a

IOQM-2024Karthik Thiagarajan

Find the smallest positive integer that does notdivide 1×2×3×4×5×6×7×8×9.n | 9! ∀ n∈{1,…,10}11 ∤ 9!11Find the number of four-digit odd numbers havingdigits 1,2,3,4 each occurring exactly once.---1---33!3!12Find the number obtained by taking thelast two digits of 52024 in the same order.5n≡25(mod 100), ∀n⩾225(1)(2)(3)

Let ABCD be a quadrilateral with ∠ADC=70∘,∠ACD=70∘,∠ACB=10∘ and ∠BAD=110∘.Find the measure of ∠CAB.(4)a

∠CAB	=∠BAD-∠CAD
	=110∘-40∘
	= 70∘

a

∠CAD	=180∘-(∠ADC+∠ACD)
	=180∘-(70∘+70∘)
	=40∘

Let a=x

y+y

z+z

x, let b=x

z+y

x+z

y and let c=(x

y+y

z)(y

z+z

x)(z

x+x

y). Find |ab-c|.X=x

y,Y=y

z,Z=z

xa

a	=X+Y+Z
b	=1 X+1 Y+1 Z
c	=(X+Y)(Y+Z)(Z+X)

XYZ=1a

ab	=(X+Y+Z)(1 X+1 Y+1 Z)
	=(X+Y+Z)(XY+YZ+ZX)

a

c	=(X+Y)(Y+Z)(Z+X)
	=(X+Y)(YZ+XY+Z2+ZX)
	=X(YZ)+X(XY)+Z(ZX)+X(ZX)
	+Y(YZ)+Y(XY)+Z(YZ)+Y(ZX)
	=X(XY+YZ+ZX)
	+Y(XY+YZ+ZX)
	+Z(YZ+ZX)+Z(XY)-Z(XY)
	=(X+Y+Z)(XY+YZ+ZX)-1
	=ab-1

try to get this pattern seenin ab when expanding cXYZ=1Setupc=ab-1⟹|ab-c|=

1

(5)

Find the number of triples of real numbers (a,b,c)such that a20+b20+c20=a24+b24+c24=1.GCD(20,24)=4A=a4,B=b4,C=c4A5+B5+C5=A6+B6+C6=10⩽A,B,C⩽1A5⩾A6B5⩾B6C5⩾C6A5+B5+C5⩾A6+B6+C6A5=A6B5=B6C5=C6(A,B,C)∈{(1,0,0),(0,1,0),(0,0,1)}(a,b,c)∈{(1,0,0),(0,1,0),(0,0,1) (-1,0,0),(0,-1,0),(0,0,-1)}

6

triples(6)

Determine the sum of all possible surface areas of acube two of whose vertices are (1,2,0) and (3,3,2).d=(3-1)2+(3-2)2+(2-0)2=3Area=6a2sidea=dA=6a2=54diagonalof faced=2aA=6a2=27diagonalof cubed=3aA=6a2=1854+27+18=

99

(7)