2013 Student Delegates Invitational _______________________________________ Geometry Team
Question 1
Find the sum of the answers to the four questions below.
a) Find the area of a square with side length 2.
b) How many sides does a pentagon have?
c) What is the maximum number of distinct points in which a line and a circle can intersect?
d) How many diagonals does an octagon have?
Question 2
Find the sum of the answers to the two questions below.
a) A hexagonal pyramid has a base with a side length 2 and a height of 5. Find its volume.
b) The greatest distance between two vertices of a rectangular prism is 5. The greatest distance
between two vertices which is less than that is 3. Find the greatest possible volume of the prism.
Question 3
Find the number of true statements below.
a) The reflections of the altitudes of a triangle about the angle bisectors from their respective
vertices intersect at the triangle’s circumcenter.
b) The circumcenter of a triangle can lie outside the triangle.
c) If two distinct planes intersect, they intersect in a line.
d) In the plane, one circle is centered at A with radius x, and another circle is centered at B with
radius y. If they intersect at the distinct points M and N, for every point P on the line MN we have
PA2 – x2 = PB2 – y2.
2013 Student Delegates Invitational _______________________________________ Geometry Team
Question 4
Find the sum of the answers to the four questions below.
a) Find the sum of measures of the interior angles of an octagon in degrees.
b) Find the measure of an interior angle of a regular hexagon in degrees.
c) Find the measure of the acute angle whose sine is 1/2, in degrees.
d) A circle is divided into 90 congruent sectors. Find the degree measure of the central angle
spanned by one of these sectors.
Question 5
Find the sum of the answers to the three questions below.
a) Triangle ABC has AB = 13, BC = 14, and CA = 15. Let D and E be the points of tangency of the
incircle of ABC with BC and CA, respectively. Find DE 5.
b) The shortest two sides of a right triangle have lengths 7 and 24. Find the length of its hypotenuse.
c) The medial triangle of a triangle ABC is the triangle with the midpoints of the sides of ABC as its
vertices. If a triangle has side lengths 9, 10, and 11, find the perimeter of its medial triangle.
Question 6
Find the sum of the answers to the two questions below.
a) Tony and Mitchell are about to eat the interior of an isosceles triangle with side lengths of 10, 10,
and 16. They have to cut it into two pieces of equal area, one piece for each of them. The Mu
Alpha Theta gods require them to cut it along a straight line parallel to the base. How far from the
base should they cut?
b) Determine the area of the largest square which can be inscribed in a semicircle of diameter 10.
Question 7
Find the sum of the answers to the two questions below.
a) ABCDE…R is a regular octadecagon (18-sided polygon). Find the measure of angle ADE in
degrees.
b) In isosceles triangle ABC, AB = AC, and BC is the shortest side. There exists a point D on
segment AC (strictly between A and C) such that AD = DB = BC. Find the measure of angle
BAC in degrees.
2013 Student Delegates Invitational _______________________________________ Geometry Team
Question 8
All four of the vertices of quadrilateral JOHN lie on the same circle.
JH bisects angle NJO and intersects ON at X.
If JO = 8, JN = 10, and OX = 2, find OH ! .
Question 9
Find the sum of the answers to the two questions below.
a) How many noncongruent triangles with integer side lengths have perimeter 9?
b) The lattice pictured below has evenly spaced rows of evenly spaced points, with alternate rows
shifted one half spacing. In how many ways can three points be chosen from this lattice such that
they are the vertices of a (non-degenerate!) triangle?
Question 10
Find the sum of the answers to the two questions below.
a) Quadrilateral JOHN has the following interior angle measures: J = 15, O = 240, H = 15. If JN =
NH = 8 6, find the area of JOHN.
b) A right triangle has legs of length 3 and 4. A rectangle has two vertices on the triangle’s
hypotenuse and the remaining two vertices on the triangle’s legs. Find the maximal area of the
rectangle.
2013 Student Delegates Invitational _______________________________________ Geometry Team
Question 11
Find the sum of the answers to the two questions below.
a) Triangle MAX has MA = 11, AX = 13, and XM = 20.
The distances from P to MA and P to AX are both 3.
Find two times the distance from P to XM.
b) Trapezoid ABCD has AB || CD and AB = 3, BC = DA = 13, and CD = 15. AC and BD intersect
at E. Let F be on AD such that EF ⊥ AD. Find 13(EF).
Question 12
Find the sum of the areas of the two convex quadrilaterals described below.
a) ABCD, where AB = 6, BC = 6, CD = 5, DA = 5, and BD = 8.
b) EFGH, where EF = FG = GH = HE = 4 and EG = 8