Found problems: 51
A rectangle has sides of length $a$ and $36$. A hinge is installed at each vertex of the rectangle and at the midpoint of each side of length $36$. The sides of length $a$ can be pressed toward each other keeping those two sides parallel so the rectangle becomes a convex hexagon as shown. When the figure is a hexagon with the sides of length $a$ parallel and separated by a distance of $24,$ the hexagon has the same area as the original rectangle. Find $a^2$.
[asy]
pair A,B,C,D,E,F,R,S,T,X,Y,Z;
dotfactor = 2;
unitsize(.1cm);
A = (0,0);
B = (0,18);
C = (0,36);
// don't look here
D = (12*2.236, 36);
E = (12*2.236, 18);
F = (12*2.236, 0);
draw(A--B--C--D--E--F--cycle);
dot(" ",A,NW);
dot(" ",B,NW);
dot(" ",C,NW);
dot(" ",D,NW);
dot(" ",E,NW);
dot(" ",F,NW);
//don't look here
R = (12*2.236 +22,0);
S = (12*2.236 + 22 - 13.4164,12);
T = (12*2.236 + 22,24);
X = (12*4.472+ 22,24);
Y = (12*4.472+ 22 + 13.4164,12);
Z = (12*4.472+ 22,0);
draw(R--S--T--X--Y--Z--cycle);
dot(" ",R,NW);
dot(" ",S,NW);
dot(" ",T,NW);
dot(" ",X,NW);
dot(" ",Y,NW);
dot(" ",Z,NW);
// sqrt180 = 13.4164
// sqrt5 = 2.236
[/asy]
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 $
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 $\geq\frac94 S$.
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 $m+n$.
(1) Let $a,b,c$ be positive real numbers satisfying $(a^2+b^2+c^2)^2>2(a^4+b^4+c^4)$. Prove that $a,b,c$ can be the lengths of three sides of a triangle respectively.
(2) Let $a_1,a_2,\dots ,a_n$ be $n$ ($n>3$) positive real numbers satisfying $(a_1^2+a_2^2+\dots +a_n^2)^2>(n-1)(a_1^4+ a_2^4+\dots +a_n^4)$. Prove that any three of $a_1,a_2,\dots ,a_n$ can be the lengths of three sides of a triangle respectively.
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
$\text{(A)}\ 1.6 \qquad \text{(B)}\ 1.9 \qquad \text{(C)}\ 2.1 \qquad \text{(D)}\ 2.7 \qquad \text{(E)}\ 2.9$
The sides of $\triangle ABC$ measure 11,20, and 21 units. We fold it along $PQ,QR,RP$ where $P,Q,R$ are the midpoints of its sides until $A,B,C$ coincide. What is the volume of the resulting tetrahedron?
Let $T$ be a triangle with side lengths $26$, $51$, and $73$. Let $S$ be the set of points inside $T$ which do not lie within a distance of $5$ of any side of $T$. Find the area of $S$.
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.
Denote by $S(a,b,c)$ the area of a triangle whose lengthes of three sides are $a,b,c$
Prove that for any positive real numbers $a_{1},b_{1},c_{1}$ and $a_{2},b_{2},c_{2}$ which can serve as the lengthes of three sides of two triangles respectively ,we have
$ \sqrt{S(a_{1},b_{1},c_{1})}+\sqrt{S(a_{2},b_{2},c_{2})}\le\sqrt{S(a_{1}+a_{2},b_{1}+b_{2},c_{1}+c_{2})}$
In a triangle $ABC$, the incircle touches the sides $BC, CA, AB$ at $D, E, F$ respectively. If the radius if the incircle is $4$ units and if $BD, CE , AF$ are consecutive integers, find the sides of the triangle $ABC$.
Prove that for every triangle the following inequality holds:
\[\frac{ab+bc+ca}{4S} \geq \cot \frac{\pi}{6}.\]
where $a, b, c$ are lengths of the sides and $S$ is the area of the triangle.
Find the radius of a circle inscribed in a triangle with side lengths $4$, $5$, and $6$
A trapezoid has side lengths $3, 5, 7,$ and $11$. The sum of all the possible areas of the trapezoid can be written in the form of $r_1 \sqrt{n_1} + r_2 \sqrt{n_2} + r_3$, where $r_1, r_2,$ and $r_3$ are rational numbers and $n_1$ and $n_2$ are positive integers not divisible by the square of a prime. What is the greatest integer less than or equal to
\[r_1 + r_2 + r_3 + n_1 + n_2?\]
$ \textbf{(A)}\ 57\qquad\textbf{(B)}\ 59\qquad\textbf{(C)}\ 61\qquad\textbf{(D)}\ 63\qquad\textbf{(E)}\ 65 $
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 $b$?
$\textbf{(A) }3\qquad\textbf{(B) }4\qquad\textbf{(C) }5\qquad\textbf{(D) }6\qquad\textbf{(E) }8$
Let $a,b,c$ be positive real numbers such that $abc(a+b+c)=3.$ Prove that we have
\[(a+b)(b+c)(c+a)\geq 8.\]
Also determine the case of equality.
In $\triangle ABC$, $AC = BC$, and point $D$ is on $\overline{BC}$ so that $CD = 3 \cdot BD$. Let $E$ be the midpoint of $\overline{AD}$. Given that $CE = \sqrt{7}$ and $BE = 3$, the area of $\triangle ABC$ can be expressed in the form $m\sqrt{n}$, where $m$ and $n$ are positive integers and $n$ is not divisible by the square of any prime. Find $m+n$.
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} $
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 $\triangle$.
Let $V_n=\sqrt{F_n^2+F_{n+2}^2}$, where $F_n$ is the Fibonacci sequence
($F_1=F_2=1,F_{n+2}=F_{n+1}+F_{n}$)
Show that $V_n,V_{n+1},V_{n+2}$ are the sides of a triangle with area $1/2$
A triangle has sides of length $\sqrt{13}$, $\sqrt{17}$, and $2 \sqrt{5}$. Compute the area of the triangle.
Prove that a triangle with angles $\alpha, \beta, \gamma$, circumradius $R$, and area $A$ satisfies
\[\tan \frac{ \alpha}{2}+\tan \frac{ \beta}{2}+\tan \frac{ \gamma}{2} \leq \frac{9R^2}{4A}.\]
[hide="Remark."]Remark. Can we determine [i]all[/i] of equality cases ?[/hide]
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 $a,b,c$).
Let $P$ be an arbitrary point inside $\triangle ABC$ with sides $3,7,8$. What is the probability that the distance of $P$ to at least one vertices of the triangle is less than $1$?
$ \textbf{(A)}\ \frac{\pi}{36}\sqrt 2
\qquad\textbf{(B)}\ \frac{\pi}{36}\sqrt 3
\qquad\textbf{(C)}\ \frac{\pi}{36}
\qquad\textbf{(D)}\ \frac12
\qquad\textbf{(E)}\ \frac 34
$
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.