FOR ONE school year, college students who had taken a problem-solving class taught by former Ateneo president Fr. Bienvenido Nebres, SJ, and myself, volunteered to coach talented elementary and secondary students in mathematics.
Lessons were not confined to the regular math textbook but focused on higher-level problems to help the students develop the discipline and determination to tackle nonroutine questions.
The students deemed the most creative and most participative during the training received medals.
The finale of the training was the Ateneo Math Olympiad held in January and February, respectively, for the high school and grade school students, who were given three hours to solve three complex problems, a la International Math Olympiad.
Speed was not the essential factor but perseverance and creativity, with a touch of ingenuity and elegance of solution.
Nonroutine problems
Grade school students faced problems like these:
There are four taps: A, B, C and D. Taps A and B will supply water to a jar while taps C and D will let water out from the jar. When only tap A or tap B is on, it takes four or eight hours, respectively, to fill the empty jar. If only tap C or tap D is on, then it takes six or 10 hours, respectively, to empty the full jar. The four taps are switched on one at a time, for an hour each time, in the order A, C, B, D, A, C… If the jar is initially 1/8 full, how long before it overflows?
A 10-digit number abcdefghij exists, such that each digit is different and that the two-digit number ab is a multiple of 2, the three-digit number abc is a multiple of 3, the four-digit number abcd is a multiple of 4… and so on, until the 10-digit number abcdefghij is a multiple of 10. What is the number?
No student merited a gold medal, but Jose Martin Macapanpan was awarded a silver. The bronze medalists were Jose Enrique Guevarra, Joshua Daniel Dabalos, Pocholo Resurreccion, Gabriel Fadri, Jake Belmonte Lim, Mark Gabriel Sandoval, Renaeus Torres and Michael Alvarez.
The high school students tackled sample problems such as:
Juan left Town A at x minutes past 6 p.m. and reached Town B at y minutes past 6 p.m. on the same day. He noticed that at both the start and the end of the trip, the minute hand made the same angle of 110 degrees with the hour hand on his watch. How many minutes did it take Juan to travel from Town A to Town B?
How many four-digit numbers have a remainder 2 when divided by 4, remainder 3 when divided by 5, remainder 4 when divided by 6 and remainder 5 when divided by 7?
The high school gold medalists were Jo Adrian del Mundo, Lorenzo Quiogue and Immanuel Gabriel Sin. Silver medalists were Frederick Corpuz, Vince Walter Domasig, Rafael Abaño IV, Rafael Dimaano, Irvin Embalsado and David Cuajunco. The bronze medalists were Martin Rodolfo Santiago, Jason Bautista, Edward John Ofilada, Daniel Luis Bautista, Hanz Manguiat and Paul Songalia.
Contest-style problems
Since the Ateneo Math Olympiad finals were held in front of other students, contestants also had a group relay round, where they not only had to answer questions in time but also work out a strategy to win the most points. Time limits, measured in seconds or minutes, were set.
Sample grade school questions were:
(Easy) What is the sum of the smallest two-digit prime and largest two-digit composite number?
(Moderate) In basketball, a shot can either be worth
two points or three points. If the Ateneo Blue Eagles scored
77 points and made exactly
33 shots, how many two-point shots did they make?
(Difficult) A father is now
50 years old while his son is
14 years old. How many years ago was the father five times as old as his son? (Answers are at the end of this column.)
The grade school champions were Carlos Bello, Marty Aguiluz, Mikko Vitug, Jericho Carlo Violago, Juan Aurelio, Enrico Ungson, Francis Ortega and Aldrin Benedict Camba. In second place were Zachary Claude Orfiano, Nicholas Barlisan, Moro Lorenzo, Karlo Marquez, Raymart Santamaria, Jose Martin Macapanpan, Ryan Delfin Encarnacion and Jose Carlo Arangurer. In third place were Miguel Carlos Liamzon, Mikael Sulit, Renaeus Torres, Lorenzo Thomas Lazaro, Yves Gabriel Lazaro, Elian Sinson and Vincent Jamias.
High school questions included:
(Easy) Ana flips three coins and a six-sided die. What is the probability that the number of heads equals the number in the die?
(Moderate) Mike is thinking of three positive numbers. The second number is the square of the first number and the third number is 990 greater than the first number. The sum of the three numbers is 2013. Find the second number.
(Difficult) The area of square ABCD is 324 square units. The midpoints of its sides are joined to form the square EFGH. The midpoints of its sides are J, K, L and M. Determine the area of pentagon JEKLM.
The champion group in high school included Lorenzo Quiogue, Ernst Jeng, Jed Golez, Manuel Montoya, Philip Libunao, Glad Danguilan, CJ Magno and Angelo Payba. The group of Irwin Embalsado, Robert Alagar, Justine Lumbo, Luigi Mallare, Patrick Eala, Ruzzel Castillo, Reuben Antonio and Kyle Nathan Lim ranked second. In third place was the group of Symone Xavier Candelaria, Kirsten Magnus Untal, Paulo Songalia, Paco Rivera, Antonio G. Callanta and Edward Joseph Ofilada.
The Ateneo Math Olympiad was conducted in cooperation with the Ateneo Math Society, Ateneo Grade School, high school, college math faculty and the Ateneo Problem Solving Group. Scholastic Philippines gave winners books and gift certificates.
Answers: Grade school: (Easy) 110, (moderate) 22, (difficult) 5 years ago. High school: (Easy) 7/48, (moderate) 961, (difficult) 101.25 square units.