Computer Science, Engineering, and Mathematics Club
CCBC-Catonsville

Faculty Advisor :

Dr. Nilotpal Ghosh
Associate Professor of Mathematics
Mathematics, Science, and Engineering

• Dr. Nilotpal Ghosh
• VOICE MAIL : 443-840-4281
• FAX : 443-840-5974
• E-mail : nghosh@ccbcmd.edu

About the CSEM Club

The CSEM Club currently sponsors "First Friday", a Mathematics seminar which takes place on the first Friday of every month. The speakers are the faculty members of the Mathematics Department of CCBC-Catonsville and sometimes invited guest speakers from outside the college.

One of the many ways this home page intends to keep interest in mathematics alive in the community is by publishing every month interesting mathematical problems or puzzles that we challenge anyone on the internet to attempt to solve. We welcome you to e-mail your solution to us so that we can publish the names of the people with correct answers together with the solutions the next month.

Problems for the month of January

Show that if n is a positive integer then
1. ( _nC₀ )² + ( _nC₁ )² + ( _nC₂ )² + . . . + ( _nC_n )² = _2nC_n .

2. ( _nC₀ )² - ( _nC₁ )² + ( _nC₂ )² - . . . + (-1) ⁿ ( _nC_n )²

   = 0   if n is odd;       and

   = (-1)^k _nC_k   if    n is even = 2k.

3. ( _nC₀ )( _nC₁ ) + ( _nC₁ )( _nC₂ ) + ( _nC₂ )( _nC₃ ) + . . . + ( _nC_{n - 1} )( _nC_n ) = _2nC_{n - 1} .

Solutions of the problems for the month of January

Use Binomial Theorem.
1. Equate the coefficient of xⁿ in (1 + x)ⁿ(1 + x)ⁿ = (1 + x)²ⁿ.
2. Equate the coefficient of xⁿ in (1 + x) ⁿ(1 - x)ⁿ = (1 - x²)ⁿ
3. Equate the coefficient of x^{n- 1} in (1 + x)ⁿ(1 + x)ⁿ = (1 + x)²ⁿ.

Problems for the month of February

Show that if n and m are positive integers, (n ³ m),
1. ( _nC_m )( _mC_m ) + ( _nC_{m + 1} )( _{m + 1}C_m ) + ( _nC_{m + 2} )( _{m + 2}C_m ) + . . . + ( _nC_n )( _nC_m ) = 2^{n - m}_nC_m.

2. ( _nC₀ )( _mC₀ ) + ( _nC₁ )( _mC₁ ) + ( _nC₂ )( _mC₂ ) + . . . + ( _nC_m )( _mC_m ) = _{n + m}C_m .

3. ( _nC_m )( _mC_m ) - ( _nC_{m + 1} )( _{m + 1}C_m ) + ( _nC_{m + 2} )( _{m + 2}C_m ) - . . . + (-1)^n-m( _nC_n )( _nC_m ) = 0     if n > m.

Solutions of the problems for the month of February

Use Binomial Theorem

1. Equate the coefficient of x^m in (2 + x)ⁿ = (1 + (1 + x))ⁿ.
2. Equate the coefficient of x^m in (1 + x)ⁿ(1 + x)^m = (1 + x)^{m + n}.
3. Equate the coefficient of x^m in (-x)ⁿ = (1 - (1 + x))ⁿ.

Problems for the month of March

If n is a positive integer, show that
1. sin (p/n) + sin (2p/n) + sin (3p/n) + . . . + sin ((n - 1)p/n) = cot (p/(2n)).
2. sin²(p/n) + sin²(2p/n) + sin²(3p/n) + . . . + sin²((n - 1)p/n) = n/2.
3. sin³(p/n) + sin³(2p/n) + sin³(3p/n) + . . . + sin³((n - 1)p/n) = (3cot (p/(2n)) - cot (3p/(2n)))/4.

Solutions of the problems for the month of March

Use the identity e^ix = cos x + i sin x .Use the formula for the sum of a (finite) geometric series and take imaginary parts of both sides.
Use the trigonometric formulas :
sin²x = (1 - cos 2x)/2,
sin³x = (3 sin x - sin 3x)/4.

Problems for the month of April

1. If n ³ k + l ,

   ( _kC_k )( _{n - k}C_l ) + ( _{k + 1}C_k )( _{n - k - 1}C_l ) + ( _{k + 2}C_k )( _{n - k - 2}C_l ) + . . . + ( _{n - l}C_k )( _lC_l ) = _{n + 1}C_{k + l + 1}.

2. ( _nC₀ )( _mC_k ) + ( _nC₁ )( _mC_{k - 1} ) + ( _nC₂ )( _mC_{k - 2} ) + . . . + ( _nC_k )( _mC₀ ) = _{n + m}C_k .

3. ( _nC₀ )( _kC_m ) - ( _nC₁ )( _{k - 1}C_m ) + ( _nC₂ )( _{k - 2}C_m ) - . . . + (-1)ⁿ( _nC_n )( _{k - n}C_m ) = _{k - n}C_{m - n}     if k ³ m ³ n.

Problems for the month of May

1. Show that
   tan²(p/7) + tan²(2p/7) + tan²(3p/7) = 21.
2. Show that
   cot²(p/7) + cot²(2p/7) + cot²(3p/7) = 5.

Problems for the month of June

If A, B, C are the angles of a triangle then
1. sin A + sin B + sin C = 4 cos (A/2) cos (B/2) cos (C/2).
2. cos A + cos B + cos C = 1 + 4 sin (A/2) sin (B/2) sin (C/2).
3. tan A + tan B + tan C = tan A tan B tan C.
4. sin 2A + sin 2B + sin 2C = 4 sin A sin B sin C.

Problems for the month of July

1. Let ABC be a triangle on the plane. If isosceles triangles ABD, BCE, and CAF are drawn on the sides of the triangle ABC such that the angles ÐADB, ÐBEC, and ÐCFA are all equal to 120°, then show that the triangle DEF is an equilateral triangle.
2. Let ABC be a triangle inscribed in a circle. If D is any point on the arc BC, show that the distances of D from the vertices saisfy the relation
   AD = BD + CD.

Problems for the month of August

1. Show that the three medians of a triangle are concurrent.
2. Show that the bisectors of the angles of a triangle are concurrent.
3. Show that the perpendicular bisectors of the three sides of a triangle are concurrent.
4. Show that the three altitudes (perpendiculars from vertices to opposite sides) of a triangle are concurrent.

Problems for the month of September

1. Show that the circle that passes through the midpoints of the three sides of a triangle also passes through the feet of the perpendiculars from the vertices to the opposite sides.
2. The orthocenter of a triangle is the point where the altitudes intersect.
   Show that the circle in problem 1 also passes through the midpoints of the line segments joining a vertex and the orthocenter. This circle is called the Nine-Point circle of the triangle.
3. Show that the radius of the nine-point circle of a triangle is half that of the circum-circle of the triangle. The circum-circle of a triangle is the circle that passes through the vertices of the triangle.

Problems for the month of October

1. Show that the center of the Nine-Point circle, the orthocenter, and the circum-center of a triangle lie on the same line.
   The circum-center is the center of the circum-circle.
2. The centroid of a triangle is the point where the three medians intersect.
   Show that the centroid of a triangle also lies on the line in Problem 1.
3. Show that the center of the Nine-Point circle bisects the line segment joining the circum-center and the ortho-center.

Problems for the month of November

1. The pedal triangle of a triangle is a triangle whose vertices are the feet of the perpendiculars from the vertices to the opposite sides of the given triangle. Show that the perpendiculars bisect the angles of the pedal triangle.
2. Show that for a given acute triangle, the pedal triangle is the one with the smallest perimeter among all the triangles with a vertex on each side of the given triangle.
3. Show that the ortho-center of a given triangle is the in-center of its pedal triangle. The in-center of a triangle is the center of the largest circle lying completely within the triangle.

Problem for the month of December

1. Show that for n a positive integer ³ 2,

   csc² (p/n) + csc² (2p/n) + . . . + csc² ((n - 1)p/n) = (n² - 1)/3.

Ashby's Web Math Problems

Last update March 5, 2009
For more information e-mail Nilotpal Ghosh.