Found problems: 663
In $\triangle{ABC}$, $\angle{C} = 90^{\circ}$ and $AB = 12$. Squares $ABXY$ and $ACWZ$ are constructed outside of the triangle. The points $X, Y, Z$, and $W$ lie on a circle. What is the perimeter of the triangle?
$ \textbf{(A)}\ 12+9\sqrt{3}\qquad\textbf{(B)}\ 18+6\sqrt{3}\qquad\textbf{(C)}\ 12+12\sqrt{2}\qquad\textbf{(D)}\ 30\qquad\textbf{(E)}\ 32 $
Let $ h$ be a triangle of perimeter $ 1$, and let $ H$ be a triangle of perimeter $ \lambda$ homothetic to $ h$. Let $ h_1,h_2,...$ be translates of $ h$ such that , for all $ i$, $ h_i$ is different from $ h_{i\plus{}2}$ and touches $ H$ and $ h_{i\plus{}1}$ (that is, intersects without overlapping). For which values of $ \lambda$ can these triangles be chosen so that the sequence $ h_1,h_2,...$ is periodic? If $ \lambda \geq 1$ is such a value, then determine the number of different triangles in a periodic
chain $ h_1,h_2,...$ and also the number of times such a chain goes around the triangle $ H$.
[i]L. Fejes-Toth[/i]
The intersection of a plane with a regular tetrahedron with edge $a$ is a quadrilateral with perimeter $P.$ Prove that $2a \leq P \leq 3a.$
Consider the set of all rectangles with a given area $S$.
Find the largest value o $ M = \frac{16-p}{p^2+2p}$ where $p$ is the perimeter of the rectangle.
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]
Draw a $2004 \times 2004$ array of points. What is the largest integer $n$ for which it is possible to draw a convex $n$-gon whose vertices are chosen from the points in the array?
The perimeter of one square is $3$ times the perimeter of another square. The area of the larger square is how many times the area of the smaller square?
$\text{(A)}\ 2 \qquad \text{(B)}\ 3 \qquad \text{(C)}\ 4 \qquad \text{(D)}\ 6 \qquad \text{(E)}\ 9$
In $\triangle ABC$ with integer side lengths,
\[
\cos A=\frac{11}{16}, \qquad \cos B= \frac{7}{8}, \qquad \text{and} \qquad\cos C=-\frac{1}{4}.
\] What is the least possible perimeter for $\triangle ABC$?
$\textbf{(A) } 9 \qquad \textbf{(B) } 12 \qquad \textbf{(C) } 23 \qquad \textbf{(D) } 27 \qquad \textbf{(E) } 44$
Jacob's analog clock has 12 equally spaced tick marks on the perimeter, but all the digits have been erased, so he doesn't know which tick mark corresponds to which hour. Jacob takes an arbitrary tick mark and measures clockwise to the hour hand and minute hand. He measures that the minute hand is 300 degrees clockwise of the tick mark, and that the hour hand is 70 degrees clockwise of the same tick mark. If it is currently morning, how many minutes past midnight is it?
[i]Ray Li[/i]
In triangle $ABC$ the angle $C$ is obtuse, $m(\angle A)=2m(\angle B)$ and the sidelengths are integers. Find the smallest possible perimeter of this triangle.
Prove that there exist infinitely many positive integers $n$ such that $p=nr$, where $p$ and $r$ are respectively the semi-perimeter and the inradius of a triangle with integer side lengths.
The sides of a triangle are three consecutive integers and its inradius is $4$. Find the circumradius.
Let $ABCD$ be a quadrilateral $AB = BC = CD = DA$. Let $MN$ and $PQ$ be two segments perpendicular to the diagonal $BD$ and such that the distance between them is $d > \frac{BD}{2}$, with $M \in AD$, $N \in DC$, $P \in AB$, and $Q \in BC$. Show that the perimeter of hexagon $AMNCQP$ does not depend on the position of $MN$ and $PQ$ so long as the distance between them remains constant.
Let $n$ be a positive integer. A regular hexagon with side length $n$ is divided into equilateral triangles with side length $1$ by lines parallel to its sides.
Find the number of regular hexagons all of whose vertices are among the vertices of those equilateral triangles.
[i]UK - Sahl Khan[/i]
In a triangle $ABC$, the excircle $\omega_a$ opposite $A$ touches $AB$ at $P$ and $AC$ at $Q$, while the excircle $\omega_b$ opposite $B$ touches $BA$ at $M$ and $BC$ at $N$. Let $K$ be the projection of $C$ onto $MN$ and let $L$ be the projection of $C$ onto $PQ$. Show that the quadrilateral $MKLP$ is cyclic.
([i]Bulgaria[/i])
Consider the set of all rectangles with a given area $S$.
Find the largest value o $ M = \frac{S}{2S+p + 2}$ where $p$ is the perimeter of the rectangle.
Pairwise distinct points $P_1,P_2,\ldots, P_{16}$ lie on the perimeter of a square with side length $4$ centered at $O$ such that $\lvert P_iP_{i+1} \rvert = 1$ for $i=1,2,\ldots, 16$. (We take $P_{17}$ to be the point $P_1$.) We construct points $Q_1,Q_2,\ldots,Q_{16}$ as follows: for each $i$, a fair coin is flipped. If it lands heads, we define $Q_i$ to be $P_i$; otherwise, we define $Q_i$ to be the reflection of $P_i$ over $O$. (So, it is possible for some of the $Q_i$ to coincide.) Let $D$ be the length of the vector $\overrightarrow{OQ_1} + \overrightarrow{OQ_2} + \cdots + \overrightarrow{OQ_{16}}$. Compute the expected value of $D^2$.
[i]Ray Li[/i]
A rectangular parking lot has a diagonal of $25$ meters and an area of $168$ square meters. In meters, what is the perimeter of the parking lot?
$ \textbf{(A)}\ 52 \qquad
\textbf{(B)}\ 58 \qquad
\textbf{(C)}\ 62 \qquad
\textbf{(D)}\ 68 \qquad
\textbf{(E)}\ 70 $
In cyclic quadrilateral $ABXC$, $\measuredangle XAB = \measuredangle XAC$. Denote by $I$ the incenter of $\triangle ABC$ and by $D$ the projection of $I$ on $\overline{BC}$. If $AI = 25$, $ID = 7$, and $BC = 14$, then $XI$ can be expressed as $\frac{a}{b}$ for relatively prime positive integers $a, b$. Compute $100a + b$.
[i]Proposed by Aaron Lin[/i]
Prove that among all possible triangles whose vertices are $3,5$ and $7$ apart from a given point $P$, the ones with the largest perimeter have $P$ as incentre.
Given an angle less than $180^o$, and a point $M$ outside the angle. Draw a line through $M$ so that the triangle, whose vertices are the vertex of the angle and the intersection points of its legs with the line drawn, has a given perimeter.
A rectangular garden 50 feet long and 10 feet wide is enclosed by a fence. To make the garden larger, while using the same fence, its shape is changed to a square. By how many square feet does this enlarge the garden?
$ \text{(A)}\ 100\qquad\text{(B)}\ 200\qquad\text{(C)}\ 300\qquad\text{(D)}\ 400\qquad\text{(E)}\ 500 $
On the extension of the angle bisector $AL$ of the triangle $ABC$, a point $P$ is placed such that $P L = AL$. Prove that the perimeter of triangle $PBC$ does not exceed the perimeter of triangle $ABC$.
Let $ABC$ be a triangle of perimeter $100$ and $I$ be the point of intersection of its bisectors. Let $M$ be the midpoint of side $BC$. The line parallel to $AB$ drawn by$ I$ cuts the median $AM$ at point $P$ so that $\frac{AP}{PM} =\frac73$. Find the length of side $AB$.
A rectangle is inscribed in a circle of area $32\pi$ and the area of the rectangle is $34$. Find its perimeter.
[i]2017 CCA Math Bonanza Individual Round #2[/i]