Conversely, a minor error that leads to an incorrect answer will not substantially reduce your credit. A Suppose three integers are added pairwise, and the results are , , and Find the integers if they exist. The convex gon P is partitioned into finitely many triangles so that each side of P belongs to one triangle, and no vertex of any triangle lies in an inner point of a side of another triangle.

Can the map thus formed be colored in two colors, red and blue, so that no triangles which share a side are the same color and all the triangles along the perimeter of P are blue? Each corner of the initial cube contains a number, and the numbers include and An averaging operation creates the second cube by replacing each number by the mean of the three numbers one edge away from it. We then repeat the averaging operation obtaining the third cube, etc. A Can it so happen that all the numbers of the th cube coincide with the corresponding numbers of the initial cube? If yes, find all possible arrays of the corner numbers.

B Can it so happen that all the numbers of the th cube coincide with the corresponding numbers of the initial cube? A Given 11 real numbers in the form of infinite decimal representations, prove that there are two of them which coincide in infinitely many decimal places. Infinitely many circular disks of radius 1 are given inside a bounded figure in the plane.

Prove that there is a circular disk of radius 0. Problems 1,2 and 4 were created by Alexander Soifer for this Olympiad. Problem 3 was adapted by Soifer from Russian mathematical folklore.

## Winning solutions (math olympiad problem book) - PDF Free Download

On each turn, they must add at least one pebble and may not add more pebbles than there are already in the pile. The player who makes the pile consist of exactly pebbles wins. Find a strategy that allows Fred or Barney to win regardless of how the other may play. Oh, there is originally just one pebble in the pile, and Fred goes first. Secrets of Tables Each cell of an 8 x 8 chessboard is filled with a 0 or 1. Prove that if we compute sums of numbers in each row, each column, and each of the two diagonals, we will get at least three equal sums.

### Stevens Math Olympiad

More Secrets of Tables a. Prove that no matter how each cell of a 5 x 41 table is filled with a 0 or 1, one can choose 3 rows and 3 columns which intersect in 9 cells filled with identical numbers. Prove that 41 in part a is the lowest possible; i. Looking for the Positive A number is placed in each angle of a regular gon so that the sum of any 10 consecutive numbers is positive. The symbol [ c ] stands for the largest whole number not exceeding c.

More Secrets of Tables a. Prove that no matter how each cell of a 5 x 41 table is filled with a 0 or 1, one can choose 3 rows and 3 columns which intersect in 9 cells filled with identical numbers.

## Past Problems & Solutions

Prove that 41 in part a is the lowest possible; i. Looking for the Positive A number is placed in each angle of a regular gon so that the sum of any 10 consecutive numbers is positive. The symbol [ c ] stands for the largest whole number not exceeding c. Problem 1 was created by Prof. Sekanina from Czechoslovakia. Dynkin, et al. On a Collision Course 93 identical balls move along a line, 59 of them from left to right with speed v; the remaining 34 balls move from right to left toward the first group of balls with speed w.

When two balls collide, they exchange their speeds and direction of motion. What is the total number of collisions that will occur? A Horse! My Kingdom for a Horse! The Good, the Bad, and the Ugly divide a pile of gold and a horse. The pile consists of gold coins, and they draw in turn 1, 2, or 3 coins from the pile.

The Good gets the first turn, the Bad draws second, and the Ugly takes last.

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The Ugly does not trust the Bad and never draws the same number of coins as the Bad has drawn immediately before him. The one who takes the last coin is left behind, while the two others cross the prairie together on horseback. Who can guarantee himself the ride out on the horse regardless of how the others draw the coins?

How can he do this? Chess Tournament Professional and amateur players, n in all, participated in a round robin chess tournament. Upon its completion, it was observed that each player earned half of his total score in games against the amateurs. Round robin is a tournament where every pair of participants plays each other once. A party is attended by at least three mathematicians, and every pair of attendees has a coauthor-chain at the party.

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Cover-up Can a square of area be covered bt squares of area 5 each? Tea Time Is it possible to arrange the numbers 1 to in a 10X10 grid so that the entries of any T-shaped figure consisting of 4 unit squares of the grid sum up to an even number? One L of a Grid What is the minimum number of squares to be colored red in a 10X10 grid so that any L-shaped figure consisting of 4 unit squares of the grid contains at least 2 red squares?

John subsequently spent two years at Amazon, first serving as vice president and technical assistant to Amazon founder Jeff Bezos. A lifelong math enthusiast, John won a silver medal for the United States at the 27th International Mathematical Olympiad.

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He earned a B. In , John was honored by the Academy of Achievement for being a pioneer in technology and investment management.

### 49 Austrian Mathematical Olympiad 2018

Below are some suggested materials for students looking to do preparation problems ahead of the Stevens Math Olympiad. We gratefully acknowledge financial support from the Mathematical Association of America and from the Alfred P. Sloan Foundation. Add to Calendar.

The Olympiad has the following goals: To stimulate enthusiasm and a love for mathematics To introduce important mathematical concepts to students To strengthen mathematical intuition and creativity To attract bright middle and high school students to Stevens Institute of Technology Students were offered 15 problems to solve in five divisions: grades , grades , grades , grades and grades Math Olympiad testing begins for students in grades Students in grades should check in between a.