# Putnam

Let $a_1, a_2, . . . , a_n$ be real numbers, and let $b_1, b_2, ..., b_n$ be distinct positive integers. Suppose that there is a polynomial f(x) satisfying the identity
$(1 - x)^n f(x) = 1 + \sum _{i=1} ^n a_i x^{b_i}$
Find the value of f(1) expressed in terms of $b_1, b_2, ..., b_n$ and $n$.

I am a fourth year undergraduate student of Computer Science at IIT Kanpur. I love doing mathematics, and science. I also do a bit of coding in my free time.

### 10 responses to “Putnam”

1. Jithendar says :

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

2. Ankit Kumar says :

f(1) = 1 ans.

3. SHUBHAM says :

SHITIKANTH ISKA ANWER “0” HAI CHECK KRR LEH

4. SHUBHAM says :

VERY GOOD QUE AISEY AUR PLZZ ….

• 7275024832 says :

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

5. SIVA NAGI REDDY says :

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

• rajiv kumar says :

f(1)=1/ncb1-ncb2+ncb3……………….ncbn

6. Karthik says :

Something like -b/n-2

• Siva Modugula says :

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.

7. Yesh says :

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