MathDB
Problems
Contests
National and Regional Contests
USA Contests
USA - High School Proof Olympiads
BAMO Contests
2007 BAMO
2007 BAMO
Part of
BAMO Contests
Subcontests
(5)
5
1
Hide problems
2007 BAMO p5 y_{n+1}/x_{n+1} > y_n / x_n => y_n > \sqrt{n}
Two sequences of positive integers,
x
1
,
x
2
,
x
3
,
.
.
.
x_1,x_2,x_3, ...
x
1
,
x
2
,
x
3
,
...
and
y
1
,
y
2
,
y
3
,
.
.
y_1,y_2,y_3,..
y
1
,
y
2
,
y
3
,
..
are given, such that
y
n
+
1
x
n
+
1
>
y
n
x
n
\frac{y_{n+1}}{x_{n+1}} > \frac{y_n}{x_n}
x
n
+
1
y
n
+
1
>
x
n
y
n
for each
n
≥
1
n \ge 1
n
≥
1
. Prove that there are infinitely many values of
n
n
n
such that
y
n
>
n
y_n > \sqrt{n}
y
n
>
n
.
4
1
Hide problems
2007 BAMO p4 integers x^2+xy+y^2 <= 2007, odd , not divisible by 3
Let
N
N
N
be the number of ordered pairs
(
x
,
y
)
(x,y)
(
x
,
y
)
of integers such that
x
2
+
x
y
+
y
2
≤
2007
x^2+xy+y^2 \le 2007
x
2
+
x
y
+
y
2
≤
2007
. Remember, integers may be positive, negative, or zero! (a) Prove that
N
N
N
is odd. (b) Prove that
N
N
N
is not divisible by
3
3
3
.
3
1
Hide problems
2007 BAMO p3 D,E \in BC, BD = CE, <BAD=<CAE => ABC isosceles
In
△
A
B
C
,
D
\vartriangle ABC, D
△
A
BC
,
D
and
E
E
E
are two points on segment
B
C
BC
BC
such that
B
D
=
C
E
BD = CE
B
D
=
CE
and
∠
B
A
D
=
∠
C
A
E
\angle BAD = \angle CAE
∠
B
A
D
=
∠
C
A
E
. Prove that
△
A
B
C
\vartriangle ABC
△
A
BC
is isosceles
2
1
Hide problems
2007 BAMO p2 points in plane bw, if 3 of # have tha same color, so the 4th
The points of the plane are colored in black and white so that whenever three vertices of a parallelogram are the same color, the fourth vertex is that color, too. Prove that all the points of the plane are the same color.
1
1
Hide problems
2007 BAMO p1 marked stick with measures at a row 2,3,5,1,4
A
15
15
15
-inch-long stick has four marks on it, dividing it into five segments of length
1
,
2
,
3
,
4
1,2,3, 4
1
,
2
,
3
,
4
, and
5
5
5
inches (although not neccessarily in that order) to make a “ruler.” Here is an example. https://cdn.artofproblemsolving.com/attachments/0/e/065d42b36083453f3586970125bedbc804b8a1.png Using this ruler, you could measure
8
8
8
inches (between the marks
B
B
B
and
D
D
D
) and
11
11
11
inches (between the end of the ruler at
A
A
A
and the mark at
E
E
E
), but there’s no way you could measure
12
12
12
inches. Prove that it is impossible to place the four marks on the stick such that the five segments have length
1
,
2
,
3
,
4
1,2,3, 4
1
,
2
,
3
,
4
, and
5
5
5
inches, and such that every integer distance from
1
1
1
inch through
15
15
15
inches could be measured.