QUESTION 144
The sum of the interior angles of a hexagon is
(A) $540^\circ$
(B) $600^\circ$
(C) $660^\circ$
(D) $720^\circ$
(E) $780^\circ$

In an isoceles $\triangle \mathrm { ABC }$ with A at the apex the height and the base are both equal to 1 cm. Points $\mathrm { D } , \mathrm { E }$ and F are chosen one from each side such that BDEF is a rhombus. Find the length of the side of this rhombus.

You are given a regular hexagon. We say that a square is inscribed in the hexagon if it can be drawn in the interior such that all the four vertices lie on the perimeter of the hexagon.
(a) A line segment has its endpoints on opposite edges of the hexagon. Show that it passes through the center of the hexagon if and only if it divides the two edges in the same ratio.
(b) Suppose a square $ABCD$ is inscribed in the hexagon such that $A$ and $C$ are on the opposite sides of the hexagon. Prove that center of the square is same as that of the hexagon.
(c) Suppose the side of the hexagon is of length 1. Then find the length of the side of the inscribed square whose one pair of opposite sides is parallel to a pair of opposite sides of the hexagon.
(d) Show that, up to rotation, there is a unique way of inscribing a square in a regular hexagon.

Suppose a rectangle $EBFD$ is given and a rhombus $ABCD$ is inscribed in it so that the point $A$ is on side $ED$ of the rectangle. The diagonals of $ABCD$ intersect at point $G$.
Statements
(5) Triangles $CGD$ and $DFB$ must be similar. (6) It must be true that $\frac { AC } { BD } = \frac { EB } { ED }$. (7) Triangle $CGD$ cannot be similar to triangle $AEB$. (8) For any given rectangle $EBFD$, a rhombus $ABCD$ as described above can be constructed.

We take for $\Omega$ (only in this question) the interior of the equilateral triangle with vertices $(1,0), (-1/2, \sqrt{3}/2)$ and $(-1/2, -\sqrt{3}/2)$. We define, for all $\lambda \in \mathbb{R}^*$ and all pairs $(x_0, y_0) \in \mathbb{R}^2$: $$\Omega_{x_0, y_0, \lambda} = \left\{ \lambda(x,y) + (x_0, y_0) \mid (x,y) \in \Omega \right\}$$ Draw a figure on which both $\Omega$ and $\Omega_{2,1,1/2}$ appear.

Let $A B C D$ be a unit square. Four points $E , F , G$ and $H$ are chosen on the sides $A B , B C , C D$ and $D A$ respectively. The lengths of the sides of the quadrilateral $E F G H$ are $\alpha , \beta , \gamma$ and $\delta$. Which of the following is always true?
(A) $1 \leq \alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } + \delta ^ { 2 } \leq 2 \sqrt { 2 }$
(B) $2 \sqrt { 2 } \leq \alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } + \delta ^ { 2 } \leq 4 \sqrt { 2 }$
(C) $2 \leq \alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } + \delta ^ { 2 } \leq 4$
(D) $\sqrt { 2 } \leq \alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } + \delta ^ { 2 } \leq 2 + \sqrt { 2 }$

Let $A B C D$ be a unit square. Four points $E , F , G$ and $H$ are chosen on the sides $A B , B C , C D$ and $D A$ respectively. The lengths of the sides of the quadrilateral $E F G H$ are $\alpha , \beta , \gamma$ and $\delta$. Which of the following is always true?
(A) $1 \leq \alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } + \delta ^ { 2 } \leq 2 \sqrt { 2 }$
(B) $2 \sqrt { 2 } \leq \alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } + \delta ^ { 2 } \leq 4 \sqrt { 2 }$
(C) $2 \leq \alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } + \delta ^ { 2 } \leq 4$
(D) $\sqrt { 2 } \leq \alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } + \delta ^ { 2 } \leq 2 + \sqrt { 2 }$

Let $A B C D$ be a unit square. Four points $E , F , G$ and $H$ are chosen on the sides $A B , B C , C D$ and $D A$ respectively. The lengths of the sides of the quadrilateral $E F G H$ are $\alpha , \beta , \gamma$ and $\delta$. Which of the following is always true?
(A) $1 \leq \alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } + \delta ^ { 2 } \leq 2 \sqrt { 2 }$
(B) $2 \sqrt { 2 } \leq \alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } + \delta ^ { 2 } \leq 4 \sqrt { 2 }$
(C) $2 \leq \alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } + \delta ^ { 2 } \leq 4$
(D) $\sqrt { 2 } \leq \alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } + \delta ^ { 2 } \leq 2 + \sqrt { 2 }$

In the figure below, $A B C D$ is a square and $\triangle C E F$ is a triangle with given sides inscribed as in the figure. Find the length $B E$.
(A) $\frac { 13 } { \sqrt { 17 } }$
(B) $\frac { 14 } { \sqrt { 17 } }$
(C) $\frac { 15 } { \sqrt { 17 } }$
(D) $\frac { 16 } { \sqrt { 17 } }$

Two sides of a rhombus are along the lines, $x - y + 1 = 0$ and $7x - y - 5 = 0$. If its diagonals intersect at $(-1, -2)$, then which one of the following is a vertex of this rhombus?
(1) $(-3, -9)$
(2) $(-3, -8)$
(3) $\left(\frac{1}{3}, -\frac{8}{3}\right)$
(4) $\left(-\frac{1}{3}, -\frac{8}{3}\right)$

Let $m_1, m_2$ be the slopes of two adjacent sides of a square of side $a$ such that $a^2 + 11a + 3(m_1^2 + m_2^2) = 220$. If one vertex of the square is $(10\cos\alpha - \sin\alpha, 10\sin\alpha + \cos\alpha)$, where $\alpha \in \left(0, \frac{\pi}{2}\right)$ and the equation of one diagonal is $(\cos\alpha - \sin\alpha)x + (\sin\alpha + \cos\alpha)y = 10$, then $72(\sin^4\alpha + \cos^4\alpha) + a^2 - 3a + 13$ is equal to
(1) 119
(2) 128
(3) 145
(4) 155

Let $\alpha , \quad \beta , \quad \gamma , \quad \delta \in Z$ and let $A(\alpha , \beta)$, $B(1, 0)$, $C(\gamma , \delta)$ and $D(1, 2)$ be the vertices of a parallelogram $ABCD$. If $AB = \sqrt { 10 }$ and the points $A$ and $C$ lie on the line $3 y = 2 x + 1$, then $2 \alpha + \beta + \gamma + \delta$ is equal to
(1) 10
(2) 5
(3) 12
(4) 8

Consider a triangle ABC having the vertices $\mathrm { A } ( 1,2 ) , \mathrm { B } ( \alpha , \beta )$ and $\mathrm { C } ( \gamma , \delta )$ and angles $\angle A B C = \frac { \pi } { 6 }$ and $\angle B A C = \frac { 2 \pi } { 3 }$. If the points B and C lie on the line $y = x + 4$, then $\alpha ^ { 2 } + \gamma ^ { 2 }$ is equal to $\_\_\_\_$

Let the distance between two parallel lines be 5 units and a point $P$ lie between the lines at a unit distance from one of them. An equilateral triangle $PQR$ is formed such that $Q$ lies on one of the parallel lines, while $R$ lies on the other. Then $( QR ) ^ { 2 }$ is equal to $\_\_\_\_$

A resident of a building displays a Christmas tree-shaped light decoration on the building's exterior wall, as shown in the figure. From a certain point $P$ on the fifth floor exterior wall, small light bulbs are pulled to the two ends $A , B$ of the fourth floor to form an isosceles triangle $P A B$, where $\overline { P A } = \overline { P B }$; light bulbs are pulled to the two ends $C , D$ of the third floor to form an isosceles triangle $P C D$; light bulbs are pulled to the two ends $E , F$ of the second floor to form an isosceles triangle $PEF$. Assume each floor has equal height and each floor has equal length. If the length of the line segment cut out by the fifth floor inside triangle $P A B$ is $\frac { 1 } { 3 }$ of the floor length, what fraction of the floor length is the length of the line segment cut out by the fifth floor inside triangle $PEF$? (Light decoration thickness is negligible)
(1) $\frac { 1 } { 7 }$
(2) $\frac { 1 } { 6 }$
(3) $\frac { 1 } { 5 }$
(4) $\frac { 2 } { 9 }$
(5) $\frac { 1 } { 4 }$

ABCD is a rectangle $| \mathrm { AD } | = 1 \mathrm {~cm}$ $| \mathrm { AE } | = | \mathrm { EB } | = 2 \mathrm {~cm}$ $|FE| = \mathrm { x }$
According to the given information, what is x in cm?
A) $\frac { \sqrt { 3 } } { 2 }$
B) $\frac { \sqrt { 5 } } { 2 }$
C) $\frac { \sqrt { 3 } } { 3 }$
D) $\frac { \sqrt { 5 } } { 3 }$
E) $\frac { \sqrt { 7 } } { 3 }$

The vertices of a parallelogram with diagonals $[ AC ]$ and $[ BD ]$ are $\mathrm { A } ( 3,1 ) , \mathrm { B } ( 5,3 ) , \mathrm { C } ( 2,5 )$ and $\mathrm { D } ( \mathrm { a } , \mathrm { b } )$. What is the length of diagonal $[ BD ]$ in units?
A) 1
B) 2
C) 3
D) 4
E) 5

OABC is a parallelogram $\mathrm { A } = ( 5,0 )$ $\mathrm { C } = ( 3,4 )$
According to the given information above, what is the sum of the diagonal lengths of parallelogram OABC in units?
A) $5 \sqrt { 5 }$
B) $6 \sqrt { 5 }$
C) $7 \sqrt { 5 }$
D) $7 \sqrt { 3 }$
E) $8 \sqrt { 3 }$

What is the perimeter of rectangle ABCD given in the coordinate plane in the figure?
A) 18
B) 21
C) 24
D) 27
E) 30

In the rectangular coordinate plane, the sides of rectangle $A B C D$ are parallel to the axes.
If the coordinates of vertices A and $C$ are $(1, -1)$ and $(3, 5)$ respectively, what is the area of rectangle ABCD in square units?
A) 8 B) 10 C) 12 D) 15 E) 16

In the rectangular coordinate plane, a parallelogram whose vertices are the intersection points of the lines $y = 2$ and $y = 6$ with the line $y = 2x$ has diagonals intersecting at the point $(0,4)$.
What is the area of this parallelogram in square units?
A) 16 B) 18 C) 20 D) 22 E) 24

In the rectangular coordinate plane, a square $ABCD$ with two vertices at $A(0, a)$ and $B(0, b)$ is given. The vertex $C$ of square $ABCD$ lies on the line $y = \frac{x}{3}$. If $a + b = 15$, what is the sum of the coordinates of point $D$?
A) 14
B) 18
C) 21
D) 24
E) 27

5 identical isosceles right triangles with right-angled side lengths of 1 unit are arranged as shown in Figure 1 such that their hypotenuses are on the same line and the vertices of adjacent triangles coincide.
Then triangle $ABC$ is rotated around point $A$ by some amount, and as shown in Figure 2, points B, C, and D become collinear.
Accordingly, what is the distance between points C and D in the final position in units?
A) 4
B) 5
C) 6
D) $3\sqrt{2}$
E) $4\sqrt{2}$

In the rectangular coordinate plane below, a red square with one side on the $y$-axis and a blue square with one side on the $x$-axis share a common vertex.
One vertex of each of the red and blue squares lies on the line $\dfrac{x}{2} + \dfrac{y}{3} = 1$.
According to this, what is the side length of the red square in units?
A) $\dfrac{14}{15}$ B) $\dfrac{15}{16}$ C) $\dfrac{16}{17}$ D) $\dfrac{17}{18}$ E) $\dfrac{18}{19}$

Let $a$ and $b$ be positive real numbers. In the rectangular coordinate plane, the region between the lines $y = -\sqrt{3}x$ and $y = ax + b$ and the $x$-axis forms an equilateral triangle with area $9\sqrt{3}$ square units.
Accordingly, what is the product $a \cdot b$?
A) 18 B) 24 C) 27 D) 30 E) 36