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National and Regional Contests
Ireland Contests
Ireland National Math Olympiad
2003 Irish Math Olympiad
3
3
Part of
2003 Irish Math Olympiad
Problems
(2)
Difference of Floors
Source: 2003 IrMO Paper 1 Problem 3
12/28/2017
For each positive integer
k
k
k
, let
a
k
a_k
a
k
be the greatest integer not exceeding
k
\sqrt{k}
k
and let
b
k
b_k
b
k
be the greatest integer not exceeding
k
3
\sqrt[3]{k}
3
k
. Calculate
∑
k
=
1
2003
(
a
k
−
b
k
)
.
\sum_{k=1}^{2003} (a_k-b_k).
k
=
1
∑
2003
(
a
k
−
b
k
)
.
algebra
Easy algebra
Source:
5/8/2016
Find all the (x,y) integer ,if
y
2
+
2
y
=
x
4
+
20
x
3
+
104
x
2
+
40
x
+
2003
y^2+2y=x^4+20x^3+104x^2+40x+2003
y
2
+
2
y
=
x
4
+
20
x
3
+
104
x
2
+
40
x
+
2003
algebra