AMC 12 Diagnostic

There are 10 people standing equally spaced around a circle. Each person knows exactly 3 of the other 9 people: the 2 people standing next to her or him, as well as the person directly across the circle. How many ways are there for the 10 people to split up into 5 pairs so that the members of each pair know each other?

In rhombus \(A B C D\), point \(P\) lies on segment \(\overline{A D}\) so that \(\overline{B P} \) perp \( \overline{A D}\), \(A P = 3\), and \(P D = 2\). What is the area of \(A B C D\)? (Note: The figure is not drawn to scale.)

Let \(c\) be a real number, and let \(z_1\) and \(z_2\) be the two complex numbers satisfying the equation \(z^2 - c z + 10 = 0\). Points \(z_1\), \(z_2\), \(\dfrac{1}{z_1}\), and \(\dfrac{1}{z_2}\) are the vertices of (convex) quadrilateral \(\mathcal{Q}\) in the complex plane. When the area of \(\mathcal{Q}\) obtains its maximum possible value, \(c\) is closest to which of the following?

Let \(x_n = \sin^2 ( n^\circ )\). What is the mean of \(x_1 , x_2 , x_3 , \cdots.c , x_90\)?

The acronym AMC is shown in the rectangular grid below with grid lines spaced \(1\) unit apart. In units, what is the sum of the lengths of the line segments that form the acronym AMC\(?\)

Suppose that \(a\), \(b\), \(c\) and \(d\) are positive integers satisfying all of the following relations. \( a b c d = 2^6 \cdot 3^9 \cdot 5^7 \) \( \text{\operatorname{lcm}} ( a , b ) = 2^3 \cdot 3^2 \cdot 5^3 \), \( \text{\operatorname{lcm}} ( a , c ) = 2^3 \cdot 3^3 \cdot 5^3 \), \( \text{\operatorname{lcm}} ( a , d ) = 2^3 \cdot 3^3 \cdot 5^3 \), \( \text{\operatorname{lcm}} ( b , c ) = 2^1 \cdot 3^3 \cdot 5^2 \), \( \text{\operatorname{lcm}} ( b , d ) = 2^2 \cdot 3^3 \cdot 5^2 \) \( \text{\operatorname{lcm}} ( c , d ) = 2^2 \cdot 3^3 \cdot 5^2 \) What is \(\text{\gcd} ( a , b , c , d )\)?

There are real numbers \(x , y , h\) and \(k\) that satisfy the system of equations \( x^2 + y^2 - 6 x - 8 y = h \) \( x^2 + y^2 - 10 x + 4 y = k \) What is the minimum possible value of \(h + k\)?

Trapezoid \(A B C D\) has \(\overline{A B} \parallel \overline{C D} , B C = C D = 43\), and \(\overline{A D} \perp \overline{B D}\). Let \(O\) be the intersection of the diagonals \(\overline{A C}\) and \(\overline{B D}\), and let \(P\) be the midpoint of \(\overline{B D}\). Given that \(O P = 11\), the length of \(A D\) can be written in the form \(m \sqrt{n}\), where \(m\) and \(n\) are positive integers and \(n\) is not divisible by the square of any prime. What is \(m + n\)?

What is the area of the region in the coordinate plane defined by \(|\| x \| - 1| + |\| y \| - 1| \leq 1 ?\)

Let \(\overline{A B}\) be a diameter in a circle of radius \(5 \sqrt{2} .\) Let \(\overline{C D}\) be a chord in the circle that intersects \(\overline{A B}\) at a point \(E\) such that \(B E = 2 \sqrt{5}\) and \(\angle A E C = 45^\circ .\) What is \(C E^2 + D E^2 ?\)