Question 2.6.21: A 10 × 10 table consists of positive integers such that for ......

Mathematical Olympiads, Problems and Solutions from Around the World, 1996-1997 [1513040]

A 10 × 10 table consists of positive integers such that for every five rows and five columns, the sum of the numbers at their intersections is even. Prove that all of the integers in the table are even.

The blue check mark means that this solution has been answered and checked by an expert. This guarantees that the final answer is accurate.
Learn more on how we answer questions.

We denote the first five entries in a row as the “head” of that row. We first show that the sum of each head is even. We are given that the sum of any five heads is even; by subtracting two such sums overlapping in four heads, we deduce that the sum of any two heads is even. Now subtracting two such relations from a sum of five heads, we determine that the sum of any head is even.

By a similar argument, the sum of any five entries in a row is even.
By the same argument as above, we deduce that each entry is even.

Related Answered Questions

Question: 2.6.27

The positive integers 1, 2, . . . , n² are placed in some fashion in the squares of an n × n table. As each number is placed in a square, the sum of the numbers already placed in the row and column containing that square is written on a blackboard. Give an arrangement of the numbers that minimizes ...

Verified Answer:

Rather than describe the arrangement, we demonstra...

Question: 2.6.26

A set of geometric figures consists of red equilateral triangles and blue quadrilaterals with all angles greater than 80° and less than 100°. A convex polygon with all of its angles greater than 60° is assembled from the figures in the set. Prove that the number of (entirely) red sides of the ...

Verified Answer:

We first enumerate the ways to decompose various a...

Question: 2.6.25

The positive integers m, n, m, n are written on a blackboard. A generalized Euclidean algorithm is applied to this quadruple as follows: if the numbers x, y, u, v appear on the board and x > y, then x − y, y, u + v, v are written instead; otherwise x, y − x, u, v + u are written instead. The ...

Verified Answer:

Note that

xv + yu

does not change u...

Question: 2.6.24

There are 2000 towns in a country, each pair of which is linked by a road. The Ministry of Reconstruction proposed all of the possible assignments of one-way traffic to each road. The Ministry of Transportation rejected each assignment that did not allow travel from any town to any other town ...

Verified Answer:

We will prove the same statement for n ≥ 6 towns. ...

Question: 2.6.23

In triangle ABC, the angle A is 60°. A point O is taken inside the triangle such that ∠AOB = ∠BOC = 120°. The points D and E are the midpoints of sides AB and AC. Prove that the quadrilateral ADOE is cyclic. ...

Verified Answer:

Since ∠OBA = 60° − ∠OAB = ∠OAC, the triangles OAB ...

Question: 2.6.22

Prove that there are no positive integers a and b such that for each pair p, q of distinct primes greater than 1000, the number ap + bq is also prime. ...

Verified Answer:

Suppose a, b are so chosen, and let m be a prime g...

Question: 2.6.20

Let BD be the bisector of angle B in triangle ABC. The circumcircle of triangle BDC meets AB at E, while the circumcircle of triangle ABD meets BC at F. Prove that AE = CF. ...

Verified Answer:

By power-of-a-point, AE · AB = AD · AC and CF · CB...

Question: 2.6.19

Two players play the following game on a 100 × 100 board. The first player marks a free square, then the second player puts a 1 × 2 domino down covering two free squares, one of which is marked. This continues until one player is unable to move. The first player wins if the entire board is covered ...

Verified Answer:

The first player has a winning strategy. Let us sa...

Question: 2.6.18

Find all quadruples of polynomials P1(x), P2(x), P3(x), P4(x) with real coefficients such that for each quadruple of integers x, y, z, t such that xy − zt = 1, one has P1(x)P2(y) − P3(z)P4(t) = 1. ...

Verified Answer:

If

P_{1}(1)\:=\:0

, then

P_{3...

Question: 2.6.17

The points A′ and C′ are chosen on the diagonal BD of a parallelogram ABCD so that AA′ || CC′. The point K lies on the segment A′C, and the line AK meets CC′ at L. A line parallel to BC is drawn through K, and a line parallel to BD is drawn through C; these meet at M. Prove that D, M, L are ...

Verified Answer:

Let M′ and M′′ be the intersections of DM with MK ...