Download March Regional

March Regional
Geometry Team: Question 1
A five-point star and a six-point star are formed by drawing all the diagonals of a regular pentagon and the
diagonals connecting every other vertex of a regular hexagon, respectively. If A and B are the sums of the
angle measures of the points of the five-point star and six-point star (respectively), find B
March Regional
A
.
Geometry Team: Question 1
A five-point star and a six-point star are formed by drawing all the diagonals of a regular pentagon and the
diagonals connecting every other vertex of a regular hexagon, respectively. If A and B are the sums of the
angle measures of the points of the five-point star and six-point star (respectively), find B
A
.
March Regional
Geometry Team: Question 2
Two side lengths of a triangle are known to measure 13 and 18 . If j and k are the smallest integral lengths
that makes the triangle acute and obtuse (respectively), and if l and m are the lengths that can make the
triangle right, find the value of j  k  l 2  m2 .
March Regional
Geometry Team: Question 2
Two side lengths of a triangle are known to measure 13 and 18 . If j and k are the smallest integral lengths
that makes the triangle acute and obtuse (respectively), and if l and m are the lengths that can make the
triangle right, find the value of j  k  l 2  m2 .
March Regional
Geometry Team: Question 3
A panda is tethered to a rope of 10 feet, and the other end is tied to the corner of a cage with dimensions 8 3
by 5 feet.
Let A be the area the panda can roam if the cage were not present (that is, presume the panda is tied to a
pole with infinitesimally small radius instead of the cage).
B
be the area the panda can roam outside the cage.
4
C  D 3
Let
be the area the panda can reach within its cage.
6
Let
Find A  B  C  D .
March Regional
Geometry Team: Question 3
A panda is tethered to a rope of 10 feet, and the other end is tied to the corner of a cage with dimensions 8 3
by 5 feet.
Let A be the area the panda can roam if the cage were not present (that is, presume the panda is tied to a
pole with infinitesimally small radius instead of the cage).
B
be the area the panda can roam outside the cage.
4
C  D 3
Let
be the area the panda can reach within its cage.
6
Let
Find A  B  C  D .
March Regional
Geometry Team: Question 4
Let A be the sum of the radius and the coordinates of the center of the circle whose equation is
x 2   y  1  25 . Let B be the sum of the coordinates of the centroid of the triangle whose vertices have
2
coordinates  0, 0  , 1, 4  and  5, 7  . Let C be the distance between the points P  2,1 and Q  5,5 .
Find A  B  C .
March Regional
Geometry Team: Question 4
Let A be the sum of the radius and the coordinates of the center of the circle whose equation is
x 2   y  1  25 . Let B be the sum of the coordinates of the centroid of the triangle whose vertices have
2
coordinates  0, 0  , 1, 4  and  5, 7  . Let C be the distance between the points P  2,1 and Q  5,5 .
Find A  B  C .
March Regional
Geometry Team: Question 5
As a detective, your skills have brought you to merely one pass code away from finishing your
investigation—a pass code you have deduced to be ABCD. Your subordinates have given you four clues they
found in their own investigations, as follow:
A = the sum of the digits of the sum of the exterior angles of a regular icosagon.
B = the number of sides of the largest regular polygon that can tessellate a plane without the use of any other
shape.
C = the minimum number of distinct angle measures formed by the intersection of three nonparallel lines, if
all three do not intersect each other at the same point.
D = the number of vertices a solid has given it has eight edges and six faces.
Solve the pass code and finish your investigation! (Note: A, B, C and D represent the digits of a number and
are not to be multiplied!)
March Regional
Geometry Team: Question 5
As a detective, your skills have brought you to merely one pass code away from finishing your
investigation—a pass code you have deduced to be ABCD. Your subordinates have given you four clues they
found in their own investigations, as follow:
A = the sum of the digits of the sum of the exterior angles of a regular icosagon.
B = the number of sides of the largest regular polygon that can tessellate a plane without the use of any other
shape.
C = the minimum number of distinct angle measures formed by the intersection of three nonparallel lines, if
all three do not intersect each other at the same point.
D = the number of vertices a solid has given it has eight edges and six faces.
Solve the pass code and finish your investigation! (Note: A, B, C and D represent the digits of a number and
are not to be multiplied!)
March Regional
A wheel with a radius of 5
Geometry Team: Question 6
30 (that is, it rises
 units is situated at the top of a hill with an inclination of
30 degrees from ground level). If the hill has a vertical height of 100 units, how many full rotations will the
wheel make before reaching the bottom of the hill?
March Regional
A wheel with a radius of 5
Geometry Team: Question 6
30 (that is, it rises
 units is situated at the top of a hill with an inclination of
30 degrees from ground level). If the hill has a vertical height of 100 units, how many full rotations will the
wheel make before reaching the bottom of the hill?
March Regional
Geometry Team: Question 7
Suppose quadrilateral ABCD has a perimeter of 60 . Let x square units be the area if ABCD is a square. Let y
square units be the area if ABCD is a rhombus whose diagonals’ lengths have a ratio of 4: 3 . Let z square
units be the area if ABCD is a rectangle whose sides’ lengths have a ratio of 7 :8 .
Find x  y  z .
March Regional
Geometry Team: Question 7
Suppose quadrilateral ABCD has a perimeter of 60 . Let x square units be the area if ABCD is a square. Let y
square units be the area if ABCD is a rhombus whose diagonals’ lengths have a ratio of 4: 3 . Let z square
units be the area if ABCD is a rectangle whose sides’ lengths have a ratio of 7 :8 .
Find x  y  z .
March Regional
Geometry Team: Question 8
The symbol that represents the Deathly Hallows is formed by drawing an equilateral triangle, an inscribed
circle, and the diameter of the circle (which serves as a portion of the triangle’s altitude). Suppose the
circumference of the circle is 8 units. What is the perimeter of the triangle, in units?
March Regional
Geometry Team: Question 8
The symbol that represents the Deathly Hallows is formed by drawing an equilateral triangle, an inscribed
circle, and the diameter of the circle (which serves as a portion of the triangle’s altitude). Suppose the
circumference of the circle is 8 units. What is the perimeter of the triangle, in units?
March Regional
Geometry Team: Question 9
The perimeters of two similar triangles are at ratio of 4: 5 . If the smaller triangle has an area of 400 square
units, the larger triangle has an area of x square units; If the larger triangle has an area of 400 square units, the
smaller triangle has an area of y square units.
Find x  y .
March Regional
Geometry Team: Question 9
The perimeters of two similar triangles are at ratio of 4: 5 . If the smaller triangle has an area of 400 square
units, the larger triangle has an area of x square units; If the larger triangle has an area of 400 square units, the
smaller triangle has an area of y square units.
Find x  y .
March Regional
Geometry Team: Question 10
A circle passes through the points  0, 0  ,  2, 4  and  4, 2  . Find the radius of this circle.
March Regional
Geometry Team: Question 10
A circle passes through the points  0, 0  ,  2, 4  and  4, 2  . Find the radius of this circle.
March Regional
Geometry Team: Question 11
For each of the following statements, assign a value of 1 if it is unconditionally true and 0 if it is false for any
given circumstance.
A) A triangle with side lengths of 4, 6, and 11 has a perimeter of 21 and is both scalene and obtuse.
B) If distinct lines m and n in space are both perpendicular to line l, then m must be parallel to n.
C) Six coplanar but nonparallel lines can divide a plane into at most 22 regions.
D) In an equilateral triangle, the circumcenter, orthocenter, incenter and centroid all coincide.
E) A regular polygon with ten sides has interior angle measures of 144 .
F) A rectangle’s diagonals are not perpendicular to each other unless it is a square.
G) In the case of a quadrilateral inscribed within a circle, two consecutive angles are supplementary.
Find the value of A  B  C  D  E  F  G .
March Regional
Geometry Team: Question 11
For each of the following statements, assign a value of 1 if it is unconditionally true and 0 if it is false for any
given circumstance.
A) A triangle with side lengths of 4, 6, and 11 has a perimeter of 21 and is both scalene and obtuse.
B) If distinct lines m and n in space are both perpendicular to line l, then m must be parallel to n.
C) Six coplanar but nonparallel lines can divide a plane into at most 22 regions.
D) In an equilateral triangle, the circumcenter, orthocenter, incenter and centroid all coincide.
E) A regular polygon with ten sides has interior angle measures of 144 .
F) A rectangle’s diagonals are not perpendicular to each other unless it is a square.
G) In the case of a quadrilateral inscribed within a circle, two consecutive angles are supplementary.
Find the value of A  B  C  D  E  F  G .
March Regional
Geometry Team: Question 12
Consider a right triangle with legs of lengths 3 and 4. What is the length of the median drawn to the triangle’s
hypotenuse?
March Regional
Geometry Team: Question 12
Consider a right triangle with legs of lengths 3 and 4. What is the length of the median drawn to the triangle’s
hypotenuse?
March Regional
Geometry Team: Question 13
A kite with integral side lengths has an area of 168 square units. Let A and B be the two possible perimeters
this kite can have.
Find A  B .
March Regional
Geometry Team: Question 13
A kite with integral side lengths has an area of 168 square units. Let A and B be the two possible perimeters
this kite can have.
Find A  B .
March Regional
Geometry Team: Question 14
Suppose a sphere is inscribed within a cube. Let A be the volume of the cube if the sphere’s great circle has
circumference 8 , and let B  be the volume of the sphere if this cube has an eighth the volume than the
3
cube found in part A. Find A  B .
March Regional
Geometry Team: Question 14
Suppose a sphere is inscribed within a cube. Let A be the volume of the cube if the sphere’s great circle has
circumference 8 , and let B  be the volume of the sphere if this cube has an eighth the volume than the
3
cube found in part A. Find A  B .
March Regional
Geometry Team: Question 15
Let A be the 100th term in the sequence 2, 5, 9, 14, 20 … and let B be the 100th term in the sequence 6, 12, 20,
30, 42 …. Find A  B .
March Regional
Geometry Team: Question 15
Let A be the 100th term in the sequence 2, 5, 9, 14, 20 … and let B be the 100th term in the sequence 6, 12, 20,
30, 42 …. Find A  B .