Is this biology?

A practice problem for Saudi Biology Olympiad

The problem

A scientist has isolated five different peptides (1 to 5) containing five amino acids (named A, B, C, D, E). He determined the mass and the sequence of each peptide. The data which he obtained is shown on the table below

Peptide	Amino Acids Sequence	Mass (Da)
1	BCDACCDEDCB	966
2	ABBCAEEDECB	1099
3	BACDAEAEECA	1357
4	CACADBACAEB	1279
5	EDDCABBCCEE	1014

The mass of individual amino acids are shown in the table below

Amino acids	Mass (Da)	Amino acids	Mass (Da)
Alanine	89	Leucine	131
Arginine	174	Lysine	146
Asparagine	132	Methionine	149
Aspartic Acid	133	Phenylalanine	165
Cysteine	121	Proline	115
Glutamic Acid	147	Serine	105
Glutamine	146	Threonine	119
Glycine	75	Tryptophan	204
Histidine	155	Tyrosine	181
Isoleucine	131	Valine	117

Note: The mass of a water molecule is 18 Da.

Indicate if each of the following statements is True or False.

A. Amino acid named C is serine B. Amino acid named A is tyrosine C. Amino acid named E is cysteine D. Amino acid named B is glycine

Shorter mathematical simplification

Given these equations:

$A + 2B + 4C + 3D + E = 1046$
$2A + 3B + 2C + D + 3E = 1139$
$3A + 1B + 2C + D + 3E = 1397$
$4A + 2B + 3C + D + E = 1319$
$A + 2B + 4C + 2D + 3E = 1054$

Indicate if each of the following statements is True or False.

$C = 105$
$A = 181$
$E = 121$
$B = 75$

Solutions

Programming:
- Gaussian elimination in $O(n^3)$, so it runs in less than $0.0001$
- Bruteforce in $O(20^5 \cdot log)$ which runs in less than 2 seconds.
Math:
- System of equations :))

Bruteforce Solution

We would use recursion, you can download the C++ code from here Input:

5
BCDACCDEDCB 966
ABBCAEEDECB 1099
BACDAEAEECA 1357
CACADBACAEB 1279
EDDCABBCCEE 1014
4
C serine
A trysine
E cysteine
B glycine

Output:

A: tryptophan
B: glycine
C: alanine
D: serine
E: cystine