# Wolfram Mathematical Olympiad Problems Database

Wolfram Mathematical Olympiad Problems Database

Miroslav Marinov

Wolfram Summer School 2022

Wolfram Summer School 2022

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

Introduction

### What does the database contain?

What does the database contain?

### Keywords for the problems

Keywords for the problems

### On the Application Interface

On the Application Interface

## Problems

Problems

### Consider the sequence a0=a1=1,an+1=14an-an-1. Prove that for all non-negative integers n the integer 2an-1 is a perfect square.

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

a

0

a

1

a

n+1

a

n

a

n-1

n

2-1

a

n

### Solve x3-y7=2 in non-negative integers.

Solve -=2 in non-negative integers.

x

3

y

7

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

Solve ++=n! in positive integers.

n-1

a

n-1

b

n-1

c

### Solve 4x+7y=1031 in integers.

Solve += in integers.

4

x

7

y

10

31

### Find all positive integers m such that every term of the sequence a1=a2=1,a3=4,an=m(an-1+an-2)-an-3 is a perfect square.

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

m

a

1

a

2

a

3

a

n

a

n-1

a

n-2

a

n-3

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

For determine the minimum possible value of

a,b,c,d>0

(a+b)(b+c)(c+d)(a+b+c+d)

abcd

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

For prove the inequality +bc++≥6

a,b,c>0

2

(b+c)

2

a

2

(a+c)

ac+

2

b

2

(a+b)

ab+

2

c

### For a,b,c>0 with a+b+c=1, show that a+bc+12a+1+b+ac+12b+1+c+ab+12c+1≤3910.

For with , show that +1++1++1≤.

a,b,c>0

a+b+c=1

a+bc+1

2

a

b+ac+1

2

b

c+ab+1

2

c

39

10

### Given x1=12 and xn=2+xn-11-2xn-1 prove that and that the terms are well defined, non-zero and distinct.

Given = and = prove that and that the terms are well defined, non-zero and distinct.

x

1

1

2

x

n

2+

x

n-1

1-2

x

n-1

### Find all functions f: such that 2f(a)+2f(b)+2f(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.

Find all functions such that ++=2(f(a)f(b)+f(b)f(c)+f(c)f(a)) for all integers such that .

f:

2

f(a)

2

f(b)

2

f(c)

a,b,c

a+b+c=0

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

The function is strictly increasing and satisfies . What is f(2022)?

f:

f(f(n))=2n+2

### If f: is such that f(x)+f11-x=arctan(x), compute 0∫1f(x)dx.

If is such that , compute f(x)dx.

f:

f(x)+f=arctan(x)

1

1-x

0

∫

1

### 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.

Choose a random -element subset of uniformly and, independently of , choose random -element subset of uniformly.Prove that the probability does not depend on .

k

X

{1,2,...,k+a}

X

n

Y

{1,2,...,k+a+n}

(min(Y)>max(X))

a

### Rational solutions to {2x}+{x}=y for y=99100 and y=1.

Rational solutions to for and .

{}+{x}=y

2

x

y=

99

100

y=1

### Random Polynomials

Random Polynomials

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

For uniformly distributed and in what is the probability that the roots of +bx+c are real?

b

c

[-∞,∞]

2

x

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

For uniformly distributed and in what is the probability that the roots of +bx+c are real?

a,b

c

[-∞,∞]

2

ax

## Keywords

Keywords

◼

Language Testing

◼

Mathematical Olympiads

◼

CellTags

◼

TaggingRules

## Acknowledgments

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

References

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