Found problems: 25757
2012-2013 SDML (Middle School), 2
A regular tetrahedron with $5$-inch edges weighs $2.5$ pounds. What is the weight in pounds of a similarly constructed regular tetrahedron that has $6$-inch edges? Express your answer as a decimal rounded to the nearest hundredth.
1902 Eotvos Mathematical Competition, 3
The area $T$ and an angle $\gamma$ of a triangle are given. Determine the lengths of the sides $a$ and $b$ so that the side $c$, opposite the angle $\gamma$, is as short as possible.
2011 Morocco TST, 3
The vertices $X, Y , Z$ of an equilateral triangle $XYZ$ lie respectively on the sides $BC, CA, AB$ of an acute-angled triangle $ABC.$ Prove that the incenter of triangle $ABC$ lies inside triangle $XYZ.$
[i]Proposed by Nikolay Beluhov, Bulgaria[/i]
1992 IMO Longlists, 3
Let $ABC$ be a triangle, $O$ its circumcenter, $S$ its centroid, and $H$ its orthocenter. Denote by $A_1, B_1$, and $C_1$ the centers of the circles circumscribed about the triangles $CHB, CHA$, and $AHB$, respectively. Prove that the triangle $ABC$ is congruent to the triangle $A_1B_1C_1$ and that the nine-point circle of $\triangle ABC$ is also the nine-point circle of $\triangle A_1B_1C_1$.
Ukrainian TYM Qualifying - geometry, II.18
Inside an acute angle is a circle. Investigate the possibility of constructing with only a compass and a ruler, a tangent to this circle that the point of contact will bisect the segment of the tangent that is cut off by the sides of the angle.
2000 All-Russian Olympiad Regional Round, 8.7
Angle bisectors $AD$ and $CE$ of triangle $ABC$ intersect at point $O$. A line symmetrical to $ AB$ with respect to $CE$ intersects the line symmetric $BC$ with respect to $AD$, at point $K$. Prove that $KO \perp AC$.
2012 Sharygin Geometry Olympiad, 5
On side $AC$ of triangle $ABC$ an arbitrary point is selected $D$. The tangent in $D$ to the circumcircle of triangle $BDC$ meets $AB$ in point $C_{1}$; point $A_{1}$ is defined similarly. Prove that $A_{1}C_{1}\parallel AC$.
2007 iTest Tournament of Champions, 4
In triangle $ABC$, points $A'$, $B'$, and $C'$ are chosen with $A'$ on segment $AB$, $B'$ on segment $BC$, and $C'$ on segment $CA$ so that triangle $A'B'C'$ is directly similar to $ABC$. The incenters of $ABC$ and $A'B'C'$ are $I$ and $I'$ respectively. Lines $BC$, $A'C'$, and $II'$ are parallel. If $AB=100$ and $AC=120$, what is the length of $BC$?
Kvant 2025, M2837
On the graphic of the function $y=x^2$ were selected $1000$ pairwise distinct points, abscissas of which are integer numbers from the segment $[0; 100000]$. Prove that it is possible to choose six different selected points $A$, $B$, $C$, $A'$, $B'$, $C'$ such that areas of triangles $ABC$ and $A'B'C'$ are equals.
[i]A. Tereshin[/i]
Ukraine Correspondence MO - geometry, 2018.9
Let $ABC$ be an acute-angled triangle in which $AB <AC$. On the side $BC$ mark a point $D$ such that $AD = AB$, and on the side $AB$ mark a point $E$ such that the segment $DE$ passes through the orthocenter of triangle $ABC$. Prove that the center of the circumcircle of triangle $ADE$ lies on the segment $AC$.
2010 Cono Sur Olympiad, 3
Let us define [i]cutting[/i] a convex polygon with $n$ sides by choosing a pair of consecutive sides $AB$ and $BC$ and substituting them by three segments $AM, MN$, and $NC$, where $M$ is the midpoint of $AB$ and $N$ is the midpoint of $BC$. In other words, the triangle $MBN$ is removed and a convex polygon with $n+1$ sides is obtained.
Let $P_6$ be a regular hexagon with area $1$. $P_6$ is [i]cut[/i] and the polygon $P_7$ is obtained. Then $P_7$ is cut in one of seven ways and polygon $P_8$ is obtained, and so on. Prove that, regardless of how the cuts are made, the area of $P_n$ is always greater than $2/3$.
1977 Czech and Slovak Olympiad III A, 2
The numbers $p,q>0$ are given. Construct a rectangle $ABCD$ with $AE=p,AF=q$ where $E,F$ are midpoints of $BC,CD,$ respectively. Discuss conditions of solvability.
2015 Greece National Olympiad, 3
Given is a triangle $ABC$ with $\angle{B}=105^{\circ}$.Let $D$ be a point on $BC$ such that $\angle{BDA}=45^{\circ}$.
A) If $D$ is the midpoint of $BC$ then prove that $\angle{C}=30^{\circ}$,
B) If $\angle{C}=30^{\circ}$ then prove that $D$ is the midpoint of $BC$
2024 Korea Winter Program Practice Test, Q1
A point $P$ lies inside $\usepackage{gensymb} \angle ABC(<90 \degree)$. Show that there exists a point $Q$ inside $\angle ABC$ satisfying the following condition:
[center]For any two points $X$ and $Y$ on the rays $\overrightarrow{BA}$ and $\overrightarrow{BC}$ respectively satisfying $\angle XPY = \angle ABC$, it holds that $\usepackage{gensymb} \angle XQY = 180 \degree - 2 \angle ABC.$[/center]
2000 Moldova National Olympiad, Problem 4
The orthocenter $H$ of a triangle $ABC$ is not on the sides of the triangle and the distance $AH$ equals the circumradius of the triangle. Find the measure of $\angle A$.
2007 Romania Team Selection Test, 2
Let $ A_{1}A_{2}A_{3}A_{4}A_{5}$ be a convex pentagon, such that
\[ [A_{1}A_{2}A_{3}] \equal{} [A_{2}A_{3}A_{4}] \equal{} [A_{3}A_{4}A_{5}] \equal{} [A_{4}A_{5}A_{1}] \equal{} [A_{5}A_{1}A_{2}].\]
Prove that there exists a point $ M$ in the plane of the pentagon such that
\[ [A_{1}MA_{2}] \equal{} [A_{2}MA_{3}] \equal{} [A_{3}MA_{4}] \equal{} [A_{4}MA_{5}] \equal{} [A_{5}MA_{1}].\]
Here $ [XYZ]$ stands for the area of the triangle $ \Delta XYZ$.
2019 Mediterranean Mathematics Olympiad, 1
Let $\Delta ABC$ be a triangle with angle $\angle CAB=60^{\circ}$, let $D$ be the intersection point of the angle bisector at $A$ and the side $BC$, and let $r_B,r_C,r$ be the respective radii of the incircles of $ABD$, $ADC$, $ABC$. Let $b$ and $c$ be the lengths of sides $AC$ and $AB$ of the triangle. Prove that
\[ \frac{1}{r_B} +\frac{1}{r_C} ~=~ 2\cdot\left( \frac1r +\frac1b +\frac1c\right)\]
2019 China Team Selection Test, 3
$60$ points lie on the plane, such that no three points are collinear. Prove that one can divide the points into $20$ groups, with $3$ points in each group, such that the triangles ( $20$ in total) consist of three points in a group have a non-empty intersection.
2006 MOP Homework, 4
Let $ABC$ be a triangle with circumcenter $O$. Let $A_1$ be the midpoint of side $BC$. Ray $AA_1$ meet the circumcircle of triangle $ABC$ again at $A_2$ (other than A). Let $Q_a$ be the foot of the perpendicular from $A_1$ to line $AO$. Point $P_a$ lies on line $Q_aA_1$ such that $P_aA_2 \perp A_2O$. Define points $P_b$ and $P_c$ analogously. Prove that points $P_a$, P_b$, and $P_c$ lie on a line.
1995 Tournament Of Towns, (466) 4
From the vertex $A$ of a triangle $ABC$, three segments are drawn: the bisectors $AM$ and $AN$ of its interior and exterior angles and the tangent $AK$ to the circumscribed circle of the triangle (the points $M$, $K$ and $N$ lie on the line $BC$). Prove that $MK = KN$.
(I Sharygin)
2012 Bosnia and Herzegovina Junior BMO TST, 1
On circle $k$ there are clockwise points $A$, $B$, $C$, $D$ and $E$ such that $\angle ABE = \angle BEC = \angle ECD = 45^{\circ}$. Prove that $AB^2 + CE^2 = BE^2 + CD^2$
2004 Purple Comet Problems, 2
In $\triangle ABC$, three lines are drawn parallel to side $BC$ dividing the altitude of the triangle into four equal parts. If the area of the second largest part is $35$, what is the area of the whole $\triangle ABC$?
[asy]
defaultpen(linewidth(0.7)); size(120);
pair B = (0,0), C = (1,0), A = (0.7,1); pair[] AB, AC;
draw(A--B--C--cycle);
for(int i = 1; i < 4; ++i) {
AB.push((i*A + (4-i)*B)/4); AC.push((i*A + (4-i)*C)/4);
draw(AB[i-1] -- AC[i-1]);
}
filldraw(AB[1]--AB[0]--AC[0]--AC[1]--cycle, gray(0.7));
label("$A$",A,N); label("$B$",B,S); label("$C$",C,S);[/asy]
2002 AMC 10, 17
A regular octagon $ ABCDEFGH$ has sides of length two. Find the area of $ \triangle{ADG}$.
$ \textbf{(A)}\ 4 \plus{} 2 \sqrt{2} \qquad
\textbf{(B)}\ 6 \plus{} \sqrt{2} \qquad
\textbf{(C)}\ 4 \plus{} 3 \sqrt{2} \qquad
\textbf{(D)}\ 3 \plus{} 4 \sqrt{2} \qquad
\textbf{(E)}\ 8 \plus{} \sqrt{2}$
2016 Mediterranean Mathematics Olympiad, 1
Let $ABC$ be a triangle. Let $D$ be the intersection point of the angle bisector at $A$ with $BC$.
Let $T$ be the intersection point of the tangent line to the circumcircle of triangle $ABC$ at point $A$ with the line through $B$ and $C$.
Let $I$ be the intersection point of the orthogonal line to $AT$ through point $D$ with the altitude $h_a$ of the triangle at point $A$.
Let $P$ be the midpoint of $AB$, and let $O$ be the circumcenter of triangle $ABC$.
Let $M$ be the intersection point of $AB$ and $TI$, and let $F$ be the intersection point of $PT$ and $AD$.
Prove: $MF$ and $AO$ are orthogonal to each other.
2012 Sharygin Geometry Olympiad, 17
A square $ABCD$ is inscribed into a circle. Point $M$ lies on arc $BC$, $AM$ meets $BD$ in point $P$, $DM$ meets $AC$ in point $Q$. Prove that the area of quadrilateral $APQD$ is equal to the half of the area of the square.