Found problems: 663
Suppose a square piece of paper is folded in half vertically. The folded paper is then cut in half along the dashed line. Three rectangles are formed-a large one and two small ones. What is the ratio of the perimeter of one of the small rectangles to the perimeter of the large rectangle?
$\text{(A)}\ \frac{1}{2} \qquad \text{(B)}\ \frac{2}{3} \qquad \text{(C)}\ \frac{3}{4} \qquad \text{(D)}\ \frac{4}{5} \qquad \text{(E)}\ \frac{5}{6}$
[asy]
draw((0,0)--(0,8)--(6,8)--(6,0)--cycle);
draw((0,8)--(5,9)--(5,8));
draw((3,-1.5)--(3,10.3),dashed);
draw((0,5.5)..(-.75,4.75)..(0,4));
draw((0,4)--(1.5,4),EndArrow);
[/asy]
How many non-congruent triangles with integer sides and perimeter 1999 can be constructed?
The centre of the coin with radius $r$ is moved along some polygon with the perimeter $P$, that is circumscribed around the circle with radius $R$ ($R>r$). Find the coin trace area (a sort of polygon ring).
The side of the square $ABCD$ has a length equal to $1$. On the sides $(BC)$ ¸and $(CD)$ take respectively the arbitrary points $M$ and $N$ so that the perimeter of the triangle $MCN$ is equal to $2$.
a) Determine the measure of the angle $\angle MAN$.
b) If the point $P$ is the foot of the perpendicular taken from point $A$ to the line $MN$, determine the locus of the points $P$.
For an integer $m\geq 1$, we consider partitions of a $2^m\times 2^m$ chessboard into rectangles consisting of cells of chessboard, in which each of the $2^m$ cells along one diagonal forms a separate rectangle of side length $1$. Determine the smallest possible sum of rectangle perimeters in such a partition.
[i]Proposed by Gerhard Woeginger, Netherlands[/i]
The the parallel lines through an inner point $P$ of triangle $\triangle ABC$ split the triangle into three parallelograms and three triangles adjacent to the sides of $\triangle ABC$.
(a) Show that if $P$ is the incenter, the perimeter of each of the three small triangles equals the length of the adjacent side.
(b) For a given triangle $\triangle ABC$, determine all inner points $P$ such that the perimeter of each of the three small triangles equals the length of the adjacent side.
(c) For which inner point does the sum of the areas of the three small triangles attain a minimum?
[i](41st Austrian Mathematical Olympiad, National Competition, part 1, Problem 4)[/i]
Given an arbitrary triangle $ ABC$, denote by $ P,Q,R$ the intersections of the incircle with sides $ BC, CA, AB$ respectively. Let the area of triangle $ ABC$ be $ T$, and its perimeter $ L$. Prove that the inequality
\[\left(\frac {AB}{PQ}\right)^3 \plus{}\left(\frac {BC}{QR}\right)^3 \plus{}\left(\frac {CA}{RP}\right)^3 \geq \frac {2}{\sqrt {3}} \cdot \frac {L^2}{T}\]
holds.
Suppose we have a simple polygon (that is it does not intersect itself, but not necessarily convex).
Show that this polygon has a diameter which is completely inside the polygon and the two arcs it creates on the polygon perimeter (the two arcs have 2 vertices in common) both have at least one third of the vertices of the polygon.
Find the number of ordered pairs $(x,y)$ of positive integers that satisfy $x\le 2y\le 60$ and $y\le 2x\le 60.$
Let $L$ be a positive constant. For a point $P(t,\ 0)$ on the positive part of the $x$ axis on the coordinate plane, denote $Q(u(t),\ v(t))$ the point at which the point reach starting from $P$ proceeds by distance $L$ in counter-clockwise on the perimeter of a circle passing the point $P$ with center $O$.
(1) Find $u(t),\ v(t)$.
(2) For real number $a$ with $0<a<1$, find $f(a)=\int_a^1 \sqrt{\{u'(t)\}^2+\{v'(t)\}^2}\ dt$.
(3) Find $\lim_{a\rightarrow +0} \frac{f(a)}{\ln a}$.
[i]2011 Tokyo University entrance exam/Science, Problem 3[/i]
Triangular grid divides an equilateral triangle with sides of length $n$ into $n^2$ triangular cells as shown in figure for $n=12$. Some cells are infected. A cell that is not yet infected, ia infected when it shares adjacent sides with at least two already infected cells. Specify for $n=12$, the least number of infected cells at the start in which it is possible that over time they will infected all the cells of the original triangle.
[asy]
unitsize(0.25cm);
path p=polygon(3);
for(int m=0; m<=11;++m){
for(int n=0 ; n<= 11-m; ++n){
draw(shift((n+0.5*m)*sqrt(3),1.5*m)*p);
}
}
[/asy]
Two of the three sides of a triangle are $20$ and $15$. Which of the following numbers is not a possible perimeter of the triangle?
$\textbf{(A) }52\qquad\textbf{(B) }57\qquad\textbf{(C) }62\qquad\textbf{(D) }67\qquad\textbf{(E) }72$
The diagram below shows a $12$-sided figure made up of three congruent squares. The figure has total perimeter $60$. Find its area.
[asy]
size(150);
defaultpen(linewidth(0.8));
path square=unitsquare;
draw(rotate(360-135)*square^^rotate(345)*square^^rotate(105)*square);
[/asy]