Found problems: 663
Let $n$ be a natural number. Given a chessboard sized $m \times n$. The sides of the small squares of chessboard are not on the perimeter of the chessboard will be colored so that each small square has exactly two sides colored. Prove that a coloring like that is possible if and only if $m \cdot n$ is even.
$P$ is an arbitrary point inside acute triangle $ABC$. Let $A_1,B_1,C_1$ be the reflections of point $P$ with respect to sides $BC,CA,AB$. Prove that the centroid of triangle $A_1B_1C_1$ lies inside triangle $ABC$.
Prove: If $x, y, z$ are the lengths of the angle bisectors of a triangle with perimeter 6, than we have:
\[\frac{1}{x^2} + \frac{1}{y^2} + \frac{1}{z^2} \geq 1.\]
Forty-two cubes with 1 cm edges are glued together to form a solid rectangular block. If the perimeter of the base of the block is 18 cm, then the height, in cm, is
$\textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ \dfrac{7}{3} \qquad \textbf{(D)}\ 3 \qquad \textbf{(E)}\ 4$
Triangles $ABC$, $ADE$, and $EFG$ are all equilateral. Points $D$ and $G$ are midpoints of $\overline{AC}$ and $\overline{AE}$, respectively. If $AB = 4$, what is the perimeter of figure $ABCDEFG$?
[asy]
pair A,B,C,D,EE,F,G;
A = (4,0); B = (0,0); C = (2,2*sqrt(3)); D = (3,sqrt(3));
EE = (5,sqrt(3)); F = (5.5,sqrt(3)/2); G = (4.5,sqrt(3)/2);
draw(A--B--C--cycle);
draw(D--EE--A);
draw(EE--F--G);
label("$A$",A,S);
label("$B$",B,SW);
label("$C$",C,N);
label("$D$",D,NE);
label("$E$",EE,NE);
label("$F$",F,SE);
label("$G$",G,SE);
[/asy]
$\text{(A)}\ 12 \qquad \text{(B)}\ 13 \qquad \text{(C)}\ 15 \qquad \text{(D)}\ 18 \qquad \text{(E)}\ 21$
Let U be the union of a finite number (not necessarily connected and not necessarily disjoint) of closed unit squares lying in the plane. Can the quotient of the perimeter and area of U be arbitrarily large?
@below: i think "single" means "connected".
In the triangle $ABC$, the side $BC$ is equal to $a$. Point $F$ is midpoint of $AB$, $I$ is the point of intersection of the bisectors of triangle $ABC$. It turned out that $\angle AIF = \angle ACB$. Find the perimeter of the triangle $ABC$.
(Grigory Filippovsky)
Point $B$ is on $\overline{AC}$ with $AB = 9$ and $BC = 21$. Point $D$ is not on $\overline{AC}$ so that $AD = CD$, and $AD$ and $BD$ are integers. Let $s$ be the sum of all possible perimeters of $\triangle ACD$. Find $s$.
If $a$, $b$ and $c$ are sides of triangle which perimeter equals $1$, prove that:
$a^2+b^2+c^2+4abc<\frac{1}{2}$
A straight line passing through the center of the circumscribed circle and the intersection point of the heights of the non-equilateral triangle $ABC$ divides its perimeter and area in the same ratio.Find this ratio.
From each of the vertices of a regular $n$-gon a car starts to move with constant speed along the perimeter of the $n$-gon in the same direction. Prove that if all the cars end up at a vertex $A$ at the same time, then they never again meet at any other vertex of the $n$-gon. Can they meet again at $A \ ?$
Points $N$ and $M$ are on the sides $CD$ and $BC$ of square $ABCD$, respectively. The perimeter of triangle $MCN$ is equal to the double of the length of the square's side. Find $\angle MAN$.
A triangle has semi-perimeter $s$, circumradius $R$ and inradius $r$. Show that it is right-angled iff $2R = s - r$.
Prove that all acute-angled triangles with the equal altitudes $h_c$ and the equal angles $\gamma$ have orthic triangles with same perimeters.
Let $ABCD$ be a square of side paper $10$ and $P$ a point on side $BC$. By folding the paper along the $AP$ line, point $B$ determines the point $Q$, as seen in the figure. The line $PQ$ cuts the side $CD$ at $R$. Calculate the perimeter of the triangle $ PCR$
[img]https://3.bp.blogspot.com/-ZSyCUznwutE/XNY7cz7reQI/AAAAAAAAKLc/XqgQnjm8DQYq6Q7fmCAKJwKt3ihoL8AuQCK4BGAYYCw/s400/may%2B2013%2Bl1.png[/img]
Excircle of triangle $ABC$ corresponding vertex $A$, is tangent to $BC$ at $P$. $AP$ intersects circumcircle of $ABC$ at $D$. Prove \[r(PCD)=r(PBD)\] whcih $r(PCD)$ and $r(PBD)$ are inradii of triangles $PCD$ and $PBD$.
Let $ABC$ be a triangle with integral side lengths such that $\angle A=3\angle B$. Find the minimum value of its perimeter.
$ABC$ is a triangle. The point A' lies on the side opposite to $A$ and $BA'/BC = k$, where $1/2 < k < 1$. Similarly, $B'$ lies on the side opposite to $B$ with $CB'/CA = k$, and $C'$ lies on the side opposite to $C$ with $AC'/AB = k$. Show that the perimeter of $A'B'C'$ is less than $k$ times the perimeter of $ABC$.
Suppose we climb a mountain that is a cone with radius $100$ and height $4$. We start at the bottom of the mountain (on the perimeter of the base of the cone), and our destination is the opposite side of the mountain, halfway up (height $z = 2$). Our climbing speed starts at $v_0=2$ but gets slower at a rate inversely proportional to the distance to the mountain top (so at height $z$ the speed $v$ is $(h-z)v_0/h$). Find the minimum time needed to get to the destination.
The quadrilateral $ABCD$ has the following equality $\angle ABC=\angle BCD=150^{\circ}$. Moreover, $AB=18$ and $BC=24$, the equilateral triangles $\triangle APB,\triangle BQC,\triangle CRD$ are drawn outside the quadrilateral. If $P(X)$ is the perimeter of the polygon $X$, then the following equality is true $P(APQRD)=P(ABCD)+32$. Determine the length of the side $CD$.
A right triangle has perimeter $2008$, and the area of a circle inscribed in the triangle is $100\pi^3$. Let $A$ be the area of the triangle. Compute $\lfloor A\rfloor$.
Consider the rectangle $ ABCD $ and the points $ M,N,P,Q $ on the segments $ AB,BC,CD, $ respectively, $ DA, $ excluding its extremities. Denote with $ p_{\square} , A_{\square} $ the perimeter, respectively, the area of $ \square. $ Prove that:
[b]a)[/b] $ p_{MNPQ}\ge AC+BD. $
[b]b)[/b] $ p_{MNPQ} =AC+BD\implies A_{MNPQ}\le \frac{A_{ABCD}}{2} . $
[b]c)[/b] $ p_{MNPQ} =AC+BD\implies MP^2 +NQ^2\ge AC^2. $
[i]Dan Brânzei[/i] and [i]Gheorghe Iurea[/i]
Quadrilateral $ABCD$ is a rhombus with perimeter $52$ meters. The length of diagonal $\overline{AC}$ is $24$ meters. What is the area in square meters of rhombus $ABCD$?
[asy]
unitsize(1cm);
draw((0,1)--(2,2)--(4,1)--(2,0)--cycle);
dot("$A$",(0,1),W);
dot("$D$",(2,2),N);
dot("$C$",(4,1),E);
dot("$B$",(2,0),S);
[/asy]
$\textbf{(A) } 60
\qquad\textbf{(B) } 90
\qquad\textbf{(C) } 105
\qquad\textbf{(D) } 120
\qquad\textbf{(E) } 144$
a) $n$ $2\times2$ squares are drawn on the Cartesian plane. The sides of these squares are parallel to the coordinate axes. It is known that the center of any square is not an inner point of any other square. Let $\Pi$ be a rectangle such that it contains all these $n$ squares and its sides are parallel to the coordinate axes.
Prove that the perimeter of $\Pi$ is greater than or equal to $4(\sqrt{n}+1)$.
b) Prove the sharp estimate: the perimeter of $\Pi$ is greater than or equal to $2\lceil \sqrt{n}+1) \rceil$
(here $\lceil a\rceil$ stands for the smallest integer which is greater than or equal to $a$).
If the length and width of a rectangle are each increased by $ 10 \%$, then the perimeter of the rectangle is increased by
\[ \textbf{(A)}\ 1 \% \qquad
\textbf{(B)}\ 10 \% \qquad
\textbf{(C)}\ 20 \% \qquad
\textbf{(D)}\ 21 \% \qquad
\textbf{(E)}\ 40 \%
\]