Found problems: 51
1993 All-Russian Olympiad, 1
The lengths of the sides of a triangle are prime numbers of centimeters. Prove that its area cannot be an integer number of square centimeters.
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
$\text{(A)}\ 1.6 \qquad \text{(B)}\ 1.9 \qquad \text{(C)}\ 2.1 \qquad \text{(D)}\ 2.7 \qquad \text{(E)}\ 2.9$
1989 AIME Problems, 15
Point $P$ is inside $\triangle ABC$. Line segments $APD$, $BPE$, and $CPF$ are drawn with $D$ on $BC$, $E$ on $AC$, and $F$ on $AB$ (see the figure at right). Given that $AP=6$, $BP=9$, $PD=6$, $PE=3$, and $CF=20$, find the area of $\triangle ABC$.
[asy]
size(200);
pair A=origin, B=(7,0), C=(3.2,15), D=midpoint(B--C), F=(3,0), P=intersectionpoint(C--F, A--D), ex=B+40*dir(B--P), E=intersectionpoint(B--ex, A--C);
draw(A--B--C--A--D^^C--F^^B--E);
pair point=P;
label("$A$", A, dir(point--A));
label("$B$", B, dir(point--B));
label("$C$", C, dir(point--C));
label("$D$", D, dir(point--D));
label("$E$", E, dir(point--E));
label("$F$", F, dir(point--F));
label("$P$", P, dir(0));[/asy]
2011 IFYM, Sozopol, 8
The lengths of the sides of a triangle are integers, whereas the radius of its circumscribed circle is a prime number. Prove that the triangle is right-angled.
1959 AMC 12/AHSME, 43
The sides of a triangle are $25,39,$ and $40$. The diameter of the circumscribed circle is:
$ \textbf{(A)}\ \frac{133}{3}\qquad\textbf{(B)}\ \frac{125}{3}\qquad\textbf{(C)}\ 42\qquad\textbf{(D)}\ 41\qquad\textbf{(E)}\ 40 $
1968 AMC 12/AHSME, 12
A circle passes through the vertices of a triangle with side-lengths of $7\tfrac{1}{2},10,12\tfrac{1}{2}$. The radius of the circle is:
$\textbf{(A)}\ \dfrac{15}{4} \qquad
\textbf{(B)}\ 5 \qquad
\textbf{(C)}\ \dfrac{25}{4} \qquad
\textbf{(D)}\ \dfrac{35}{4} \qquad
\textbf{(E)}\ \dfrac{15\sqrt2}{2} $
1969 Canada National Olympiad, 4
Let $ABC$ be an equilateral triangle, and $P$ be an arbitrary point within the triangle. Perpendiculars $PD,PE,PF$ are drawn to the three sides of the triangle. Show that, no matter where $P$ is chosen, \[ \frac{PD+PE+PF}{AB+BC+CA}=\frac{1}{2\sqrt{3}}. \]
2014 Harvard-MIT Mathematics Tournament, 6
In quadrilateral $ABCD$, we have $AB = 5$, $BC = 6$, $CD = 5$, $DA = 4$, and $\angle ABC = 90^\circ$. Let $AC$ and $BD$ meet at $E$. Compute $\dfrac{BE}{ED}$.
2017 India National Olympiad, 6
Let $n\ge 1$ be an integer and consider the sum $$x=\sum_{k\ge 0} \dbinom{n}{2k} 2^{n-2k}3^k=\dbinom{n}{0}2^n+\dbinom{n}{2}2^{n-2}\cdot{}3+\dbinom{n}{4}2^{n-k}\cdot{}3^2 + \cdots{}.$$
Show that $2x-1,2x,2x+1$ form the sides of a triangle whose area and inradius are also integers.
2000 AMC 12/AHSME, 19
In triangle $ ABC$, $ AB \equal{} 13$, $ BC \equal{} 14$, and $ AC \equal{} 15$. Let $ D$ denote the midpoint of $ \overline{BC}$ and let $ E$ denote the intersection of $ \overline{BC}$ with the bisector of angle $ BAC$. Which of the following is closest to the area of the triangle $ ADE$?
$ \textbf{(A)}\ 2 \qquad \textbf{(B)}\ 2.5 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 3.5 \qquad \textbf{(E)}\ 4$
2005 AIME Problems, 15
Triangle $ABC$ has $BC=20$. The incircle of the triangle evenly trisects the median $AD$. If the area of the triangle is $m \sqrt{n}$ where $m$ and $n$ are integers and $n$ is not divisible by the square of a prime, find $m+n$.
2014 Harvard-MIT Mathematics Tournament, 3
$ABC$ 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.
1964 IMO, 3
A circle is inscribed in a triangle $ABC$ with sides $a,b,c$. Tangents to the circle parallel to the sides of the triangle are contructe. Each of these tangents cuts off a triagnle from $\triangle ABC$. In each of these triangles, a circle is inscribed. Find the sum of the areas of all four inscribed circles (in terms of $a,b,c$).
2015 AMC 12/AHSME, 20
Isosceles triangles $T$ and $T'$ are not congruent but have the same area and the same perimeter. The sides of $T$ have lengths $5$, $5$, and $8$, while those of $T'$ have lengths $a$, $a$, and $b$. Which of the following numbers is closest to $b$?
$\textbf{(A) }3\qquad\textbf{(B) }4\qquad\textbf{(C) }5\qquad\textbf{(D) }6\qquad\textbf{(E) }8$
2008 AMC 12/AHSME, 18
A pyramid has a square base $ ABCD$ and vertex $ E$. The area of square $ ABCD$ is $ 196$, and the areas of $ \triangle{ABE}$ and $ \triangle{CDE}$ are $ 105$ and $ 91$, respectively. What is the volume of the pyramid?
$ \textbf{(A)}\ 392 \qquad
\textbf{(B)}\ 196\sqrt{6} \qquad
\textbf{(C)}\ 392\sqrt2 \qquad
\textbf{(D)}\ 392\sqrt3 \qquad
\textbf{(E)}\ 784$
1964 AMC 12/AHSME, 35
The sides of a triangle are of lengths $13$, $14$, and $15$. The altitudes of the triangle meet at point $H$. If $AD$ is the altitude to the side length $14$, what is the ratio $HD:HA$?
$\textbf{(A) } 3 : 11\qquad
\textbf{(B) } 5 : 11\qquad
\textbf{(C) } 1 : 2\qquad
\textbf{(D) }2 : 3\qquad
\textbf{(E) }25 : 33$
2003 India National Olympiad, 5
Let a, b, c be the sidelengths and S the area of a triangle ABC. Denote $x=a+\frac{b}{2}$, $y=b+\frac{c}{2}$ and $z=c+\frac{a}{2}$. Prove that there exists a triangle with sidelengths x, y, z, and the area of this triangle is $\geq\frac94 S$.
2012 AMC 8, 23
An equilateral triangle and a regular hexagon have equal perimeters. If the area of the triangle is 4, what is the area of the hexagon?
$\textbf{(A)}\hspace{.05in}4 \qquad \textbf{(B)}\hspace{.05in}5 \qquad \textbf{(C)}\hspace{.05in}6 \qquad \textbf{(D)}\hspace{.05in}4\sqrt3 \qquad \textbf{(E)}\hspace{.05in}6\sqrt3 $
2010 Stanford Mathematics Tournament, 2
Find the radius of a circle inscribed in a triangle with side lengths $4$, $5$, and $6$
2011 Kosovo National Mathematical Olympiad, 4
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?
1995 AIME Problems, 14
In a circle of radius 42, two chords of length 78 intersect at a point whose distance from the center is 18. The two chords divide the interior of the circle into four regions. Two of these regions are bordered by segments of unequal lenghts, and the area of either of them can be expressed uniquley in the form $m\pi-n\sqrt{d},$ where $m, n,$ and $d$ are positive integers and $d$ is not divisible by the square of any prime number. Find $m+n+d.$
1961 IMO, 2
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?
1963 AMC 12/AHSME, 35
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:
$\textbf{(A)}\ 8 \qquad
\textbf{(B)}\ 10\qquad
\textbf{(C)}\ 12 \qquad
\textbf{(D)}\ 14 \qquad
\textbf{(E)}\ 16$
1961 IMO Shortlist, 2
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?
2007 China Northern MO, 4
The inradius of triangle $ ABC$ is $ 1$ and the side lengths of $ ABC$ are all integers. Prove that triangle $ ABC$ is right-angled.