MathDB
Problems
Contests
National and Regional Contests
Israel Contests
Grosman Mathematical Olympiad
2023 Grosman Mathematical Olympiad
2023 Grosman Mathematical Olympiad
Part of
Grosman Mathematical Olympiad
Subcontests
(7)
7
1
Hide problems
Existence of monochromatic triangle
The plane is colored with two colors so that the following property holds: for each real
a
>
0
a>0
a
>
0
there is an equilateral triangle of side length
a
a
a
whose
3
3
3
vertices are of the same color.Prove that for any three numbers
a
,
b
,
c
>
0
a,b,c>0
a
,
b
,
c
>
0
for which the sum of any two is greater than the third there is a triangle with sides
a
a
a
,
b
b
b
, and
c
c
c
whose
3
3
3
vertices are of the same color.
6
1
Hide problems
Guessing a number from prime tests
Adam has a secret natural number
x
x
x
which Eve is trying to discover. At each stage Eve may only ask questions of the form "is
x
+
n
x+n
x
+
n
a prime number?" for some natural number
n
n
n
of her choice.Prove that Eve may discover
x
x
x
using finitely many questions.
5
1
Hide problems
Parities aren't periodic
Consider the sequence of natural numbers
a
n
a_n
a
n
defined as
a
0
=
4
a_0=4
a
0
=
4
and
a
n
+
1
=
a
n
(
a
n
−
1
)
2
a_{n+1}=\frac{a_n(a_n-1)}{2}
a
n
+
1
=
2
a
n
(
a
n
−
1
)
for each
n
≥
0
n\geq 0
n
≥
0
. Define a new sequence
b
n
b_n
b
n
as follows:
b
n
=
0
b_n=0
b
n
=
0
if
a
n
a_n
a
n
is even, and
b
n
=
1
b_n=1
b
n
=
1
if
a
n
a_n
a
n
is odd. Prove that for each natural
m
m
m
, the sequence
b
m
,
b
m
+
1
,
b
m
+
2
,
b
m
+
3
,
…
b_m, b_{m+1}, b_{m+2},b_{m+3}, \dots
b
m
,
b
m
+
1
,
b
m
+
2
,
b
m
+
3
,
…
is not periodic.
4
1
Hide problems
All values of a quadratic are semiprime
Let
q
q
q
be an odd prime number. Prove that it is impossible for all
(
q
−
1
)
(q-1)
(
q
−
1
)
numbers
1
2
+
1
+
q
,
2
2
+
2
+
q
,
…
,
(
q
−
1
)
2
+
(
q
−
1
)
+
q
1^2+1+q, 2^2+2+q, \dots, (q-1)^2+(q-1)+q
1
2
+
1
+
q
,
2
2
+
2
+
q
,
…
,
(
q
−
1
)
2
+
(
q
−
1
)
+
q
to be products of two primes (not necessarily distinct).
3
1
Hide problems
Product of polynomials equals composition
Find all pairs of polynomials
p
p
p
,
q
q
q
with complex coefficients so that
p
(
x
)
⋅
q
(
x
)
=
p
(
q
(
x
)
)
.
p(x)\cdot q(x)=p(q(x)).
p
(
x
)
⋅
q
(
x
)
=
p
(
q
(
x
))
.
2
1
Hide problems
Numbers on lottery ticket
A "Hishgad" lottery ticket contains the numbers
1
1
1
to
m
n
mn
mn
, arranged in some order in a table with
n
n
n
rows and
m
m
m
columns. It is known that the numbers in each row increase from left to right and the numbers in each column increase from top to bottom. An example for
n
=
3
n=3
n
=
3
and
m
=
4
m=4
m
=
4
:[asy] size(3cm); Label[][] numbers = {{"
1
1
1
", "
2
2
2
", "
3
3
3
", "
9
9
9
"}, {"
4
4
4
", "
6
6
6
", "
7
7
7
", "
10
10
10
"}, {"
5
5
5
", "
8
8
8
", "
11
11
11
", "
12
12
12
"}}; for (int i=0; i<5;++i) { draw((i,0)--(i,3)); } for (int i=0; i<4;++i) { draw((0,i)--(4,i)); } for (int i=0; i<4;++i){ for (int j=0; j<3;++j){ label(numbers[2-j], (i+0.5, j+0.5)); }} [/asy] When the ticket is bought the numbers are hidden, and one must "scratch" the ticket to reveal them. How many cells does it always suffice to reveal in order to determine the whole table with certainty?
1
1
Hide problems
Product of an arithmetic progression
An arithmetic progression of natural numbers of length
10
10
10
and with difference
11
11
11
is given. Prove that the product of the numbers in this progression is divisible by
10
!
10!
10
!
.