MathDB
Problems
Contests
National and Regional Contests
Singapore Contests
Singapore Junior Math Olympiad
2024 Singapore Junior Maths Olympiad
2024 Singapore Junior Maths Olympiad
Part of
Singapore Junior Math Olympiad
Subcontests
(5)
Q4
1
Hide problems
2^n and 5^n have same starting digit
Suppose for some positive integer
n
n
n
, the numbers
2
n
2^n
2
n
and
5
n
5^n
5
n
have equal first digit. What are the possible values of this first digit?Note: solved [url=https://artofproblemsolving.com/community/c6h312638p1685546]here
Q3
1
Hide problems
7 triangles in a square
Seven triangles of area
7
7
7
lie in a square of area
27
27
27
. Prove that among the
7
7
7
triangles there are
2
2
2
that intersect in a region of area not less than
1
1
1
.
Q2
1
Hide problems
Triangle is similar to two others
Let
A
B
C
D
ABCD
A
BC
D
be a parallelogram and points
E
,
F
E,F
E
,
F
be on its exterior. If triangles
B
C
F
BCF
BCF
and
D
E
C
DEC
D
EC
are similar, i.e.
△
B
C
F
∼
△
D
E
C
\triangle BCF \sim \triangle DEC
△
BCF
∼
△
D
EC
, prove that triangle
A
E
F
AEF
A
EF
is similar to these two triangles.
Q1
1
Hide problems
Halving isosceles right triangle with one line
Let
A
B
C
ABC
A
BC
be an isosceles right-angled triangle of area 1. Find the length of the shortest segment that divides the triangle into 2 parts of equal area.
Q5
1
Hide problems
Easy diophantine
Find all integer solutions of the equation
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.
Note: has appeared many times before, see [url=https://artofproblemsolving.com/community/q1_%22x%5E4%2B20x%5E3%2B104x%5E2%22]here