## NDSU Mathematics

### North Dakota Mathematics Talent Search 2013-2014

The Department of Mathematics at NDSU is happy to announce the start of the annual North Dakota Mathematics Talent Search. The goals of the talent search are to locate high school students in North Dakota and surrounding areas with a talent for solving mathematical problems, to reward these students and their teachers for their efforts, and to encourage these students to attend NDSU and major in the mathematical sciences or engineering.

The Talent Search poses sets of challenging mathematical problems throughout the year which will be posted as follows:

Interested students are strongly encouraged to send in solutions even if they only solve one problem in a set; finding a good solution to a problem is always an achievement. The problems do not require advanced mathematical knowledge -- just creativity and a feeling or taste for problem solving. It is not necessary to have participated in previous rounds. Everybody is welcome to send solutions to Round 3.

The students who submit a significant number of mathematically sound solutions for each of the three rounds will be rewarded with various prizes, including a one-year subscription to a science/mathematics magazine of their choice. The best participants who decide to attend NDSU and major in the mathematical sciences will also be rewarded with scholarships.

Please submit your solutions by email to maria.alfonseca@ndsu.edu. Alternatively, solutions may be sent by regular mail to:

Talent Search
c/o Maria Alfonseca-Cubero
Mathematics NDSU Dept.# 2750
PO BOX 6050
Fargo, ND 58108-6050

North Dakota Mathematics Talent Search Questions (2013-2014)

• Find z/(y+z), given that z/(x+y)=a and y/(x+z)=b.

• We write all the irreducible fractions p/q, where p,q are positive and q<100 in ascending order of value (for example, 4/9=0.444... comes after 2/5=0.4). Between which two fractions does 5/8 stand?

• The police department of a city is investigating a burglary. There are four suspects, A, B, C, and D. The police have established that the following facts are true:
• Fact 1: If both A and B are guilty, then C was their accomplice.
• Fact 2: If A is guilty, then either B or C were accomplices.
• Fact 3: If C is guilty, then D was his accomplice.
• Fact 4: If A is innocent, then D is guilty.
From these facts it can be proven that one of the four suspects is necessarily guilty. Who? (Note: An accomplice is considered guilty).

• Let ABCD be a rhombus (vertices are named consecutively, hence the two vertices closer to A are B and D). Consider all the points M on the plane of the rhombus, verifying that the sum of the angles AMB and CMD equals 180 degrees. Describe how such points are located.

• Prove: 4 a b <= (a + b)² <= (a + b)². Under what circumstances does equality hold?

• A family of four (father, mother, son, and daughter) are hiking in the evening and reach a shaky bridge when it is already dark. The bridge can hold no more than two people at a time, and those crossing need to have the lantern with them. There is just one lantern for the whole family. Suppose that it takes the son 1 minute to cross the bridge, the daughter 3 minutes, the father 8 minutes and the mother 10 minutes. Can the entire family cross the bridge in 20 minutes? If so, how? (When any two persons cross the bridge, their speed is equal to that of the slower one).

 Address for US Postal Services Letters / Billing / small PO Box type packages etc. :NDSU Mathematics Dept #2750 Attn: Melanie PO Box 6050 Fargo ND 58108-6050 Address for UPS / Fed Ex / Large packages:Department of Mathematics - #2750 Attn: Melanie 1210 Albrecht Boulevard Minard 412 North Dakota State University Fargo ND 58102 Tel: 701.231.8171 Fax: 701.231.7598 Email: ndsu.math@ndsu.edu Office Hours: Monday - Friday 8:00 - 5:00

### Previous Years

Student Focused. Land Grant. Research University.