# Another Cube Slice Problem

This is a problem from the UKMT Senior Challenge for 2019.  (It has been slightly edited to reflect the colors I added to the diagram.)

“The edge-length of the solid cube shown is 2.  A single plane cut goes through the points Y, T, V and W which are midpoints of the edges of the cube, as shown.

What is the area of the cross-section?

A_√3_____B_3√3_____C_6_____D_6√3_____E_8”

# Circle in Slot Problem

Here is another UKMT Senior Challenge problem from 2017, which has a straight-forward solution:

“The diagram shows a circle of radius 1 touching three sides of a 2 x 4 rectangle. A diagonal of the rectangle intersects the circle at P and Q, as shown.

What is the length of the chord PQ?

__A_√5____B_4/√5____C_√5 – 2/√5____D_5√5/6____E_2”

See the Circle in Slot Problem

# Flipping Parabolas

This is a stimulating problem from the UKMT Senior Math Challenge for 2017. The additional problem “for investigation” is particularly challenging. (I have edited the problem slightly for clarity.)

“The parabola with equation y = x² is reflected about the line with equation y = x + 2. Which of the following is the equation of the reflected parabola?

A_x = y² + 4y + 2_____B_x = y² + 4y – 2_____C_x = y² – 4y + 2
D_x = y² – 4y – 2_____E_x = y² + 2

For investigation: Find the coordinates of the point that is obtained when the point with coordinates (x, y) is reflected about the line with equation y = mx + b.”

# Cube Slice Angle Problem

This is from the UKMT Senior Challenge of 2004.

“L, M, and N are midpoints of a skeleton cube, as shown. What is the value of angle LMN?

_____A_90°_____B_105°_____C_120°_____D_135°_____E_150°”

# Star Polygon Problem

This is problem #25 from the UKMT 2014 Senior Challenge.

“Figure 1 shows a tile in the form of a trapezium [trapezoid], where a = 83⅓°. Several copies of the tile placed together form a symmetrical pattern, part of which is shown in Figure 2. The outer border of the complete pattern is a regular ‘star polygon’. Figure 3 shows an example of a regular ‘star polygon’.
How many tiles are there in the complete pattern?
_____A_48_____B_54_____C_60_____D_66_____E_72”

See the Star Polygon Problem.

# Fibonacci Fandango

This is from the UKMT Senior Challenge of 1999.

What is the sum to infinity of the convergent series

____________________________________A_7/4_____B_2_____C_√5_____D_9/4_____E_7/3”

# Rising Sun

Here is a problem from the UKMT Senior (17-18 year-old) Mathematics Challenge for 2012:

“A semicircle of radius r is drawn with centre V and diameter UW. The line UW is then extended to the point X, such that UW and WX are of equal length. An arc of the circle with centre X and radius 4r is then drawn so that the line XY is tangent to the semicircle at Z, as shown. What, in terms of r, is the area of triangle YVW?”

See the Rising Sun

# Six Squares Problem

This is a problem from the UKMT Senior Challenge for 2001. (It has been slightly edited to reflect the colors I added to the diagram.)

“The [arbitrary] blue triangle is drawn, and a square is drawn on each of its edges. The three green triangles are then formed by drawing their lines which join vertices of the squares and a square is now drawn on each of these three lines. The total area of the original three squares is A1, and the total area of the three new squares is A2. Given that A2 = k A1, then

_____A_ k = 1_____B_ k = 3/2_____C_ k = 2_____D_ k = 3_____E_ more information is needed.”

I solved this problem using a Polya principle to simplify the situation, but UKMT’s solution was direct (and more complicated). See the Six Squares Problem.

# Hitting the Target

Here is a problem from the UKMT Senior (17-18 year-old) Mathematics Challenge for 2012:

“Tom and Geri have a competition. Initially, each player has one attempt at hitting a target. If one player hits the target and the other does not then the successful player wins. If both players hit the target, or if both players miss the target, then each has another attempt, with the same rules applying. If the probability of Tom hitting the target is always 4/5 and the probability of Geri hitting the target is always 2/3, what is the probability that Tom wins the competition?

______A 4/15______B 8/15______C 2/3______D 4/5______E 13/15”

# Challenging Sum

Here is a problem from the UKMT Senior (17-18 year-old) Mathematics Challenge for 2009:

“Four positive integers a, b, c, and d are such that
_________abcd + abc + bcd + cda + dab + ab + bc + cd + da + ac + bd + a + b + c + d = 2009.
What is the value of a + b + c + d?
_________A 73_________B 75_________C 77_________D 79_________E 81”

See the Challenging Sum