WOLFRAM SUMMER SCHOOL 2022

Wolfram Mathematical Olympiad Problems Database

Miroslav Marinov
Wolfram Summer School 2022
m.marinov1617@gmail.com
We build a database of problems from mathematical olympiads which are interesting not only when attempted by hand, but also when the Wolfram language is used as an assistant. These should be of interest not only to those who wish to train in certain areas of mathematics, but also to people interested in how the language performs against such problems. In this manner strengths or weaknesses of particular functions of the language could be exposed and one could have a thought of how to make a more automated approach to obtain a nice solution which really relies on the language, but also hopefully gives insight on how would a human find it on their own. Included also are problems for which we do not know how to utilize the language to help, even though it seems likely to be possible. We also intend to build an application which generates training problem sets based on keywords input by the user.

Introduction

What does the database contain?


Keywords for the problems


On the Application Interface


Problems

Consider the sequence
a
0
=
a
1
=1,
a
n+1
=14
a
n
-
a
n-1
. Prove that for all non-negative integers
n
the integer
2
a
n
-1
is a perfect square.


Solve
x
3
-
y
7
=2
in non-negative integers.


Solve
n-1
a
+
n-1
b
+
n-1
c
=n!
in positive integers.


Solve
4
x
+
7
y
=
10
31
in integers.


Find all positive integers
m
such that every term of the sequence
a
1
=
a
2
=1,
a
3
=4,
a
n
=m(
a
n-1
+
a
n-2
)-
a
n-3
is a perfect square.


For
a,b,c,d>0
determine the minimum possible value of
(a+b)(b+c)(c+d)(a+b+c+d)
abcd


For
a,b,c>0
prove the inequality
2
(b+c)
2
a
+bc
+
2
(a+c)
ac+
2
b
+
2
(a+b)
ab+
2
c
≥6


For
a,b,c>0
with
a+b+c=1
, show that
a+bc+1
2
a
+1
+
b+ac+1
2
b
+1
+
c+ab+1
2
c
+1
≤
39
10
.


Given
x
1
=
1
2
and
x
n
=
2+
x
n-1
1-2
x
n-1
prove that and that the terms are well defined, non-zero and distinct.


Find all functions
f:
such that
2
f(a)
+
2
f(b)
+
2
f(c)
=2(f(a)f(b)+f(b)f(c)+f(c)f(a))
for all integers
a,b,c
such that
a+b+c=0
.


The function
f:
is strictly increasing and satisfies
f(f(n))=2n+2
. What is f(2022)?


If
f:
is such that
f(x)+f
1
1-x
=arctan(x)
, compute
0
∫
1
f(x)dx
.


Choose a random
k
-element subset
X
 of
{1,2,...,k+a}
 uniformly and, independently of
X
, choose random
n
-element subset
Y
of
{1,2,...,k+a+n}
 uniformly.​​Prove that the probability
(min(Y)>max(X))
does not depend on 
a
.


Rational solutions to
{
2
x
}+{x}=y
for
y=
99
100
and
y=1
.


Random Polynomials

For uniformly distributed
b
and
c
in
[-∞,∞]
what is the probability that the roots of
2
x
+bx+c
are real?


For uniformly distributed
a,b
and
c
in
[-∞,∞]
what is the probability that the roots of
2
ax
+bx+c
are real?​


Keywords

◼
  • Language Testing
    • ◼
  • Mathematical Olympiads
    • ◼
  • CellTags
    • ◼
  • TaggingRules

    • Acknowledgments

    I would like to thank my project mentors Daniel Robinson and Paul Abbott for suggestion of the topic, as well as for helpful comments and discussions on technical details. I also express my gratitude to Stephen Wolfram and the whole Wolfram community for the possibility to be in Wolfram Summer School 2022.

    References

    ◼
  • LinkedIn Mathematical Olympiads group
    • ◼
  • Automated Generation of Planar Geometry Olympiad Problems
    • ◼
  • Art of Problem Solving