The Small One

The maximum sum of 10 distinct numbers chosen from 1 to 100 is 91+92+...+99+100 = 955.

For sum to be 954, the numbers chosen must be 90, (92,93,...,99,100).

Hence, the smallest number in the group is 90.

100 + 99 + 98 + 97 + 96 + 95 +94 + 93 + 92 + 90 = 954. So, the smallest number is 90

954/10 = (90 x 10) + 54

so, the integers must be 90's to 100

0+1+2+3+......+10 = 55

54= 55-1

so, 10 integers is = 90 + (0,2,3,4,........10) = 90, 92, 93,94,95,96,97,98,99,100

the smallest = 90

The largest possible sum of the first 9 would be 100+99+98+97+96+95+94+93+92=864

Then to find the smallest number in the set we subtract the sum of the current numbers from 954

954-864=90

Let's add up 91 to 100

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91+92+93+.....100

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= (91+100)*10/2

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= 191*5

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=955

Since the sum is 954, we replace 91 with 90.

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Thus, the smallest number is 90

let the choosen numbers are $a_{1}, a_{2}, ... a_{10}$ where $a_{1}<a_{2}<a_{3}<a_{4}<a_{5}<a_{6}<a_{7}<a_{8}<a_{8}<a_{9}<a_{10}$ . we want to find min( $a_{1}$ ). $a_{1}$ is minimum if and only if $a_{2} + a_{3} + ... + a_{10}$ is maximum, and the maximum value of $a_{2} + a_{3} + ... + a_{10}$ reach when $a_{2} + a_{3} + ... + a_{10} = 92+93+94+...+100=864$ .

so, min( $a_{1}=954-864=90$ )

as 10 distinct integers for [1,2,.......,100],

smallest of these 10 numbers=954-100-99-98-97-96-95-94-93-92=90(the smallest)...

• 100 + 99 +98+97+96+95+94+93+92+90 = 954

O menor é $\boxed{90}$

SINCE WE EXAMINE THAT 91+92+93+94+95+96+97+98+99+100=954. SO IF WE TAKE 1 LESS NUMBER THAT IS 90 AND WHICH IS ALSO DISTINCT IN THE PLACE OF 91 WE WILL GET THE SUM = 954 AND SINCE 90 IS THE SMALLEST NUMBER SO ANSWER IS 90

100+99+98+....+91=955.BUT WE WANT SUM TO BE 954.THEREFORE WE WILL SUBTRACT 1 FROM ANY ONE NUMBER.SINCE,WE ARE ASKED SMALLEST NO. WE WILL SUBTRACT 1 FROM SMALLEST NUMBER i.e. 91-1=90

The maximum possible numbers which can be selected are 100, 99, 98, 97, 96, 95, 94, 93 and 92. Subtracting the sum of these from 954, we get \boxed{90}

x+100+99+98+..+92 = 954, where x is the smallest number possible. (The other 9 distinct numbers have to be as large as possible for x to be the smallest.)

100+99+98+..+92 = 864 Therefore, x = 954-864 = 90 .

"n"be a number then it will n,n+1,n+2,n+3.......,n+10...now if we add them it would be 10n+45....it should be equal to 954.......then after solving it comes out to be 90. something ...so it should be 90....

In this case, it is impossible if the 10 numbers are less than 90.

100 + 99 + 98 + 97 + 96 + 95 + 94 + 93 + 92 + 91 = 955. (1 more than the sum).

If we change 91 with 90, then:

100 + 99 +98 + 97 + 96 + 95 + 94 + 93 + 92 + 90 = 954

Thus, the answer is 90.