MathDB

Round 1
p1. The Queen of Bees invented a new language for her hive. The alphabet has only

6

letters: A, C, E, N, R, T; however, the alphabetic order is different than in English. A word is any sequence of

6

different letters. In the dictionary for this language, the word TRANCE immediately follows NECTAR. What is the last word in the dictionary?
p2. Is it possible to solve the equation

\frac{1}{x}= \frac{1}{y} +\frac{1}{z}

with

x,y,z

integers (positive or negative) such that one of the numbers

x,y,z

has one digit, another has two digits, and the remaining one has three digits?
p3. The

10,000

dots in a

100\times 100

square grid are all colored blue. Rekha can paint some of them red, but there must always be a blue dot on the line segment between any two red dots. What is the largest number of dots she can color red? The picture shows a possible coloring for a

5\times 7

grid. https://cdn.artofproblemsolving.com/attachments/0/6/795f5ab879938ed2a4c8844092b873fb8589f8.jpg
p4. Six flies rest on a table. You have a swatter with a checkerboard pattern, much larger than the table. Show that there is always a way to position and orient the swatter to kill at least five of the flies. Each fly is much smaller than a swatter square and is killed if any portion of a black square hits any part of the fly.
p5. Maryam writes all the numbers

1-81

in the cells of a

9\times 9

table. Tian calculates the product of the numbers in each of the nine rows, and Olga calculates the product of the numbers in every column. Could Tian's and Olga's lists of nine products be identical?
Round 2
p6. A set of points in the plane is epic if, for every way of coloring the points red or blue, it is possible to draw two lines such that each blue point is on a line, but none of the red points are. The figure shows a particular set of

4

points and demonstrates that it is epic. What is the maximum possible size of an epic set? https://cdn.artofproblemsolving.com/attachments/e/f/44fd1679c520bdc55c78603190409222d0b721.jpg
p7. Froggy Chess is a game played on a pond with lily pads. First Judit places a frog on a pad of her choice, then Magnus places a frog on a different pad of his choice. After that, they alternate turns, with Judit moving first. Each player, on his or her turn, selects either of the two frogs and another lily pad where that frog must jump. The jump must reduce the distance between the frogs (all distances between the lily pads are different), but both frogs cannot end up on the same lily pad. Whoever cannot make a move loses. The picture below shows the jumps permitted in a particular situation. Who wins the game if there are

2017

lily pads? https://cdn.artofproblemsolving.com/attachments/a/9/1a26e046a2a614a663f9d317363aac61654684.jpg
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.

Problem Statement