# Putnam

Let be real numbers, and let be distinct positive integers. Suppose that there is a polynomial f(x) satisfying the identity

Find the value of f(1) expressed in terms of and .

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Let be real numbers, and let be distinct positive integers. Suppose that there is a polynomial f(x) satisfying the identity

Find the value of f(1) expressed in terms of and .

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Hoping to see more good problems here if you have any. Also, if I may ask, can you post some elegant solutions too.

f(1) = 1 ans.

SHITIKANTH ISKA ANWER “0” HAI CHECK KRR LEH

VERY GOOD QUE AISEY AUR PLZZ ….

deepak yadav says:Sir,iska ans. (0/0) ka form aa raha hai.

Answer will not depend upon b1,b2…..bn’s. Answer is 1+a1+a2+…an (unless I made calculation errors).

Take the nth roots of unity 1,w,w^2,…w^(n-1). Substitute into the equation and add all of them. We will get f(1)- constant term of f =a1+a2…+an. Since f(0)=1. f(1)=1+a1+a2+..+an

f(1)=1/ncb1-ncb2+ncb3……………….ncbn

Something like -b/n-2

Differentiate both sides n times using the product rule and Leibniz formula, then x=1. We get f(1) = sum of (a_i.max(0, b_i.(b_i-1)……(b_i-n+1))) from i=1 to n.