MathDB

5

Part of 2015 Caucasus Mathematical Olympiad

Problems(5)

palindromes from 10,000 - 999,999 with even sum of digits, and how many

Source: Caucasus 2015 9.5

4/26/2019
Let's call a natural number a palindrome, the decimal notation of which is equally readable from left to right and right to left (decimal notation cannot start from zero; for example, the number 12211221 is a palindrome, but the numbers 1231,12121231, 1212 and 10101010 are not). Which palindromes among the numbers from 10,00010,000 to 999,999999,999 have an odd sum of digits, which have an one even, and how many times are the ones with odd sum more than the ones with the even sum?
number theorycombinatoricspalindromessum of digits
smallest no of 3-cell corners in a 6x6 square grid so that

Source: Caucasus 2015 7.5

4/26/2019
What is the smallest number of 33-cell corners needed to be painted in a 6×66\times 6 square so that it was impossible to paint more than one corner of it? (The painted corners should not overlap.)
combinatoricscombinatorial geometrysquare grid
game with 300 coins, 3 player, only two may win, can it happen?

Source: Caucasus 2015 8.5

4/26/2019
On the table are 300300 coins. Petya, Vasya and Tolya play the next game. They go in turn in the following order: Petya, Vasya, Tolya, Petya. Vasya, Tolya, etc. In one move, Petya can take 1,2,31, 2, 3, or 44 coins from the table, Vasya, 11 or 22 coins, and Tolya, too, 11 or 22 coins. Can Vasya and Tolya agree so that, as if Petya were playing, one of them two will take the last coin off the table?
combinatoricsgame strategygame
if <C_1MA_1=<ABC then C_1 K=A_1L (Caucasus 2015 for the 10th grade)

Source: I Caucasus 2015 10.5

9/6/2018
Let AA1AA_1 and CC1CC_1 be the altitudes of the acute-angled triangle ABCABC. Let K,LK,L and MM be the midpoints of the sides AB,BCAB,BC and CACA respectively. Prove that if ∠C1MA1=∠ABC\angle C_1MA_1 =\angle ABC, then C1K=A1LC_1 K = A_1L.
geometryaltitudesmidpointsequal anglesequal segments
a,b,c are not triangle sidelengths when c is a perfect square, a,b>1000

Source: Caucasus 2015 11.5

4/26/2019
Are there natural a,b>1000a, b >1000 , such that for any cc that is a perfect square, the three numbers a,ba, b and cc are not the lengths of the sides of a triangle?
number theoryPerfect SquareSides of a triangle