Construct a rectangle with compasses and a straightedge.

Solution 1: Drawing a circle and two lines that are diagonals of a rectangle

Solution 2: Drawing four circles to determine the four vertices of a rectangle

Solution 3: Drawing two pairs of concentric circles

With compasses and a straightedge, first draw a line segment, and then construct a point that divides the line segment internally in the ratio 1:2.

Solution 1: Constructing, a line segment that is parallel to the initial line, and a point that divides the line segment externally 2:3

Solution 2: Constructing a pair of parallel lines each of which passes through either end of the initial line segment

Solution 3: Constructing a triangle whose one side is the initial line segment, and the other sides are 1:2 in length

Solution 4: Utilizing an isosceles triangle whose sides are 2:3:3 in length (1) (An isosceles triangle of 1:3:3 can be utilized instead.)

Solution 5: Utilizing an isosceles triangle whose sides are 2:3:3 in length (2) (An isosceles triangle of 1:3:3 can be utilized instead.)

Solution 6: Determining points that are on the circumference of a circle whose center is a point to be constructed

Solution 7: Constructing an equilateral triangle and its centroid, with its one side the initial line segment

Solution 8: Utilizing that the centroid of a triangle cuts a line segment between one vertex of the triangle and the midpoint of the opposite side in the ratio 2:1

With compasses and a straightedge, first draw two line segments, and then, letting 'a' and 'b' be the lengths of them, draw a line segment whose length is √(ab).

Solution 1: Utilizing (a+b)^{2}-(a-b)^{2}=4ab

Solution 2: Utilizing (a-b/2)^{2}-(b/2)^{2}=a^{2}-ab, and a^{2}-(a^{2}-ab)=ab

Solution 3: Utilizing (a-b)^{2}+b^{2}-2(a-b)b*cos120°=a^{2}+b^{2}-ab, and (a^{2}+b^{2}-ab)-(a-b)^{2}=ab

Solution 4: Utilizing a right triangle that is divided into two similar right triangles

Solution 5: Utilizing, a line that intersects a circle at two points, and another line that touches the circle

With compasses and a straightedge, first draw a circular sector with its central angle less than 180°, and then construct an inscribed circle of the circular sector.

Solution 1: Constructing, a line segment that is parallel to the bisector of the central angle, and a point that divides the line segment internally in the ratio "the radius" to "half the chord" (cf. Solution 1 of construction problem 2)

Solution 2: Constructing, a line that passes through the center of the circular sector, and another line that is parallel to the line and that passes through the midpoint of the arc (cf. Solution 2 of construction problem 2)

Solution 3: Constructing a triangle whose one side is the line segment between the center of the circular sector and the midpoint of the arc, and the other two sides are "the radius" to "half the chord" in length (cf. Solution 3 of construction problem 2)

Solution 4: Utilizing an isosceles triangle whose sides are x:y:y in length (1) (x="the radius of the circular sector" and y="the radius plus half the chord") (cf. Solution 4 of construction problem 2)

Solution 5: Utilizing an isosceles triangle whose sides are x:y:y in length (2) (x="the radius of the circular sector" and y="the radius plus half the chord") (cf. Solution 5 of construction problem 2)

Solution 6: Determining points that are on the circumference of a circle concentric with a circle to be constructed (cf. Solution 6 of construction problem 2)

Solution 7: Determining tangent points between the circular sector and its inscribed circle

Solution 8: Constructing the inscribed circle of an isosceles triangle

What day of the week is today? Then, what day will it be, 2^{30} days from today?

Solution 1: Calculating 2^{30}

Solution 2: Examining the remainders of 2, 2^{2}, 2^{3}, ... divided by 7s

Solution 3: Utilizing 2^{30}=8^{10}

Solution 4: Factorizing 2^{30}-1

What is the smallest positive integer that has just 15 factors?

(Note: 1 and the integer itself are included in the factors. For example, the number 6 has 4 factors, that are 1, 2, 3, and 6.)Solution 1: Noticing what integer has odd number of factors

Solution 2: Utilizing the relation between a factorized integer and the number of factors the integer has

What is the smallest positive integer whose remainder is 3 when divided by 7, whose remainder is 4 when divided by 9, and whose remainder is 2 when divided by 16?

Solution 1: Examining positive integers that satisfy the given conditions

Solution 2: Setting up equations from the given conditions

What are the last 4 digits of 7^{100}?

Solution 1: Calculating the last 4 digits of 7^{4}, 7^{8}, 7^{16}, ...

Solution 2: Utilizing 7^{2}=49

Solution 3: Utilizing 7^{4}=2401

Factorize 111111111111, which is a product of 7 primes.

Solution 1: Factorizing the given integer in two ways

Solution 2: Multiplying the given integer by 9 before prime factorizing

A positive integer is represented, as 21022200 in a base 'n' system, and as 10112121 in a base n+1 system. What is the 'n', that is larger than 2, in the decimal system?

(Note: for example, the decimal number 37 means 3*10+7. 102 in the base 3 system is equivalent to 1*3Solution 1: Inspecting a 7th-degree polynomial of 'n' by taking its derivatives

Solution 2: Deriving two inequalities from an equation to limit n's range

Solution 3: Factorizing a 7th-degree polynomial of 'n'

Solution 4: Making the most of n's being an integer

Mathematical Problems

Updated: April 14, 2013