Problem 01
![$p$](https://latex.artofproblemsolving.com/3/6/f/36f73fc1312ee0349b3f3a0f3bd9eb5504339011.png)
![$q$](https://latex.artofproblemsolving.com/0/6/1/0615acc3725de21025457e7d6f7694dab8e2f758.png)
![\[\frac{p}{q}=1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+ \ldots -\frac{1}{1318}+\frac{1}{1319},\]](https://latex.artofproblemsolving.com/f/e/7/fe7a47dd9fd28d623ff2b2b7186626d70073ea29.png)
![$p$](https://latex.artofproblemsolving.com/3/6/f/36f73fc1312ee0349b3f3a0f3bd9eb5504339011.png)
![$1979$](https://latex.artofproblemsolving.com/7/b/6/7b6cdf62a4c800756c74d0fac1e13ec127b1d146.png)
Find all real roots of the equation
where is a real parameter.
Real numbers are given. For each
(
) define
and let
.
(a) Prove that, for any real numbers ,
(b) Show that there are real numbers such that equality holds in (*)
Let us substitute in for
to get
Now, since the domain and range of are the same, we can let
and
equal some constant
to get
Determine all three-digit numbers having the property that
is divisible by 11, and
is equal to the sum of the squares of the digits of
.
Let for some digits
and
. Then
for some
. We also have
. Substituting this into the first equation and simplification, we get
Cambodian Olympiad Math is written by Lim Sovanvichet . This book is shared on social media by many students which is very important document for all Khmer students. In this book, there are many International Math Problems which he translated from other foreign books and wrote them in to Khmer language.
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