MathDB

6

Part of 2014 China Team Selection Test

Problems(3)

2x2 square diagonal same sum

Source: 2014 China TST Day 2 Q6

3/18/2014
Let n≥2n\ge 2 be a positive integer. Fill up a n×nn\times n table with the numbers 1,2,...,n21,2,...,n^2 exactly once each. Two cells are termed adjacent if they have a common edge. It is known that for any two adjacent cells, the numbers they contain differ by at most nn. Show that there exist a 2×22\times 2 square of adjacent cells such that the diagonally opposite pairs sum to the same number.
IMO Shortlistcombinatorics proposedcombinatorics
Difference between number of factors of different types.

Source: 2014 China TST 2 Day 2 Q6

3/20/2014
Let kk be a fixed even positive integer, NN is the product of kk distinct primes p1,...,pkp_1,...,p_k, a,ba,b are two positive integers, a,b≤Na,b\leq N. Denote S1={d∣S_1=\{d| d∣N,a≤d≤b,dd|N, a\leq d\leq b, d has even number of prime factors}\}, S2={d∣S_2=\{d| d∣N,a≤d≤b,dd|N, a\leq d\leq b, d has odd number of prime factors}\}, Prove: ∣S1∣−∣S2∣≤Ckk2|S_1|-|S_2|\leq C^{\frac{k}{2}}_k
number theory proposednumber theoryposet
Representations of k as product of integers is bounded

Source: 2014 China TST 3 Day 2 Q6

4/5/2014
For positive integer k>1k>1, let f(k)f(k) be the number of ways of factoring kk into product of positive integers greater than 11 (The order of factors are not countered, for example f(12)=4f(12)=4, as 1212 can be factored in these 44 ways: 12,2⋅6,3⋅4,2⋅2⋅312,2\cdot 6,3\cdot 4, 2\cdot 2\cdot 3. Prove: If nn is a positive integer greater than 11, pp is a prime factor of nn, then f(n)≤npf(n)\leq \frac{n}{p}
inductionnumber theory proposednumber theory