ATOMIC GENETIC CODE: PROBLEMS FOR 46th INTERNATIONAL MATHEMATICS OLYMPIAD IN MEXICO

Problem 1

Magic codes 19 and 7

In group of all natural numbers from X to Y there are two codes which interconnect all those numbers. Those are codes A and B.

Formula of codes A and B

{[SA(R1,2,3,n) x B] - [SB(R1,2,3,n) x A] + (AxB)} = (AxBxA);

A = ?

B = ?

SA, SB = Groups of AB numbers in group of all natural numbers from X to Y

R1,2,3,n = Natural numbers from X to Y

.......

Help in resolving this problem:

Example 1

SA(R1,2,3,n)

A= 6; R = 114;

S6(114) = (109+110+111+112+113+114) ) = 669;

.........

Example 2

SA(R1,2,3,n)

A= 25; R = 233;

S25(233) = (209+210+211+212+213+214+215+216+217+218+219+220+221+222+223+

+224+225+226+227+228+229+230+231+232+233 ) = 5525;

........

Example 3

SB(R1,2,3,n)

B= 8; R = 98;

S8(98) = (91+92+93+94+95+96+97+98) = 756;

Etc.

.................................................................

SOLUTION:

{[SA(R1,2,3,n) x B] - [SB(R1,2,3,n) x A] + (AxB)} = (AxBxA);

A = 7; B = 19;

............

{[S7(R1,2,3,n) x 19] - [S19(R1,2,3,n) x 7] + (7x19)} = (7x19x7);

...........

Example 1

R = 35;

{[S7(35) x 19] - [S19(35) x 7] + (7x19)} = (7x19x7);

S7(35) = (29+30+31+32+33+34+35) = 224;

S19(35) = (17+18+19+20+21+22+23+24+25+26+27+28+29+30+31+32+33+34+35) = 494;

(224 x 19) – (494 x 7) + (7 x 19) = (7 x 19 x 7);

...............

Example 2

R = 114;

{[S7(114) x 19] - [S19(114) x 7] + (7x19)} = (7x19x7);

S7(114) = (108+109+110+111+112+113+114) = 777;

S19(114) = (96+97+98...+114) = 1995;

(777 x 19) – (1995 x 7) + (7 x 19) = (7 x 19 x 7);

..............

Example 3

R = 1070;

{[S7(1070) x 19] - [S19(1070) x 7] + (7x19)} = (7x19x7);

S7(1070) = (1064+1065+1066...+1070) = 7469;

S19(1070) = (1052+1053+1054...+1070) = 20159;

(7469 x 19) – (20159 x 7) + (7 x 19) = (7 x 19 x 7);

...............

Example 4

R = Y;

{[S7(Y) x 19] - [S19(Y) x 7] + (7x19)} = (7x19x7);

*********

PROBLEM 2

Magic codes 19 and 7 in magic square

8	9	15	24	25
26	30	36	43	44
45	46	47	49	59
60	61	62	63	64
65	79	87	88	93

In this square, a group of numbers which are interconnected by magic codes 19 and 7 need to be found.

SOLUTION:

Example 1

Groups of even and odd numbers in Square

8	9	15	24	25
26	30	36	43	44
45	46	47	49	59
60	61	62	63	64
65	79	87	88	93

Even numbers = (8 + 24 + 26 + 30 + 36 + 44 + 46 + 60 + 62 + 64 + 88) = 488;

Odd numbers = (9+15+25+43+45+47+49+59+61+63+65+79+87+93) = 740;

Analog code of number 488 is number 884;

Analog code of number 740 is number 047;

(884+47) = (AxBxA) = (7x19x7);

............

Example 2

Groups of numbers that are distributed in the squares with even and odd numbers

8	9	15	24	25
26	30	36	43	44
45	46	47	49	59
60	61	62	63	64
65	79	87	88	93

Group of numbers that are distributed in the squares with even number

(2,4,6,8 etc.) = (9+24+26+36+ 44+46+49+60+62+64+79+88) = 587;

Group of numbers that are distributed in the squares with odd number

(1,3,5,7,9 etc.) = (8+15+25+30+43+45+47+59+61+63+65+87+ 93) = 641;

Analog code of number 587 is number 785;

Analog code of number 641 is number 146;

(785+146) = (AxBxA) = (7x19x7);

............

Example 3

Numbers in even and odd columns

8	9	15	24	25
26	30	36	43	44
45	46	47	49	59
60	61	62	63	64
65	79	87	88	93

Numbers in even columns ( 2 and 4) = (9+24+30+43+46+49+61+63+79+88) = 492;

Numbers in odd columns (1, 3, 5) = (8+15+25+26+36+44+45+47+59+60+62+64+65+87+93) =736;

Analog code of number 492 is number 294;

Analog code of number 736 is number 637;

(294+637) = (AxBxA) = (7x19x7);

............

Example 4

Outer and inner numbers in quadrant

8	9	15	24	25
26	30	36	43	44
45	46	47	49	59
60	61	62	63	64
65	79	87	88	93

Outer numbers in quadrant = (8+9+15+24+25+26+44+45+59+60+64+65+79+87+88+93) = 791;

Inner numbers in quadrant = (30+36+43+46+47+49+61+62+63) = 437;

Analog code of number 791 is number 197;

Analog code of number 437 is number 734;

(197+734) = (AxBxA) = (7x19x7);

............

Example 5

Numbers in columns

8	9	15	24	25
26	30	36	43	44
45	46	47	49	59
60	61	62	63	64
65	79	87	88	93

Columns 1,2,4,5 = (8+9+24+25+26+30+43+44+45+46+49+59+60+61+63+64+65+79+88+93)=981;

Column 3 = (15+36+47+62+87) = 247;

Analog code of number 981 is number 189;

Analog code of number 247 is number 742;

(189+742) = (AxBxA) = (7x19x7);

............

Example 6

Numbers in diagonal “A”

8	9	15	24	25
26	30	36	43	44
45	46	47	49	59
60	61	62	63	64
65	79	87	88	93

Diagonal “A” = (8+30+47+63+93) = 241;

Other numbers = (9+15+24+25+26+36+43+44+45+46+49+59+60+61+62+64+65+79+87+88) = 987;

Analog code of number 241 is number 142;

Analog code of number 987 is number 789;

(142+789) = (AxBxA) = (7x19x7);

............

Example 7

Groups of numbers

8	9	15	24	25
26	30	36	43	44
45	46	47	49	59
60	61	62	63	64
65	79	87	88	93

Group 1 = (8+9+15+36+47+62+87+88+93) = 445;

Group 2 =(24+25+26+30+43+44+45+46+49+59+60+61+63+64+65+79) = 783;

Analog code of number 445 is number 544;

Analog code of number 783 is number 387;

(544+387) = (AxBxA) = (7x19x7);

............

Example 8

Groups of numbers

8	9	15	24	25
26	30	36	43	44
45	46	47	49	59
60	61	62	63	64
65	79	87	88	93

Group 1 = (8+9+15+24+25+44+59+64+65+79+87+88+93) = 660;

Group 2 = (26+30+36+43+45+46+47+49+60+61+62+63) = 568;

Analog code of number 660 is number 066;

Analog code of number 568 is number 865;

(66+865) = (AxBxA) = (7x19x7);

............

Example 9

Groups of numbers

8	9	15	24	25
26	30	36	43	44
45	46	47	49	59
60	61	62	63	64
65	79	87	88	93

Group 1 = (8+25+30+36+43+45+46+47+49+59+61+62+63+65+93) = 732;

Group 2 = (9+15+24+26+44+60+64+79+87+88) = 496;

Analog code of number 732 is number 237;

Analog code of number 496 is number 694;

(237+694) = (AxBxA) = (7x19x7);

............

Example 10

Groups of numbers

8	9	15	24	25
26	30	36	43	44
45	46	47	49	59
60	61	62	63	64
65	79	87	88	93

Group 1 = (8+25+30+43+47+61+63+65+93) = 435;

Group 2 = (9+15+24+26+36+44+45+46+49+59+60+62+64+79+87+88) = 793;

Analog code of number 435 is number 534;

Analog code of number 793 is number 397;

(534+397) = (AxBxA) = (7x19x7);

............

Example 11

Groups of numbers

8	9	15	24	25
26	30	36	43	44
45	46	47	49	59
60	61	62	63	64
65	79	87	88	93

Group 1 = (8+9+15+24+25+30+43+61+63+65+79+87+88+93) = 690;

Group 2 = (26+36+44+45+46+47+49+59+60+62+64) = 538;

Analog code of number 690 is number 096;

Analog code of number 538 is number 835;

(96+835) = (AxBxA) = (7x19x7);

............

Example 12

Groups of numbers

8	9	15	24	25
26	30	36	43	44
45	46	47	49	59
60	61	62	63	64
65	79	87	88	93

Group 1 = (8+9+15+25+26+36+44+45+47+59+60+62+64+65+87+88+93) = 833;

Group 2 = (24+30+43+46+49+61+63+79) = 395;

Analog code of number 833 is number 338;

Analog code of number 395 is number 593;

(338+593) = (AxBxA) = (7x19x7);

............

Example 13

Groups of numbers

8	9	15	24	25
26	30	36	43	44
45	46	47	49	59
60	61	62	63	64
65	79	87	88	93

Group 1 = (8+25+30+36+43+46+49+61+62+63+65+93) = 581;

Group 2 = (9+15+24+26+44+45+47+59+60+64+79+87+88) = 647;

Analog code of number 581 is number 185;

Analog code of number 647 is number 746;

(185+746) = (AxBxA) = (7x19x7);

............

Example 13

Groups of numbers

8	9	15	24
26	30	36	43
45	46	47	49
60	61	62	63
65	79	87	88

Group 1 = (8+9+24+25+30+43+46+49+61+63+65+79+88+93) = 683;

Group 2 = (15+26+36+44+45+47+59+60+62+64+87) = 545;

Analog code of number 683 is number 386;

Analog code of number 545 is number 545;

(386+545) = (AxBxA) = (7x19x7);

etc.

PROBLEM 3

Codes 19 and 7 in

equation with 19 unknown values

(N1+N2+N3+N4+N5+N6+N7+N8+N9+N10+N11+N12+N13+N14+N15 +

+ N16+N17+N18+N19) = Y;

Try to find unknown values using codes 19 and 7.

Solution

In this equation we have 1 group with 19 unknown values and 13 groups with 7 unknown values:

1 and 19 è 119;

13 and 07 = 1307;

Y = (119 + 1307) è 1426;

(A1+A2+A3+A4+A5+A6+A7+A8+A9+A10+A11+A12+A13+A14+15 +

+ A16+A17+A18+A19) =1426;

Groups with 7 numbers

G1 = (N1,N2,N3,N4,N5,N6,N7);

G2 = (N2,N3,N4,N5,N6,N7,N8);

G3 = (N3,N4,N5,N6,N7,N8,N9);

G4 = (N4,N5,N6,N7,N8,N9,N10);

G5 = (N5,N6,N7,N8,N9,N10,N11);

G6 = (N6,N7,N8,N9,N10,N11,N12);

G7 = (N7,N8,N9,N10,N11,N12,N13);

G8 = (N8,N9,N10,N11,N12,N13,N14);

G9 = (N9,N10,N11,N12,N13,N14,N15);

G10 = (N10,N11,N12,N13,N14,N15,N16);

G 11 = (N11,N12,N13,N14,N15,N16,N17);

G12 = (N12,N13,N14,N15,N16,N17,N18);

G13 = (N13,N14,N15,N16,N17,N18,N19);

(G1+G2+G3+G4+G5+G6+G7+G8+G9+G10+G11+G12+G13) =1426;

Solution:

G1 = 112; G2=111; G3=122; G4=108; G5=113; G6 =114; G7=116; G8=90;

G9=112; G10=99; G11=95; G12=117; G13=117;

(112+111+122+108+113+114+116+90+112+99+95+117+117) = 1426;

G1 = (N1+N2+N3+N4+N5+N6+N7) = 112;

G2 = (N2+N3+N4+N5+N6+N7+N8) = 111;

G3 = (N3+N4+N5+N6+N7+N8+N9) = 122;

G4 = (N4+N5+N6+N7+N8+N9+N10) = 108;

G5 = (N5+N6+N7+N8+N9+N10+N11) = 113;

G6 = (N6+N7+N8+N9+N10+N11+N12) = 114;

G7 = (N7+N8+N9+N10+N11+N12+N13) = 116;

G8 = (N8+N9+N10+N11+N12+N13+N14) = 90;

G9 = (N9+N10+N11+N12+N13+N14+N15) = 112;

G10 = (N10+N11+N12+N13+N14+N15+N16) = 99;

G11 = (N11+N12+N13+N14+N15+N16+N17) = 95;

G12 = (N12+N13+N14+N15+N16+N17+N18) = 117;

G13 = (N13+N14+N15+N16+N17+N18+N19) = 117;

_______________________________________

N1 = 2; N2 = 12; N3 = 24; N4 = 1; N5 = 23; N6 = 23; N7 = 27; N8 = 1;

N9 = 23; N10 = 10; N11 = 6; N12 = 24; N13 = 25; N14 = 1; N15 = 23;

N16 = 10; N17 = 6; N18 = 28; N19 = 24;

G1 = (2 +12 +24 +1+23 + 23 + 27) = 112;

G2 = (12 +24 +1+23 + 23 + 27+1) = 111;

G3 = (24 +1+23 + 23 + 27+1 +23) = 122;

G4 = (1+23 + 23 + 27+1 +23 +10) = 108;

G5 = (23 + 23 + 27+1 +23 +10 +6) = 113;

G6 = (23 + 27+1 +23 +10 +6 +24) = 114;

G7 = (27+1 +23 +10 +6 +24 + 25) = 116;

G8 = (1 +23 +10 +6 +24 + 25 +1) = 90;

G9 = (23 +10 +6 +24 + 25 +1+ 23) = 112;

G10 = (10 +6 +24 + 25 +1+ 23 +10) = 99;

G11 = (6 +24 + 25 +1+ 23 +10 + 6) = 95;

G12 = (24 + 25 +1+ 23 +10 + 6 + 28) = 117;

G13 = (25 +1+ 23 +10 + 6 + 28 + 24) = 117;

............

PROBLEM 4

Try to decode Pascal triangle using the code 101

Row Number


Binomial Expansion
0	Â	Â	Â	Â	Â	1	Â	Â	Â	Â	Â
1	Â	Â	Â	Â	1	Â	1	Â	Â	Â	Â
2	Â	Â	Â	1	Â	2	Â	1	Â	Â	Â
3	Â	Â	1	Â	3	Â	3	Â	1	Â	Â
4	Â	1	Â	4	Â	6	Â	4	Â	1	Â
5	1	Â	5	Â	10	Â	10	Â	5	Â	1

We are looking at rows 0 through 5 of Pascal's triangle! The numbers in the rows of Pascal's triangle are the same as the coefficients generated by raising the binomial (x+y) to a power.

1) Math Forum: Pascal's Triangle (http://mathforum.org/)

SOLUTION

A binomial is a polynomial expression with two terms, such as:

Row Number

Binomial Expansion

Connection

Code 101

101⁰

101

101¹

10201

101²

1030301

101³

104060401

101⁴

10510100501

101⁵

A0 = 1;

..........

A1 = (A0 x 101); A1 = (1 x 101) = 101;

............

A2 = (A1 x 101); A2 = (101 x 101) = 10201;

...........

A3 = (A2 x 101); A3 = (10201 x 101) = 1030301;

.........

A4 = (A3 x 101); A4 = (1030301 x 101) = 104060401;

..........

A5 = (A4 x 101); A5 = (104060401 x 101) = 10510100501;

............

A6 = (A5 x 101); A6 = (10510100501x 101) = 1061520150601;

..........

A7 = (A6 x 101); A7 = (1061520150601x 101) = 107213535210701;

............

A8 = (A7 x 101); A8 = (107213535210701x 101) = 10828567056280801;

A9 = (A8 x 1001);

A8 = (1008028056070056028008001 x 1001) = 1009036084126126084036009001;

Etc.

............

(A1 : A0) = 101; (A2 : A1) = 101;

(A3 : A2) = 101; (A4 : A3) = 101;

(A5 : A4) = 101; (A6 : A5) = 101;

(An : An-1) = 101;

Etc.

The numbers in the rows of Pascal's triangle are the same as the coefficients generated by raising the binomial [(An : An-1) = 101)] to a power.

...............

PROBLEM 5

In the Pascal triangle try to develop diagonal 2 from diagonal 1, diagonal 3 from diagonal 2, diagonal 4 from diagonal 3, etc.

Diagonals in Pascal’s triangle

·	·	·	·	·	·	·	·	· Sum
· D1	· 1	· 1	· 1	· 1	· 1	· 1	· 1.....	· 7
· D2	· 1	· 2	· 3	· 4	· 5	· 6	· 7.....	· 28
· D3	· 1	· 3	· 6	· 10	· 15	· 21	· 28....	· 84
· D4	· 1	· 4	· 10	· 20	· 35	· 56	· 84....	· 210
· D5	· 1	· 5	· 15	· 35	· 70	· 126	· 210....	· 462
· D6	· 1	· 6	· 21	· 56	· 126	· 252	· 462....	· 924
· D7	· 1	· 7	· 28	· 84	· 210	· 462	· 924....	· 1716
· D8	· 1	· 8	· 36	· 120	· 330	· 792	· 1716....	· 3003
· D9	· 1	· 9	· 45	· 165	· 495	· 1287	· 3003....	· 5005
· D10	· 1	· 10	· 55	· 220	· 715	· 2002	· 5005....	· 8008
· D11	· 1	· 11	· 66	· 286	· 1001	· 3003	· 8008....	· 12376
· D12	· 1	· 12	· 78	· 364	· 1365	· 4368	· 12376....	· 18564
· D13	· 1	· 13	· 91	· 455	· 1820	· 6188	· 18564....	· 27132
· D14	· 1	· 14	· 105	· 560	· 2380	· 8568	· 27132....	· 38760
· D15	· 1	· 15	· 120	· 680	· 3060	· 11628	· 38760....	· 54264
· D16	· 1	· 16	· 136	· 816	· 3876	· 15504	· 54264....	· 74613
· D17	· 1	· 17	· 153	· 969	· 4845	· 20349	· 74613....	· 100947
· D18	· 1	· 18	· 171	· 1140	· 5985	· 26334	· 100947....	· 134596
· D19	· 1	· 19	· 190	· 1330	· 7315	· 33649	· 134596....	· 177100
· D20	· 1	· 20	· 210	· 1540	· 8855	· 42504	· 177100....	· 230230
· D21	· 1	· 21	· 231	· 1771	· 10626	· 53130	· 230230....	· 296010
·	·	·	·	·	·	·	·	· Etc.

Diagonal D1

(1+1+1+1+1+1+1) = 7;

.............

Diagonal D2

Input = Diagonal D1

Output = Diagonal D2

(1+1+1+1+1+1+1) è (1+(1+1)+(1+1+1)+(1+1+1+1)+(1+1+1+1+1)+(1+1+1+1+1+1)+

+ (1+1+1+1+1+1+1) = (1+2+3+4+5+6+7) =28:

Diagonal D3

Input = Diagonal D2

Output = Diagonal D3

(1+2+3+4+5+6+7) è (1+(1+2)+(1+2+3)+(1+2+3+4)+(1+2+3+4+5)+ +(1+2+3+4+5+6)+(1+2+3+4+5+6+7) = (1+3+6+10+15+21+28) = 84;

.............

Diagonal D4

Input = Diagonal D3

Output = Diagonal D4

(1+3+6+10+15+21+28) è (1+(1+3)+(1+3+6)+(1+3+6+10)+(1+3+6+10+15) +

+ (1+3+6+10+15+21) + (1+3+6+10+15 + +21+28) = (1+4+10+20+35+56+84)= 210:

............

Diagonal D5

Input = Diagonal D4

Output = Diagonal D5

(1+4+10+20+35+56+84) è (1+(1+4)+(1+4+10)+(1+4+10+20)+(1+4+10+20+35)+

+(1+4+10+20+35+56)+(1+4+10+20+35+56+84) = (1+5+15+35+70+126+210) = 462;

Etc.

.........

Row 1 = 1,3,6,10,15,21,28;

Row 2, column 1 = 1;

Row 2, column 2 = (Row 1, column 1 + row 2) = (1+3) = 4;

Row 2, column 3 = (Row 2, column 2 + row 1, column 3) = (4+6) = 10;

Row 2, column 4 = (Row 2, column 3 + row 1, column 4) = (10+10) = 20;

Etc.

...................

PROBLEM 6

Try to code Polynomial expression with three terms

SOLUTION:

A polynomial expression with three terms, such as:

A0 è1; 1 = 10101⁰

A1 è1,01,01; 10101 = 10101¹

A2 è1,02,03,02,01; 102030201 = 10101²

A3 è 1,03,06,07,06,03,01; 1030607060301= 10101³

A4 è 1,04,10,16,19,16,10,04,01; 10410161916100401= 10101⁴

A5 è1,05,15,30,45,51,45,30,15,05,01; 105153045514530150501= 10101⁵

^Etc.

^{..............}

CODE JESUS

Numerical value of letters of name Jesus and Maria in Arabic is:

Jesus Maria

ع	ى	س	ى	م	ر	ى	م
ê	ê	ê	ê	ê	ê	ê	ê
18	28	12	28	24	10	28	24

Names Jesus and Maria are interconnected by various codes. Our task is to find those codes and explain how mathematics interconnects those two names.

SOLUTION:

Code 86

Jesus

Maria

(18+28+12+28) = 86;

(24+10+28+24) = 86;

........

Code 1061

Jesus

Maria

S18

S28

S12

S28

S24

S10

S28

S24

171

406

300

406

300

1061

S18 = (1+2+3...+18) = 171; S28 = (1+2+3...+28) = 406;

S12 = (1+2+3...+12) = 78;

Etc.

(171+406+78+406) = 1061;

(300+55+406+300) = 1061;

**************

Combinatorics of the codes 86 and 1061

(86 + (1061x Y) = (A1,A2,A3,A4);

A1 = 32;

A2 = 63;

A3 = 53;

A4 = 85;

(A1,A2,A3,A4) è (32,63,53,85);

(32,63,53,85) è 3263 5385;

(32635385) è 3263 5385;

.......

Decoding scheme

Example 1

3263 5385 è 3263 and 5385;

(5385 – 3263) = (1061 + 1061);

***********

Example 2

3263 5385 = (86 + (1061x Y);

Y = 30759;

*************

Example 3

5385 and 3263 è 53853263;

5385 3263 = (86 + (1061x Y);

Y = 50757;

************

Example 4

3263 and 3263 è3263 3263;

3263 3263 = (86 + (1061x Y);

Y = 30757;

**********

Example 5

5385 and 5385 è5385 5385;

5385 5385 = (86 + (1061x Y);

Y = 50759;

*************

Example 6

(3263 5385 – (1061 1061) = (86 + (1061x Y);

Y = 20 758;

*********

Example 7

(3263 5385 + (1061 1061) (86 + (1061x Y);

Y = 40760;

*********

Code 949

Example 1

(3263 5385) – (18 28 12 28) = (949 + (1061 x Y);

Y = 13528;

**********

Example 2

(5385 3263 –(18 28 12 28) = (949 + (1061 x Y);

Y = 33526;

*********

Example 3

(3263 3263) – (18 28 12 28) = (949 + (1061 x Y);

Y = 13 526;

***********

Example 4

(5385 5385) – (18 28 12 28) = (949 + (1061 x Y);

Y = 33 528;

**********

[(18X28X12X28)+(24X10X28X24)] : [(18X28X12X28)-(24X10X28X24)] =

(3263 5385 : y)

y = 795 985;

Etc.

Decoding scheme 3263 5385

		3263 5385
í	í		î	î
32	63		53	85
ê
19 , 13	28, 09, 26		18, 21, 14	23,13,14,09,26

(19+(19+13)+(19+13+28)+(19+13+28+09)+(19+13+28+09+26)+ (19+13+28+09+26+18)+

(19+13+28+09+26+18+21)+(19+13+28+09+26+18+21+14)+ (19+13+28+09+26+18+21+14+23)+

(19+13+28+09+26+18+21+14+23+13)+ (19+13+28+09+26+18+21+14+23+13+14)+

(19+13+28+09+26+18+21+14+23+13+14+9)+ (19+13+28+09+26+18+21+14+23+13+14+9+26)=1663;

1663 = (1061 + 86 x y );

Y = 7;

*******

Code 3263

	3263
í		î
32		63
ê		ê
19, 13		28, 09, 26
ê		ê
NI		VET

When we translate words: Ni and Vet to Swedish we will get:

New acknowledgement!

ATOMIC GENETIC CODE

srijeda, 1. travnja 2015.

PROBLEMS FOR 46th INTERNATIONAL MATHEMATICS OLYMPIAD IN MEXICO

Nema komentara:

Objavi komentar