The Ateneo Math Olympiad (AMO) was launched 10 years ago to help train talented elementary and secondary students in complex problem solving.
Last week, we looked at sample questions and problems for grade school. Now let us discuss sample problems for high school.
Written round
In the written round, students have three hours to solve three nonroutine problems as best they can, showing detailed explanations and solutions. Patterned after the problems in the International Math Olympiad (IMO), the problems here require not so much speed, but facility with proving and problem-solving techniques.
Following are sample problems:
(First and second year) The price of a second-hand computer is displayed (in pesos) on four cards on a stand. Each card shows one digit. If the card with the thousands digit is blown away, the apparent price of the computer would drop to 1/49 of the intended value. What number is on the card (that was lost)?
(Third and fourth year) Paul and Andrew have a whole number of dollars each. Paul says to Andrew, “If you give me $3, I will have n times as much as you.” Andrew says to Paul, “If you give me n dollars, then I will have three times as much as you.” Given that all these statements are true and that n is a positive integer, what are the possible values of n?
Rafael Dimaano, Lorenzo Quiogue and Immanuel Gabriel Sin bagged the gold. Ralph Antonio, Chino Cornel, Jo Adrian del Mundo, Irvin Embalsado, Hans Rhenzo Manguiat and John Aris Reyes won silver. Bryan Emmanuel Agosto, Justin Jared Aguinaldo, Alfredo Gabriel Aldaba, Jose Daniel Berba, Gino Bermudez, David Cuajunco, Patrick Eala and Paulo Songalia got bronze.
Oral round
In the oral round, contestants are required to answer questions within a specified time period, usually seconds or minutes. Sample questions include:
(20 seconds) Five coins are tossed simultaneously. What is the probability that there are more heads than tails?
Answer: ½
(20 seconds) Given two similar triangles, the area of the larger triangle is 16 times the area of the smaller triangle. Find the ratio in simplest form of the perimeter of the larger triangle to the smaller triangle.
Answer: 4:1
(20 seconds) The sum of the interior angles of an n-gon equals the sum of the interior angles of a pentagon plus the sum of the interior angles of an octagon. What is n?
Answer: 11
(45 seconds) Find the next number in the sequence: 2. 3. 5. 14. 69. ___ .
Answer: 965
(45 seconds) You want to buy two scoops in an ice cream store. Five regular flavors cost P15 per scoop and six special flavors cost P20 per scoop. How many combinations of two scoops can you make if you only have P37 and you want the two scoops to be of different flavors?
Answer: 40
(45 seconds) A four-by-four-by-four cube is made from exactly 64 one-by- one-by-one cubes and the six faces of the four-by-four-by-four cube are painted red. If the larger cube is disassembled into the 64 smaller cubes, what fraction of the 64 smaller cubes have exactly two faces painted red?
Answer: 3/8
(60 seconds) Grass is growing at a constant rate in a pasture. It takes eight goats 20 days or 14 goats 10 days to consume the grass. X goats have been eating the grass for four days before six more goats joined them to consume the grass in two more days. Find x.
Answer: 20
(60 seconds) How many seven-digit numbers (whose digits are distinct and nonzero) are divisible by 25?
Answer: 11760
(60 seconds) How many nonnegative solutions are there to the equation x + y + z = 12 where x, y and z are nonnegative integers?
Answer: 91
The team of Quiogue, Jeosiah Cainglet, Darwin Tesion, Enrique Miguel Bautista, Juztin Roel Alvarez, Diego Ortiz, Raul Alberto Arellano III and Pio Lorenzo Valdez was first.
The team of Berba, Evan Merel Escalaw, Earl Roy del Rosario, Alexander Julius Sunodan, Kevinno Zaño and Rufino Valera III ranked second.
The team of Manguiat, Joseph John Milla, Aaron Clyde Dublin, Gabriel-Ross Santiago, del Mundo, Elouie Ryan Galan and Aguinaldo was third.
We acknowledge the efforts of the Ateneo grade school, high school and college mathematics departments, together with the Ateneo Math Society and the Ateneo Problem Solvers Group, in making the AMO a reality.
Scholastic Book Fairs, which provided tokens to winners, has been our staunch partner for a decade.
E-mail the author at blessbook@yahoo.com.