Blogs about: Pen Solutions Archive
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20S. Sequences of Consecutive Integers
A37 If is a natural number, prove that the number is not a perfect square. A9 Prove that among any ten consecutive positive integers at least one is relatively prime to the product of the others. O51 … more →
Project PEN
20S. Sequences of Consecutive Integers
danielkohen wrote 6 months ago: A37 If is a natural number, prove that the number is not a perfect square. A9 Prove that among any t … more →
19S. Binomial Sum Divisible by Primes
cpohoata wrote 7 months ago: E16 Prove that for any prime in the interval , divides Here is the official solution file (written b … more →
18S.Fractions Mod p and Wolstenholme's Theorem
danielkohen wrote 8 months ago: A23 Prove that if is expressed as a fraction, where is a prime, then divides the numerator. A24 Let … more →
17S. A Hidden Divisibility
cpohoata wrote 9 months ago: A13 Show that for all prime numbers , is an integer. Here is the official solution file: pen17.pdf. … more →
16S. Using Quadratic Residues
siuhochung wrote 11 months ago: C2 The positive integers and are such that the numbers and are both squares of positive integers. Wh … more →
15. Exponential Congruence Sequence
compactorange wrote 12 months ago: D5. Prove that for , D6. Show that, for any fixed integer the sequence is eventually constant. Sorry … more →
14S. Different Approaches to an Intuitive Problem
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gorofir wrote 1 year ago: The fourth problem of the second season of PEN is as follows: N17. Suppose that and are distinct rea … more →
13S Minimum prime divisors
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compactorange wrote 1 year ago: 13. [PEN A14 A71] A14. Let be an integer. Show that does not divide . A71. Determine all integers su … more →
12S A Generalization of an Identity
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compactorange wrote 1 year ago: 12. PEN I10 Show that for all primes , . Here is the official solution file: pen12.pdf You can also … more →
11S Three ways to attack a functional equation
compactorange wrote 1 year ago: 11. PEN K11 (Canada 2002) Find all functions such that for all : Here, denote the set of all nonnega … more →
10S Partitions
compactorange wrote 1 year ago: 10. PEN B6 Consider the set of all five-digit numbers whose decimal representation is a permutation … more →
09S Primitive Roots: Revisited
Yimin Ge wrote 1 year ago: 09. PEN B6 Suppose that does not have a primitive root. Show that for every relatively prime to . He … more →
08S An arithmetic partition
compactorange wrote 1 year ago: 08. PEN O 35 ( Romania TST 1998 ) Let be a prime number and be integers. Prove that is an arithmetic … more →
S07 A combinatorial congruence
compactorange wrote 1 year ago: 07. PEN D2 (Putnam 1991/B4) Suppose that is an odd prime. Prove that Here is the official solution f … more →
S06 A historical divisibility.
compactorange wrote 1 year ago: 06. PEN A3 ( IMO 1988 ) Let and be positive integers such that divides . Show that is the square of … more →
S05 On the monotonicity of the divisor function.
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compactorange wrote 1 year ago: 05a. [Saint-Petersburg 1998] Let denote the number of positive divisors of the number . Prove that t … more →
S04 A hidden symmetry
compactorange wrote 1 year ago: 04. PEN I11 (Korea 2000) Let be a prime number of the form . Show that Here is the official solution … more →
S03 A theorem on sum-free subsets
compactorange wrote 1 year ago: 03. PEN O53 (Schur Theorem) Suppose the set is partitioned into disjoint subsets . Show that if then … more →
S02 Three ways to reach a Diophantine equation
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compactorange wrote 1 year ago: 02. PEN H15 (Balkan Mathematical Olympiad 1998) Prove that there are no integers and satisfying . He … more →
