM$, $AK$ are concurrent.
2000 Polish MO Finals, 2
Let a triangle $ABC$ satisfy $AC = BC$; in other words, let $ABC$ be an isosceles triangle with base $AB$. Let $P$ be a point inside the triangle $ABC$ such that $\angle PAB = \angle PBC$. Denote by $M$ the midpoint of the segment $AB$. Show that $\angle APM + \angle BPC = 180^{\circ}$.
2025 NEPALTST, 1
Consider a triangle $\triangle ABC$ and some point $X$ on $BC$. The perpendicular from $X$ to $AB$ intersects the circumcircle of $\triangle AXC$ at $P$ and the perpendicular from $X$ to $AC$ intersects the circumcircle of $\triangle AXB$ at $Q$. Show that the line $PQ$ does not depend on the choice of $X$.
[i](Shining Sun, USA)[/i]
1985 Poland - Second Round, 6
There are various points in space $ A, B, C_0, C_1, C_2 $, with $ |AC_i| = 2 |BC_i| $ for $ i = 0,1,2 $ and $ |C_1C_2|=\frac{4}{3}|AB| $. Prove that the angle $ C_1C_0C_2 $ is right and the points $ A, B, C_1, C_2 $ lie on one plane.
2010 CHMMC Fall, 2
Let $A, B, C$, and $D$ be points on a circle, in that order, such that $\overline{AD}$ is a diameter of the circle. Let $E$ be the intersection of $\overleftrightarrow{AB}$ and $\overleftrightarrow{DC}$, let $F$ be the intersection of $\overleftrightarrow{AC}$ and $\overleftrightarrow{BD}$, and let $G$ be the intersection of $\overleftrightarrow{EF}$ and $\overleftrightarrow{AD}$. If $AD = 8$, $AE = 9$, and $DE = 7$, compute $EG$.
2012 BMT Spring, 3
Let $ABC$ be a triangle with side lengths $AB = 2011$, $BC = 2012$, $AC = 2013$. Create squares $S_1 =ABB'A''$, $S_2 = ACC''A'$ , and $S_3 = CBB''C'$ using the sides $AB$, $AC$, $BC$ respectively, so that the side $B'A''$ is on the opposite side of $AB$ from $C$, and so forth. Let square $S_4$ have side length $A''A' $, square $S_5$ have side length $C''C'$, and square $S_6$ have side length $B''B'$. Let $A(S_i)$ be the area of square $S_i$ . Compute $\frac{A(S_4)+A(S_5)+A(S_6)}{A(S_1)+A(S_2)+A(S_3)}$?
1948 Putnam, B5
The pairs $(a,b)$ such that $|a+bt+ t^2 |\leq 1$ for $0\leq t \leq 1$ fill a certain region in the plane. What is the area of this region?
1967 AMC 12/AHSME, 34
Points $D$, $E$, $F$ are taken respectively on sides $AB$, $BC$, and $CA$ of triangle $ABC$ so that $AD:DB=BE:CE=CF:FA=1:n$. The ratio of the area of triangle $DEF$ to that of triangle $ABC$ is:
$\textbf{(A)}\ \frac{n^2-n+1}{(n+1)^2}\qquad
\textbf{(B)}\ \frac{1}{(n+1)^2}\qquad
\textbf{(C)}\ \frac{2n^2}{(n+1)^2}\qquad
\textbf{(D)}\ \frac{n^2}{(n+1)^2}\qquad
\textbf{(E)}\ \frac{n(n-1)}{n+1}$
2006 BAMO, 3
In triangle $ABC$, choose point $A_1$ on side $BC$, point $B_1$ on side $CA$, and point $C_1$ on side $AB$ in such a way that the three segments $AA_1, BB_1$, and $CC_1$ intersect in one point $P$. Prove that $P$ is the centroid of triangle $ABC$ if and only if $P$ is the centroid of triangle $A_1B_1C_1$.
Note: A median in a triangle is a segment connecting a vertex of the triangle with the midpoint of the opposite side. The centroid of a triangle is the intersection point of the three medians of the triangle. The centroid of a triangle is also known by the names ”center of mass” and ”medicenter” of the triangle.
2007 Bulgarian Autumn Math Competition, Problem 12.2
All edges of the triangular pyramid $ABCD$ are equal in length. Let $M$ be the midpoint of $DB$, $N$ is the point on $\overline{AB}$, such that $2NA=NB$ and $N\not\in AB$ and $P$ is a point on the altitude through point $D$ in $\triangle BCD$. Find $\angle MPD$ if the intersection of the pyramid with the plane $(NMP)$ is a trapezoid.
2021 EGMO, 4
Let $ABC$ be a triangle with incenter $I$ and let $D$ be an arbitrary point on the side $BC$. Let the line through $D$ perpendicular to $BI$ intersect $CI$ at $E$. Let the line through $D$ perpendicular to $CI$ intersect $BI$ at $F$. Prove that the reflection of $A$ across the line $EF$ lies on the line $BC$.
2016 IFYM, Sozopol, 5
A convex quadrilateral is cut into smaller convex quadrilaterals so that they are adjacent to each other only by whole sides.
a) Prove that if all small quadrilaterals are inscribed in a circle, then the original one is also inscribed in a circle.
b) Prove that if all small quadrilaterals are cyclic, then the original one is also cyclic.
1953 AMC 12/AHSME, 28
In triangle $ ABC$, sides $ a,b$ and $ c$ are opposite angles $ A,B$ and $ C$ respectively. $ AD$ bisects angle $ A$ and meets $ BC$ at $ D$. Then if $ x \equal{} \overline{CD}$ and $ y \equal{} \overline{BD}$ the correct proportion is:
$ \textbf{(A)}\ \frac {x}{a} \equal{} \frac {a}{b \plus{} c} \qquad\textbf{(B)}\ \frac {x}{b} \equal{} \frac {a}{a \plus{} c} \qquad\textbf{(C)}\ \frac {y}{c} \equal{} \frac {c}{b \plus{} c} \\
\textbf{(D)}\ \frac {y}{c} \equal{} \frac {a}{b \plus{} c} \qquad\textbf{(E)}\ \frac {x}{y} \equal{} \frac {c}{b}$
Croatia MO (HMO) - geometry, 2013.3
Given a pointed triangle $ABC$ with orthocenter $H$. Let $D$ be the point such that the quadrilateral $AHCD$ is parallelogram. Let $p$ be the perpendicular to the direction $AB$ through the midpoint $A_1$ of the side $BC$. Denote the intersection of the lines $p$ and $AB$ with $E$, and the midpoint of the length $A_1E$ with $F$. The point where the parallel to the line $BD$ through point $A$ intersects $p$ denote by $G$. Prove that the quadrilateral $AFA_1C$ is cyclic if and only if the lines $BF$ passes through the midpoint of the length $CG$.
2017 IFYM, Sozopol, 3
$ABC$ is a triangle with a circumscribed circle $k$, center $I$ of its inscribed circle $\omega$, and center $I_a$ of its excircle $\omega _a$, opposite to $A$. $\omega$ and $\omega _a$ are tangent to $BC$ in points $P$ and $Q$, respectively, and $S$ is the middle point of the arc $\widehat{BC}$ that doesn’t contain $A$. Consider a circle that is tangent to $BC$ in point $P$ and to $k$ in point $R$. Let $RI$ intersect $k$ for a second time in point $L$. Prove that, $LI_a$ and $SQ$ intersect in a point that lies on $k$.
2010 Serbia National Math Olympiad, 2
In an acute-angled triangle $ABC$, $M$ is the midpoint of side $BC$, and $D, E$ and $F$ the feet of the altitudes from $A, B$ and $C$, respectively. Let $H$ be the orthocenter of $\Delta ABC$, $S$ the midpoint of $AH$, and $G$ the intersection of $FE$ and $AH$. If $N$ is the intersection of the median $AM$ and the circumcircle of $\Delta BCH$, prove that $\angle HMA = \angle GNS$.
[i]Proposed by Marko Djikic[/i]
2012 JHMT, 9
Let $ABC$ be a triangle with incircle $O$ and side lengths $5, 8$, and $9$. Consider the other tangent line to $O$ parallel to $BC$, which intersects $AB$ at $B_a$ and $AC$ at $C_a$. Let $r_a$ be the inradius of triangle $AB_aC_a$, and define $r_b$ and $r_c$ similarly. Find $r_a + r_b + r_c$.
2009 AMC 10, 21
Many Gothic cathedrals have windows with portions containing a ring of congruent circles that are circumscribed by a larger circle, In the figure shown, the number of smaller circles is four. What is the ratio of the sum of the areas of the four smaller circles to the area of the larger circle?
[asy]unitsize(6mm);
defaultpen(linewidth(.8pt));
draw(Circle((0,0),1+sqrt(2)));
draw(Circle((sqrt(2),0),1));
draw(Circle((0,sqrt(2)),1));
draw(Circle((-sqrt(2),0),1));
draw(Circle((0,-sqrt(2)),1));[/asy]$ \textbf{(A)}\ 3\minus{}2\sqrt2 \qquad
\textbf{(B)}\ 2\minus{}\sqrt2 \qquad
\textbf{(C)}\ 4(3\minus{}2\sqrt2) \qquad
\textbf{(D)}\ \frac12(3\minus{}\sqrt2)$
$ \textbf{(E)}\ 2\sqrt2\minus{}2$
2009 Today's Calculation Of Integral, 424
Let $ n$ be positive integer. For $ n \equal{} 1,\ 2,\ 3,\ \cdots n$, let denote $ S_k$ be the area of $ \triangle{AOB_k}$ such that $ \angle{AOB_k} \equal{} \frac {k}{2n}\pi ,\ OA \equal{} 1,\ OB_k \equal{} k$. Find the limit $ \lim_{n\to\infty}\frac {1}{n^2}\sum_{k \equal{} 1}^n S_k$.
2013 Moldova Team Selection Test, 3
Consider the triangle $\triangle ABC$ with $AB \not = AC$. Let point $O$ be the circumcenter of $\triangle ABC$. Let the angle bisector of $\angle BAC$ intersect $BC$ at point $D$. Let $E$ be the reflection of point $D$ across the midpoint of the segment $BC$. The lines perpendicular to $BC$ in points $D,E$ intersect the lines $AO,AD$ at the points $X,Y$ respectively. Prove that the quadrilateral $B,X,C,Y$ is cyclic.
2002 Turkey Team Selection Test, 2
In a triangle $ABC$, the angle bisector of $\widehat{ABC}$ meets $[AC]$ at $D$, and the angle bisector of $\widehat{BCA}$ meets $[AB]$ at $E$. Let $X$ be the intersection of the lines $BD$ and $CE$ where $|BX|=\sqrt 3|XD|$ ve $|XE|=(\sqrt 3 - 1)|XC|$. Find the angles of triangle $ABC$.