Found problems: 663
Find the perimeter of the convex polygon whose coordinates of the vertices are the set of pairs of the integer solutions of the equation $x^2+xy = x + 2y + 9$.
Do there exist a convex quadrilateral and a point $P$ inside it such that the sum of distances from $P$ to the vertices of the quadrilateral is greater than its perimeter?
(A.Akopyan)
Inscribe the equilateral triangle of the largest perimeter in a given semicircle.
Let us divide by straight lines a quadrangle of unit area into $ n$ subpolygons and draw a circle into each subpolygon. Show that the sum of the perimeters of the circles is at most $ \pi \sqrt{n}$ (the lines are not allowed to cut the interior of a subpolygon).
[i]G. and L. Fejes-Toth[/i]
We are given a triangle $ABC$ and three rectangles $R_1,R_2,R_3$ with sides parallel to two fixed perpendicular directions and such that their union covers the sides $AB,BC$, and $CA$; i.e., each point on the perimeter of $ABC$ is contained in or on at least one of the rectangles. Prove that all points inside the triangle are also covered by the union of $R_1,R_2,R_3.$
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 $b$?
$\textbf{(A) }3\qquad\textbf{(B) }4\qquad\textbf{(C) }5\qquad\textbf{(D) }6\qquad\textbf{(E) }8$
Consider a closed broken line of perimeter $1$ on a plane. Prove that a disc of radius $\frac14$ can cover this line.
A square of perimeter $20$ is inscribed in a square of perimeter $28$. What is the greatest distance between a vertex of the inner square and a vertex of the outer square?
$ \textbf{(A)}\ \sqrt{58} \qquad\textbf{(B)}\ \frac{7\sqrt{5}}{2} \qquad\textbf{(C)}\ 8 \qquad\textbf{(D)}\ \sqrt{65} \qquad\textbf{(E)}\ 5\sqrt{3} $
Let $ABC$ a triangle and $P$ a point such that the triangles $PAB, PBC, PCA$ have the same area and the same perimeter. Prove that if:
a) $P$ is in the interior of the triangle $ABC$ then $ABC$ is equilateral.
b) $P$ is in the exterior of the triangle $ABC$ then $ABC$ is right angled triangle.
An acute triangle $ ABC$ has three heights $ AD, BE$ and $ CF$ respectively. Prove that the perimeter of triangle $ DEF$ is not over half of the perimeter of triangle $ ABC.$
A square is called [i]proper[/i] if its sides are parallel to the coordinate axes. Point $P$ is randomly selected inside a proper square $S$ with side length 2012. Denote by $T$ the largest proper square that lies within $S$ and has $P$ on its perimeter, and denote by $a$ the expected value of the side length of $T$. Compute $\lfloor a \rfloor$, the greatest integer less than or equal to $a$.
[i]Proposed by Lewis Chen[/i]
A convex quadrilateral is inscribed in a circle of radius $1$. Prove that the difference between its perimeter and the sum of the lengths of its diagonals is greater than zero and less than $2.$
Let $C$ be a closed plane curve that has a continuously turning tangent and bounds a convex region. If $T$ is a triangle inscribed in $C$ with maximum perimeter, show that the normal to $C$ at each vertex of $T$ bisects the angle of $T$ at that vertex. If a triangle $T$ has the property just described, does it necessarily have maximum perimeter? What is the situation if $C$ is a circle?
$O$ is a point in triangle $ABC$. We draw perpendicular from $O$ to $BC,AC,AB$ which intersect $BC,AC,AB$ at $A_{1},B_{1},C_{1}$. Prove that $O$ is circumcenter of triangle $ABC$ iff perimeter of $ABC$ is not less than perimeter of triangles $AB_{1}C_{1},BC_{1}A_{1},CB_{1}A_{1}$.
Prove that if $ M $, $ N $, $ P $ are the feet of the altitudes of acute-angled triangle $ ABC $, then the ratio of the perimeter of triangle $ MNP $ to the perimeter of triangle $ ABC $ is equal to the ratio of the radius of the circle inscribed in triangle $ ABC $ to the radius of the circle circumscribed about triangle $ ABC $.
[asy]
size(120);
real t = 2/sqrt(3);
real x = 1 + sqrt(3);
pair A = t*dir(90), D = x*A;
pair B = t*dir(210), E = x*B;
pair C = t*dir(330), F = x*C;
draw(D--E--F--cycle);
draw(Circle(A, 1));
draw(Circle(B, 1));
draw(Circle(C, 1));
//Credit to MSTang for the diagram[/asy]
Each of the three circles in the adjoining figure is externally tangent to the other two, and each side of the triangle is tangent to two of the circles. If each circle has radius three, then the perimeter of the triangle is
$\textbf{(A) }36+9\sqrt{2}\qquad\textbf{(B) }36+6\sqrt{3}\qquad\textbf{(C) }36+9\sqrt{3}\qquad\textbf{(D) }18+18\sqrt{3}\qquad \textbf{(E) }45$
How many non-congruent triangles with perimeter $ 7$ have integer side lengths?
$ \textbf{(A)}\ 1 \qquad
\textbf{(B)}\ 2 \qquad
\textbf{(C)}\ 3 \qquad
\textbf{(D)}\ 4 \qquad
\textbf{(E)}\ 5$
Triangle $ ABC$ has side lengths $ AB \equal{} 5$, $ BC \equal{} 6$, and $ AC \equal{} 7$. Two bugs start simultaneously from $ A$ and crawl along the sides of the triangle in opposite directions at the same speed. They meet at point $ D$. What is $ BD$?
$ \textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 5$
The altitude drawn to the base of an isosceles triangle is $ 8$, and the perimeter $ 32$. The area of the triangle is:
$ \textbf{(A)}\ 56\qquad
\textbf{(B)}\ 48\qquad
\textbf{(C)}\ 40\qquad
\textbf{(D)}\ 32\qquad
\textbf{(E)}\ 24$
How many noncongruent integer-sided triangles with positive area and perimeter less than $15$ are neither equilateral, isosceles, nor right triangles?
$\textbf{(A) }3\qquad\textbf{(B) }4\qquad\textbf{(C) }5\qquad\textbf{(D) }6\qquad\textbf{(E) }7$
On the plane $S$ in a space, given are unit circle $C$ with radius 1 and the line $L$. Find the volume of the solid bounded by the curved surface formed by the point $P$ satifying the following condition $(a),\ (b)$.
$(a)$ The point of intersection $Q$ of the line passing through $P$ and perpendicular to $S$ are on the perimeter or the inside of $C$.
$(b)$ If $A,\ B$ are the points of intersection of the line passing through $Q$ and pararell to $L$, then $\overline{PQ}=\overline{AQ}\cdot \overline{BQ}$.
In the complex plane, consider squares having the following property: the complex numbers its vertex correspond to are exactly the roots of integer coefficients equation $ x^4 \plus{} px^3 \plus{} qx^2 \plus{} rx \plus{} s \equal{} 0$. Find the minimum of square areas.
On a circumference of a unit radius, take points $A$ and $B$ such that section $AB$ has length one. $C$ can be any point on the longer arc of the circle between $A$ and $B$. How do we take $C$ to make the perimeter of the triangle $ABC$ as large as possible?
The points $C_1, A_1, B_1$ belong to $[AB], [BC], [CA]$ sides, respectively, of the triangle $ABC$ .
$$\frac{|AC_1|}{|C_1B| }=\frac{ |BA_1|}{|A_1C| }= \frac{|CB_1|}{|B_1A| }= \frac{1}{3}$$
Prove that the perimeter $P$ of the triangle $ABC$ and the perimeter $p$ of the triangle $A_1B_1C_1$ , satisfy inequality $$\frac{P}{2} < p < \frac{3P}{4}$$
In the adjoining diagram, $BO$ bisects $\angle CBA$, $CO$ bisects $\angle ACB$, and $MN$ is parallel to $BC$. If $AB=12$, $BC=24$, and $AC=18$, then the perimeter of $\triangle AMN$ is
[asy]
size(200);
defaultpen(linewidth(0.7)+fontsize(10));
pair B=origin, C=(24,0), A=intersectionpoints(Circle(B,12), Circle(C,18))[0], O=incenter(A,B,C), M=intersectionpoint(A--B, O--O+40*dir(180)), N=intersectionpoint(A--C, O--O+40*dir(0));
draw(B--M--O--B--C--O--N--C^^N--A--M);
label("$A$", A, dir(90));
label("$B$", B, dir(O--B));
label("$C$", C, dir(O--C));
label("$M$", M, dir(90)*dir(B--A));
label("$N$", N, dir(90)*dir(A--C));
label("$O$", O, dir(90));[/asy]
$\textbf {(A) } 30 \qquad \textbf {(B) } 33 \qquad \textbf {(C) } 36 \qquad \textbf {(D) } 39 \qquad \textbf {(E) } 42$