Found problems: 259
2023 pOMA, 2
Let $\triangle ABC$ be an acute triangle, and let $D,E,F$ respectively be three points on sides $BC,CA,AB$ such that $AEDF$ is a cyclic quadrilateral. Let $O_B$ and $O_C$ be the circumcenters of $\triangle BDF$ and $\triangle CDE$, respectively. Finally, let $D'$ be a point on segment $BC$ such that $BD'=CD$. Prove that $\triangle BD'O_B$ and $\triangle CD'O_C$ have the same surface.
1989 Austrian-Polish Competition, 8
$ABC$ is an acute-angled triangle and $P$ a point inside or on the boundary. The feet of the perpendiculars from $P$ to $BC, CA, AB$ are $A', B', C'$ respectively. Show that if $ABC$ is equilateral, then $\frac{AC'+BA'+CB'}{PA'+PB'+PC'}$ is the same for all positions of $P$, but that for any other triangle it is not.
1994 Bundeswettbewerb Mathematik, 3
Given a triangle $A_1 A_2 A_3$ and a point $P$ inside. Let $B_i$ be a point on the side opposite to $A_i$ for $i=1,2,3$, and let $C_i$ and $D_i$ be the midpoints of $A_i B_i$ and $P B_i$, respectively. Prove that the triangles $C_1 C_2 C_3$ and $D_1 D_2 D_3$ have equal area.
1976 Vietnam National Olympiad, 3
$P$ is a point inside the triangle $ABC$. The perpendicular distances from $P$ to the three sides have product $p$. Show that $p \le \frac{ 8 S^3}{27abc}$, where $S =$ area $ABC$ and $a, b, c$ are the sides. Prove a similar result for a tetrahedron.
1995 Nordic, 4
Show that there exist infinitely many mutually non- congruent triangles $T$, satisfying
(i) The side lengths of $T $ are consecutive integers.
(ii) The area of $T$ is an integer.
2012 Czech And Slovak Olympiad IIIA, 2
Find out the maximum possible area of the triangle $ABC$ whose medians have lengths satisfying inequalities $m_a \le 2, m_b \le 3, m_c \le 4$.
Denmark (Mohr) - geometry, 1995.1
A trapezoid has side lengths as indicated in the figure (the sides with length $11$ and $36$ are parallel). Calculate the area of the trapezoid.[img]https://1.bp.blogspot.com/-5PKrqDG37X4/XzcJtCyUv8I/AAAAAAAAMY0/tB0FObJUJdcTlAJc4n6YNEaVIDfQ91-eQCLcBGAsYHQ/s0/1995%2BMohr%2Bp1.png[/img]
2020 Nordic, 3
Each of the sides $AB$ and $CD$ of a convex quadrilateral $ABCD$ is divided into three equal parts, $|AE| = |EF| = |F B|$ , $|DP| = |P Q| = |QC|$. The diagonals of $AEPD$ and $FBCQ$ intersect at $M$ and $N$, respectively. Prove that the sum of the areas of $\vartriangle AMD$ and $\vartriangle BNC$ is equal to the sum of the areas of $\vartriangle EPM$ and $\vartriangle FNQ$.
2012 National Olympiad First Round, 1
Find the perimeter of a triangle whose altitudes are $3,4,$ and $6$.
$ \textbf{(A)}\ 12\sqrt\frac35 \qquad \textbf{(B)}\ 16\sqrt\frac35 \qquad \textbf{(C)}\ 20\sqrt\frac35 \qquad \textbf{(D)}\ 24\sqrt\frac35 \qquad \textbf{(E)}\ \text{None}$
Estonia Open Senior - geometry, 2000.1.3
In the plane, the segments $AB$ and $CD$ are given, while the lines $AB$ and $CD$ intersect. Prove that the set of all points $P$ in the plane such that triangles $ABP$ and $CDP$ have equal areas , form two lines intersecting at the intersection of the lines $AB$ and $CD$.
2018 Yasinsky Geometry Olympiad, 2
Let $P$ the intersection point of the diagonals of a convex quadrilateral $ABCD$. It is known that the area of triangles $ABC$, $BCD$ and $DAP$ is equal to $8 cm^2$, $9 cm^2$ and $10 cm^2$. Find the area of the quadrilateral $ABCD$.
1965 AMC 12/AHSME, 16
Let line $ AC$ be perpendicular to line $ CE$. Connect $ A$ to $ D$, the midpoint of $ CE$, and connect $ E$ to $ B$, the midpoint of $ AC$. If $ AD$ and $ EB$ intersect in point $ F$, and $ \overline{BC} \equal{} \overline{CD} \equal{} 15$ inches, then the area of triangle $ DFE$, in square inches, is:
$ \textbf{(A)}\ 50 \qquad \textbf{(B)}\ 50\sqrt {2} \qquad \textbf{(C)}\ 75 \qquad \textbf{(D)}\ \frac {15}{2}\sqrt {105} \qquad \textbf{(E)}\ 100$
2024 Bundeswettbewerb Mathematik, 3
Let $ABC$ be a triangle. For a point $P$ in its interior, we draw the threee lines through $P$ parallel to the sides of the triangle. This partitions $ABC$ in three triangles and three quadrilaterals.
Let $V_A$ be the area of the quadrilateral which has $A$ as one vertex. Let $D_A$ be the area of the triangle which has a part of $BC$ as one of its sides. Define $V_B, D_B$ and $V_C, D_C$ similarly.
Determine all possible values of $\frac{D_A}{V_A}+\frac{D_B}{V_B}+\frac{D_C}{V_C}$, as $P$ varies in the interior of the triangle.
2012 India Regional Mathematical Olympiad, 5
Let $ABC$ be a triangle. Let $D,E$ be points on the segment $BC$ such that $BD=DE=EC$. Let $F$ be the mid-point of $AC$. Let $BF$ intersect $AD$ in $P$ and $AE$ in $Q$ respectively. Determine the ratio of the area of the triangle $APQ$ to that of the quadrilateral $PDEQ$.
1992 IMTS, 5
Let $T = (a,b,c)$ be a triangle with sides $a,b$ and $c$ and area $\triangle$. Denote by $T' = (a',b',c')$ the triangle whose sides are the altitudes of $T$ (i.e., $a' = h_a, b' = h_b, c' = h_c$) and denote its area by $\triangle '$. Similarly, let $T'' = (a'',b'',c'')$ be the triangle formed from the altitudes of $T'$, and denote its area by $\triangle ''$. Given that $\triangle ' = 30$ and $\triangle '' = 20$, find $\triangle$.
1995 IMO Shortlist, 3
Determine all integers $ n > 3$ for which there exist $ n$ points $ A_{1},\cdots ,A_{n}$ in the plane, no three collinear, and real numbers $ r_{1},\cdots ,r_{n}$ such that for $ 1\leq i < j < k\leq n$, the area of $ \triangle A_{i}A_{j}A_{k}$ is $ r_{i} \plus{} r_{j} \plus{} r_{k}$.
Ukrainian TYM Qualifying - geometry, 2017.5
The Fibonacci sequence is given by equalities $$F_1=F_2=1, F_{k+2}=F_k+F_{k+1}, k\in N$$.
a) Prove that for every $m \ge 0$, the area of the triangle $A_1A_2A_3$ with vertices $A_1(F_{m+1},F_{m+2})$, $A_2 (F_{m+3},F_{m+4})$, $A_3 (F_{m+5},F_{m+6})$ is equal to $0.5$.
b) Prove that for every $m \ge 0$ the quadrangle $A_1A_2A_4$ with vertices $A_1(F_{m+1},F_{m+2})$, $A_2 (F_{m+3},F_{m+4})$, $A_3 (F_{m+5},F_{m+6})$, $A_4 (F_{m+7},F_{m+8})$ is a trapezoid, whose area is equal to $2.5$.
c) Prove that the area of the polygon $A_1A_2...A_n$ , $n \ge3$ with vertices does not depend on the choice of numbers $m \ge 0$, and find this area.
2013 Hanoi Open Mathematics Competitions, 7
Let $ABC$ be a triangle with $\angle A = 90^o, \angle B = 60^o$ and $BC = 1$ cm. Draw outside of $\vartriangle ABC$ three equilateral triangles $ABD,ACE$ and $BCF$. Determine the area of $\vartriangle DEF$.
2014 Junior Balkan Team Selection Tests - Moldova, 3
Let $ABC$ be a right triangle with $\angle ABC = 90^o$ . Points $D$ and $E$, located on the legs $(AC)$ and $(AB)$ respectively, are the legs of the inner bisectors taken from the vertices $B$ and $C$, respectively. Let $I$ be the center of the circle inscribed in the triangle $ABC$. If $BD \cdot CE = m^2 \sqrt2$ , find the area of the triangle $BIC$ (in terms of parameter $m$)
1993 ITAMO, 1
Let be given points $A,B,C$ on a line, with $C$ between $A$ and $B$. Three semicircles with diameters $AC,BC,AB$ are drawn on the same side of line $ABC$. The perpendicular to $AB$ at $C$ meets the circle with diameter $AB$ at $H$. Given that $CH =\sqrt2$, compute the area of the region bounded by the three semicircles.
1998 Tuymaada Olympiad, 3
The segment of length $\ell$ with the ends on the border of a triangle divides the area of that triangle in half. Prove that $\ell >r\sqrt2$, where $r$ is the radius of the inscribed circle of the triangle.
1996 AMC 12/AHSME, 28
On a $4 \times 4 \times 3$ rectangular parallelepiped, vertices $A$, $B$, and $C$ are adjacent to vertex $D$. The perpendicular distance from $D$ to the plane containing
$A$, $B$, and $C$ is closest to
$\text{(A)}\ 1.6 \qquad \text{(B)}\ 1.9 \qquad \text{(C)}\ 2.1 \qquad \text{(D)}\ 2.7 \qquad \text{(E)}\ 2.9$
2013 AIME Problems, 12
Let $\triangle PQR$ be a triangle with $\angle P = 75^\circ$ and $\angle Q = 60^\circ$. A regular hexagon $ABCDEF$ with side length 1 is drawn inside $\triangle PQR$ so that side $\overline{AB}$ lies on $\overline{PQ}$, side $\overline{CD}$ lies on $\overline{QR}$, and one of the remaining vertices lies on $\overline{RP}$. There are positive integers $a$, $b$, $c$, and $d$ such that the area of $\triangle PQR$ can be expressed in the form $\tfrac{a+b\sqrt c}d$, where $a$ and $d$ are relatively prime and $c$ is not divisible by the square of any prime. Find $a+b+c+d$.
1980 Spain Mathematical Olympiad, 1
Among the triangles that have a side of length $5$ m and the angle opposite of $30^o$, determine the one with maximum area, calculating the value of the other two angles and area of triangle.
2023-24 IOQM India, 5
In a triangle $A B C$, let $E$ be the midpoint of $A C$ and $F$ be the midpoint of $A B$. The medians $B E$ and $C F$ intersect at $G$. Let $Y$ and $Z$ be the midpoints of $B E$ and $C F$ respectively. If the area of triangle $A B C$ is 480 , find the area of triangle $G Y Z$.