Found problems: 259
The inradius of triangle $ ABC$ is $ 1$ and the side lengths of $ ABC$ are all integers. Prove that triangle $ ABC$ is right-angled.
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.
Point $P$ is in the interior of $\triangle ABC$. The side lengths of $ABC$ are $AB = 7$, $BC = 8$, $CA = 9$. The three foots of perpendicular lines from $P$ to sides $BC$, $CA$, $AB$ are $D$, $E$, $F$ respectively. Suppose the minimal value of $\frac{BC}{PD} + \frac{CA}{PE} + \frac{AB}{PF}$ can be written as $\frac{a}{b}\sqrt{c}$, where $\gcd(a,b) = 1$ and $c$ is square free, calculate $abc$.
[asy]
size(120); pathpen = linewidth(0.7); pointfontpen = fontsize(10); // pointpen = black;
pair B=(0,0), C=(8,0), A=IP(CR(B,7),CR(C,9)), P = (2,1.6), D=foot(P,B,C), E=foot(P,A,C), F=foot(P,A,B);
D(A--B--C--cycle); D(P--D); D(P--E); D(P--F);
D(MP("A",A,N)); D(MP("B",B)); D(MP("C",C)); D(MP("D",D)); D(MP("E",E,NE)); D(MP("F",F,NW)); D(MP("P",P,SE));
[/asy]
In a triangle $ ABC,$ choose any points $ K \in BC, L \in AC, M \in AB, N \in LM, R \in MK$ and $ F \in KL.$ If $ E_1, E_2, E_3, E_4, E_5, E_6$ and $ E$ denote the areas of the triangles $ AMR, CKR, BKF, ALF, BNM, CLN$ and $ ABC$ respectively, show that
\[ E \geq 8 \cdot \sqrt [6]{E_1 E_2 E_3 E_4 E_5 E_6}.
\]
Two medians of a triangle with unequal sides are $ 3$ inches and $ 6$ inches. Its area is $ 3 \sqrt{15}$ square inches. The length of the third median in inches, is:
$ \textbf{(A)}\ 4 \qquad
\textbf{(B)}\ 3 \sqrt{3} \qquad
\textbf{(C)}\ 3 \sqrt{6} \qquad
\textbf{(D)}\ 6 \sqrt{3} \qquad
\textbf{(E)}\ 6 \sqrt{6}$
Let $ ABCD $ be a rectangle of area $ S $, and $ P $ be a point inside it. We denote by $ a, b, c, d $ the distances from $ P $ to the vertices $ A, B, C, D $ respectively. Prove that $ a ^ 2 + b ^ 2 + c ^ 2 + d ^ 2\ge 2S $. When there is equality?
In a triangle $ ABC,$ choose any points $ K \in BC, L \in AC, M \in AB, N \in LM, R \in MK$ and $ F \in KL.$ If $ E_1, E_2, E_3, E_4, E_5, E_6$ and $ E$ denote the areas of the triangles $ AMR, CKR, BKF, ALF, BNM, CLN$ and $ ABC$ respectively, show that
\[ E \geq 8 \cdot \sqrt [6]{E_1 E_2 E_3 E_4 E_5 E_6}.
\]
Let $ a$, $ b$, $ c$ be the sides of a triangle, and $ S$ its area. Prove:
\[ a^{2} \plus{} b^{2} \plus{} c^{2}\geq 4S \sqrt {3}
\]
In what case does equality hold?
We draw two lines $(\ell_1) , (\ell_2)$ through the orthocenter $H$ of the triangle $ABC$ such that each one is dividing the triangle into two figures of equal area and equal perimeters. Find the angles of the triangle.
The diagonals $AD, BE$, and $CF$ of a convex hexagon $ABCDEF$ intersect in a common point.
Show that $a(ABE) a(CDA) a(EFC) = a(BCE) a(DEA) a(FAC)$,
where $a(KLM)$ is the area of the triangle $KLM$.
[img]https://cdn.artofproblemsolving.com/attachments/0/a/bcbbddedde159150fe3c26b1f0a2bfc322aa1a.png[/img]
Let $ABCD$ be a square with side $20$ and $T_1, T_2, ..., T_{2000}$ are points in $ABCD$ such that no $3$ points in the set $S = \{A, B, C, D, T_1, T_2, ..., T_{2000}\}$ are collinear. Prove that there exists a triangle with vertices in $S$, such that the area is less than $1/10$.
For a triangle $ ABC,$ let $ k$ be its circumcircle with radius $ r.$ The bisectors of the inner angles $ A, B,$ and $ C$ of the triangle intersect respectively the circle $ k$ again at points $ A', B',$ and $ C'.$ Prove the inequality
\[ 16Q^3 \geq 27 r^4 P,\]
where $ Q$ and $ P$ are the areas of the triangles $ A'B'C'$ and $ABC$ respectively.
A clock has an hour, minute, and second hand, all of length $1$. Let $T$ be the triangle formed by the ends of these hands. A time of day is chosen uniformly at random. What is the expected value of the area of $T$?
[i]Proposed by Dylan Toh[/i]
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]
In a convex quadrilateral (in the plane) with the area of $32 \text{ cm}^{2}$ the sum of two opposite sides and a diagonal is $16 \text{ cm}$. Determine all the possible values that the other diagonal can have.
The triangle $ABC$ is right in $A$ and $R$ is the midpoint of the hypotenuse $BC$ . On the major leg $AB$ the point $P$ is marked such that $CP = BP$ and on the segment $BP$ the point $Q$ is marked such that the triangle $PQR$ is equilateral. If the area of triangle $ABC$ is $27$, calculate the area of triangle $PQR$ .
A triangle has side lengths $a$, $b$, $c$, and the altitudes have lengths $h_a$, $h_b$, $h_c$. Prove that
\[ \left( \frac{a}{h_a} \right)^2
+ \left( \frac{b}{h_b} \right)^2
+ \left( \frac{c}{h_c} \right)^2 \ge 4. \]
Suppose $ \vartriangle ABC$ is similar to $\vartriangle DEF$, with $ A$, $ B$, and $C$ corresponding to $D, E$, and $F$ respectively. If $\overline{AB} = \overline{EF}$, $\overline{BC} = \overline{FD}$, and $\overline{CA} = \overline{DE} = 2$, determine the area of $ \vartriangle ABC$.
Let $ABCD$ be a parallelogram, and let $P$ be a point on the side $AB$. Let the line through $P$ parallel to $BC$ intersect the diagonal $AC$ at point $Q$. Prove that $$|DAQ|^2 = |PAQ| \times |BCD| ,$$ where $|XY Z|$ denotes the area of triangle $XY Z$.
Let $ABCD$ be a rectangle and let $E$ and $F$ be points on $CD$ and $BC$ respectively such that area $(ADE) = 16$, area $(CEF) = 9$ and area $(ABF) = 25$. What is the area of triangle $AEF$ ?
Rob is helping to build the set for a school play. For one scene, he needs to build a multi-colored tetrahedron out of cloth and bamboo. He begins by fitting three lengths of bamboo together, such that they meet at the same point, and each pair of bamboo rods meet at a right angle. Three more lengths of bamboo are then cut to connect the other ends of the first three rods. Rob then cuts out four triangular pieces of fabric: a blue piece, a red piece, a green piece, and a yellow piece. These triangular pieces of fabric just fill in the triangular spaces between the bamboo, making up the four faces of the tetrahedron. The areas in square feet of the red, yellow, and green pieces are $60$, $20$, and $15$ respectively. If the blue piece is the largest of the four sides, find the number of square feet in its area.
$ABC$ is a triangle such that $BC = 10$, $CA = 12$. Let $M$ be the midpoint of side $AC$. Given that $BM$ is parallel to the external bisector of $\angle A$, find area of triangle $ABC$. (Lines $AB$ and $AC$ form two angles, one of which is $\angle BAC$. The external angle bisector of $\angle A$ is the line that bisects the other angle.
The lengths of the sides of a triangle are integers, and its area is also an integer. One side is $21$ and the perimeter is $48$. The shortest side is:
$\textbf{(A)}\ 8 \qquad
\textbf{(B)}\ 10\qquad
\textbf{(C)}\ 12 \qquad
\textbf{(D)}\ 14 \qquad
\textbf{(E)}\ 16$
The side lengths of a triangle are the roots of a cubic polynomial with rational coefficients. Prove that the altitudes of this triangle are roots of a polynomial of sixth degree with rational coefficients.
Triangle $AMC$ is isoceles with $AM = AC$. Medians $\overline{MV}$ and $\overline{CU}$ are perpendicular to each other, and $MV=CU=12$. What is the area of $\triangle AMC?$
[asy]
draw((-4,0)--(4,0)--(0,12)--cycle);
draw((-2,6)--(4,0));
draw((2,6)--(-4,0));
draw((-2,6)--(2,6));
label("M", (-4,0), W);
label("C", (4,0), E);
label("A", (0, 12), N);
label("V", (2, 6), NE);
label("U", (-2, 6), NW);
draw(rightanglemark((-2,6),(0,4),(-4,0),17));
[/asy]
$\textbf{(A) } 48 \qquad \textbf{(B) } 72 \qquad \textbf{(C) } 96 \qquad \textbf{(D) } 144 \qquad \textbf{(E) } 192$