MathDB
Problems
Contests
National and Regional Contests
USA Contests
Withdrawn USA Contests
Kettering Mathematics Olympiad
2003 Kettering HS MO
2003 Kettering HS MO
Part of
Kettering Mathematics Olympiad
Subcontests
(1)
1
Hide problems
2003 Kettering University Mathematics Olympiad for High School Students
p1. How many real solutions does the following system of equations have? Justify your answer.
x
+
y
=
3
x + y = 3
x
+
y
=
3
3
x
y
−
z
2
=
9
3xy -z^2 = 9
3
x
y
−
z
2
=
9
p2. After the first year the bank account of Mr. Money decreased by
25
%
25\%
25%
, during the second year it increased by
20
%
20\%
20%
, during the third year it decreased by
10
%
10\%
10%
, and during the fourth year it increased by
20
%
20\%
20%
. Does the account of Mr. Money increase or decrease during these four years and how much? p3. Two circles are internally tangent. A line passing through the center of the larger circle intersects it at the points
A
A
A
and
D
D
D
. The same line intersects the smaller circle at the points
B
B
B
and
C
C
C
. Given that
∣
A
B
∣
:
∣
B
C
∣
:
∣
C
D
∣
=
3
:
7
:
2
|AB| : |BC| : |CD| = 3 : 7 : 2
∣
A
B
∣
:
∣
BC
∣
:
∣
C
D
∣
=
3
:
7
:
2
, find the ratio of the radiuses of the circles. p4. Find all integer solutions of the equation
1
x
+
1
y
=
1
19
\frac{1}{x}+\frac{1}{y}=\frac{1}{19}
x
1
+
y
1
=
19
1
p5. Is it possible to arrange the numbers
1
,
2
,
.
.
.
,
12
1, 2, . . . , 12
1
,
2
,
...
,
12
along the circle so that the absolute value of the difference between any two numbers standing next to each other would be either
3
3
3
, or
4
4
4
, or
5
5
5
? Prove your answer. p6. Nine rectangles of the area
1
1
1
sq. mile are located inside the large rectangle of the area
5
5
5
sq. miles. Prove that at least two of the rectangles (internal rectangles of area
1
1
1
sq. mile) overlap with an overlapping area greater than or equal to
1
9
\frac19
9
1
sq. mile PS. You should use hide for answers.