February Regional
Geometry Team: Question #1
A = the area of a circle with radius 10
B = the area of a square with side length 10
C = the area of an equilateral triangle with side length 10
D = the area of a regular hexagon with side length 10
Compute the value of
!
!
− !
!!
.
February Regional
Geometry Team: Question #1
A = the area of a circle with radius 10
B = the area of a square with side length 10, multiplied by π
C = the area of an equilateral triangle with side length 10
D = the area of a regular hexagon with side length 10
Compute the value of
!
!
− !
!!
.
February Regional
Geometry Team: Question #2
The value of each sentence appears to its left in parentheses. Let the sum the values of all the true
statements be A. Let the sum of the values of all the false statements be B.
(1) The contrapositive of the converse of a statement is its inverse.
(2) The contrapositive of a statement is always true.
(3) “In Euclidean Geometry, if a triangle’s angles sum to 360°, then a quadrilateral’s angles sum to 180°.”
(4) This statement is not true.
(5) The converse of the inverse of the inverse of a statement is its converse.
Compute AB.
February Regional
Geometry Team: Question #2
The value of each sentence appears to its left in parentheses. Let the sum the values of all the true
statements be A. Let the sum of the values of all the false statements be B.
(1) The contrapositive of the converse of a statement is its inverse.
(2) The contrapositive of a statement is always true.
(3) “In Euclidean Geometry, if a triangle’s angles sum to 360°, then a quadrilateral’s angles sum to 180°.”
(4) This statement is not true.
(5) The converse of the inverse of the inverse of a statement is its converse.
Compute AB.
February Regional
Geometry Team: Question #3
Let ABCDEFGH be a regular octagon.
A = the measure of
B = the measure of
C = the measure of
D = the measure of
∠ DEF
∠ ACF
∠ DBG
∠ CHE
Find B + C + D – A.
February Regional
Let ABCDEFGH be a regular octagon.
A = the measure of
B = the measure of
C = the measure of
D = the measure of
∠ DEF
∠ ACF
∠ DBG
∠ CHE
Find B + C + D – A.
Geometry Team: Question #3
February Regional
Geometry Team: Question #4
Each portion of this problem other than the first requires the answer from the previous portion of the
problem to answer. Submit only the answer of the third and final portion of the problem to be scored.
Four people are in a room. Each person shakes hands with each of the 3 other people in the room exactly
once. Let A be the number of handshakes that take place.
There are A couples are in a room. Each person shakes hands with each of the other people in the room,
except his or her partner, exactly once. Let B be one-fifth of the number of handshakes that take place.
A regular polygon has B sides. Let C be number of diagonals of this polygon.
Submit the value of C as your answer to this problem.
February Regional
Geometry Team: Question #4
Each portion of this problem other than the first requires the answer from the previous portion of the
problem to answer. Submit only the answer of the third and final portion of the problem to be scored.
Four people are in a room. Each person shakes hands with each of the 3 other people in the room exactly
once. Let A be the number of handshakes that take place.
There are A couples are in a room. Each person shakes hands with each of the other people in the room,
except his or her partner, exactly once. Let B be one-fifth of the number of handshakes that take place.
A regular polygon has B sides. Let C be number of diagonals of this polygon.
Submit the value of C as your answer to this problem.
February Regional
Geometry Team: Question #5
In square ABCD, M is the midpoint of AB. N is the midpoint of AM. Given the square of the length of DM
is 100, compute the square of the length of DN.
February Regional
Geometry Team: Question #5
In square ABCD, M is the midpoint of AB. N is the midpoint of AM. Given the square of the length of DM
is 100, compute the square of the length of DN.
February Regional
Geometry Team: Question #6
A = the number of diagonals of a convex octagon
B = the number of sides of a regular polygon with angle measure 179°
C = the interior angle measure of a regular 18-gon
D = the measure of an external angle of a regular 36-gon
Compute
!!!
!
− 𝐷.
February Regional
A = the number of diagonals of an octagon
B = the number of sides of a regular polygon with angle measure 179°
C = the angle measure of a regular 18-gon
D = the measure of an external angle of a regular 36-gon
Compute
!!!
!
− 𝐷.
Geometry Team: Question #6
February Regional
Geometry Team: Question #7
In square ABCD, we have point E on side CD. Let F be the foot of a perpendicular from B to AE. Given
𝐴𝐸 = 9 and 𝐵𝐹 = 4, compute the side length of ABCD.
February Regional
Geometry Team: Question #7
In square ABCD, we have point E on side CD. Let F be the foot of a perpendicular from B to AE. Given
𝐴𝐸 = 9 and 𝐵𝐹 = 4, compute the side length of ABCD.
February Regional
Geometry Team: Question #8
Triangles ABC, DEF and GHI are similar. The ratios of the length of the corresponding sides are 1:2:3. If
the area of the smallest triangle is 10, what is the sum of the areas of the three triangles?
February Regional
Geometry Team: Question #8
Triangles ABC, DEF and GHI are similar. The ratios of the length of the corresponding sides are 1:2:3. If
the area of the smallest triangle is 10, what is the sum of the areas of the three triangles?
February Regional
Geometry Team: Question #9
There are 9 shaded triangles in the chart below. All the squares have the same length of 1.
A = number of shaded regions that are similar to shaded region 1 (not counting region 1)
B = number of shaded regions that are similar to shaded region 3 (not counting region 3)
C = number of shaded regions that are similar to shaded region 4 (not counting region 4)
D = number of shaded regions that are similar to shaded region 6 (not counting region 6)
Compute A+B+C+D.
February Regional
Geometry Team: Question #9
There are 9 shaded triangles in the chart below. All the squares have the same length of 1.
A = number of shaded regions that are similar to shaded region 1 (not counting region 1)
B = number of shaded regions that are similar to shaded region 3 (not counting region 3)
C = number of shaded regions that are similar to shaded region 4 (not counting region 4)
D = number of shaded regions that are similar to shaded region 6 (not counting region 6)
Compute A+B+C+D.
February Regional
Geometry Team: Question #10
Compute ∠ 1 + ∠ 2 + ∠ 3 + ∠ 4+ ∠ 5 + ∠ 6 + ∠ 7 in the following figure.
February Regional
Geometry Team: Question #10
Compute ∠ 1 + ∠ 2 + ∠ 3 + ∠ 4+ ∠ 5 + ∠ 6 + ∠ 7 in the following figure.
February Regional
Geometry Team: Question #11
EFGH is a rectangle with 𝐸𝐻 = 12 and 𝐸𝐹 = 5. P is a point on EH such that 𝐸𝑃 = 4.
A = length of FP
B = length of GP
C = length of FH
D = area of triangle PFG
Compute A2 + B2 - C2 + D.
February Regional
Geometry Team: Question #11
EFGH is a rectangle with 𝐸𝐻 = 12 and 𝐸𝐹 = 5. P is a point on EH such that 𝐸𝑃 = 4.
A = length of FP
B = length of GP
C = length of FH
D = area of triangle PFG
Compute A2 + B2 - C2 + D.
February Regional
Geometry Team: Question #12
A rectangle and a regular hexagon intersect in the following way: the area outside of the rectangle but
inside the hexagon is equal to the area outside the hexagon but inside the rectangle. If the side length of
the hexagon is 2, find the area of the rectangle.
February Regional
Geometry Team: Question #12
A rectangle and a regular hexagon intersect in the following way: the area outside of the rectangle but
inside the hexagon is equal to the area outside the hexagon but inside the rectangle. If the side length of
the hexagon is 6, find the area of the rectangle.
February Regional
Geometry Team: Question #13
Triangle ABC has ∠ A = 70° and ∠ C = 50°. Let H be the foot of the altitude to side AC. Extend BH past
H to point D such that BC = CD.
X = the measure of ∠ ABC
Y = the measure of ∠ BDA
Z = the measure of ∠ BDC
Find X+Y+Z.
February Regional
Geometry Team: Question #13
Triangle ABC has ∠ A = 70° and ∠ C = 50°. Let H be the foot of the altitude to side AC. Extend BH past
H to point D such that BC = CD.
X = the measure of ∠ ABC
Y = the measure of ∠ BDA
Z = the measure of ∠ BDC
Find X+Y+Z.
February Regional
Geometry Team: Question #14
Let A be the number of true statements below and B be the number of false statements. Find
A –B.
1. The diagonals of a parallelogram could be perpendicular to each other.
2. The locus of points equidistant from two distinct points in the plane is a line.
3. Two distinct circles may intersect at more than one point.
4. The common interior angle measure and the common exterior angle measure of a regular polygon are
never equal.
5. The area of a circle with radius r is greater than the area of a square with side r.
6. If two figures are similar with ratio 3:15, then their areas have the ratio 1:25.
7. A circle’s area is always numerically greater than its circumference.
February Regional
Geometry Team: Question #14
Let A be the number of true statements below and B be the number of false statements. Find
A –B.
1. The diagonals of a parallelogram could be perpendicular to each other.
2. The locus of points equidistant from two points in the plane is a line.
3. Two distinct circles may intersect at more than one point.
4. The common interior angle measure and the common exterior angle measure of a regular polygon are
never equal.
5. The area of a circle with radius r is greater than the area of a square with side r.
6. If two figures are similar with ratio 3:15, then their areas have the ratio 1:25.
7. A circle’s area is always greater numerically than its circumference.
February Regional
Geometry Team: Question #15
Every triangle has a nine-point circle. This is defined as the circle passing through the midpoints of the
sides of the triangles (this means a nine-point circle is also the circumcircle of the triangle formed by the
midpoints of the original triangle). This circle also has the property of passing through the feet of the
altitudes of the triangle and the midpoints of the line segments connecting each vertex of the triangle to
the triangle’s orthocenter. These nine points give the circle its name.
A = the ratio of the area of an arbitrary triangle’s nine-point circle to the ratio of the area of the triangle’s
circumcircle (Hint: the ratio of the areas of similar triangles is equal to the ratio of the areas of their
circumcircles)
B = the ratio of the area of a given equilateral triangle’s incircle to the area of its nine-point circle
Compute the product AB.
February Regional
Geometry Team: Question #15
Every triangle has a nine-point circle. This is defined as the circle passing through the midpoints of the
sides of the triangles. This circle also has the property of passing through the feet of the altitudes of the
triangle and the midpoints of the line segments connecting each vertex of the triangle to the triangle’s
orthocenter. These nine points give the circle its name.
A = the ratio of the area of an arbitrary triangle’s nine-point circle to the ratio of the area of the triangle’s
circumcircle (Hint: the ratio of the areas of similar triangles is equal to the ratio of the areas of their
circumcircles)
B = the ratio of the area of a given equilateral triangle’s incircle to the area of its nine-point circle
Compute the product AB.