International Youth Math Challenge Pre-Final Round 2024: Problems and Solutions, Exercises of Mathematics

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• Pre-Final Round
  • DATE OF RELEASE: 23 January

Important: Read all the information on this page carefully!

General Information

Please read all problems carefully!

The six problems are separated into three categories: 2x basic problems (A; 4 points), 2x advanced problems (B; 6 points), 2x special-creativity problems (C; 8 points).

You can write your solution in the blank space under the problems or type your solution digitally on a computer. If you need more space for your solution, you can use extra sheets of paper.

You receive points for the correct solution as well as for the performed steps. Example: Despite a wrong solution, if the described approach is accurate, you will still receive points.

You can reach up to 36 points in total. You qualify for the Final Round if you reach at least 16 Points (Junior), 20 points (Youth), or 24 points (Senior).

It is not allowed to work in groups on the problems. Assistance from teachers, friends, family, or the internet is prohibited. Cheating will result in immediate disqualification!

Please consider the following notation used for the problems:

• x, y 2 R are real numbers; n, k 2 N are positive integers.
• f, g, h are functions. The domain and co-domain follow from the context.
• The roots of a function f are all x such that f (x) = 0.
• ⇡ = 3. 141 ... is the circle constant and e = 2. 718 ... is Euler’s number.
• The natural logarithm of x is written as log(x).

Solution Requirements

Please upload a PDF document containing clear pictures of the problem sheet pages with your written solutions in the blank space, or upload a digitally typed document.

You can upload your solution online via your account: https://iymc.info/en/login

Only upload one single PDF file! If you have multiple pictures, please merge them into one file. Do not upload your solution in any di↵erent format. (Example: no Word and no Zip files)

Upload your solution before the submission deadline. You may modify your submission as needed until the deadline passes.

The submission deadline is Sunday, 26 January 2025, 23:59 UTC+0.

The results of the Pre-Final Round will be announced on Tuesday, 4 February 2025.

Good luck!

Problem A.

Find the derivative f 0 (x) of the following function with respect to x:

f (x) =

r

sin(x) +

q cos(x) +

p tan(x)

t

(sin(x) + i) in

I

inte

• sinx (^) +
  • (^) sinx.^2 +^ secy Cost (^) +2 x-

2 Fant

• 2 cos^ iun

I

4 cost

-2Sinx F^ +Sex

4 Fant

Sinxt costum

1

4 cost

-2Sinx F^ +Sex

(I

2. (^) Fant-)

Sinxt costum

Problem B.

(a) Determine the probability for a short sequence of k digits^1 to occur in a longer sequence of

n randomly selected digits.

(b) The digits of ⇡ appear to be randomly distributed. How many digits of ⇡ are needed to expect the sequence 9876543210 with 50% probability?

(c) The sequence of digits 9876543210 first appears in ⇡ starting at the 21,981,157,633rd decimal place. Is this finding expected?

(^1) digit: any numeral between 0 to 9

need r. (^) g3x10" (^) digit number

an

k

the

d.

I K

.

but we need for m ,

h-k+]

combine two be

multiply and : -I-n/

for probability

digit

↓

b) 9878543210 =^ 10 h (^) = nth^ digit of it

=^1

• /

-myn

=

n-g

enc

-+ =x

&byusingstor

-?

In (^) (2) =^ ( - 1)(in) - 10

%)

th

+g =^ n

693x189 =^ h

Problem B.

A circle with circumference C contains three smaller circles having half the radius of the outer

circle. The circles are equally spaced in steps of 90^ along the circumference of the outer circle. Determine the area of the outer circle not covered by any of the smaller circles (white area).

C =^ zar^

let r be

=

r (^) rudius of

big circle,

94a

1 = r

,

be the radius of

4 small circlee.

o

-because

of the^ formula, small circle^ isof big circle,

area of^ part^ in^ between^ of^ circle^ ,

a is^ same as part overlap, O^ is^ same.

C

3

e

Wπ

area of^

white (^) part of^ the

big

circle is

Problem C.

You are given three sequences of numbers:

(a) For each sequence, find recursive relations between each number and

i. the numbers in the row above. ii. the numbers in the same sequence.

(b) Find an equation that yields the n-th element in each sequence, without recursion.

(c) Let an and bn be the n-th element of the left and middle sequence, respectively. Which

value does the fraction bn/an approximate? Explain and prove it.

f, ++ z

= (^) +

I 2 3

fi +^ +y =^ +s

4 j

2 p

. 5

z I

2 b^ V 12 3 e

It 15

S (^) IS

fr =^ fan^ +^ fis^ (R > 3 ,

and U =^ R^ ,^ fu^

= (^1) , fzF]^ ,fo

= 2

An =^ 2(n) +^ Any (^) (h) 2 and^ a^ , =^1 and^ a= 2)

(9) Sum^ of^ first^ and^ second^

number is fourth^ number, so

fu =^ fi^

• (^) +s (^). Difference between^ band^13 is^3 ,

I, so

En =^ fresEnz.

Problem C.

This problem requires you to read following scientific article:

On the distribution of ((n)).

Dixit, A. B., Bhattacharjee, S. Electronic Journal of Combinatorial Number Theory (2024).

Link: https://math.colgate.edu/ integers/y65/y65.pdf

Use the content of the article to work on the problems (a-f) below. Problem (f) is a bonus problem

that can give you extra points. However, it is not possible to get more than 36 points in total.

(a) What is the numerical value of (8), (12) and ((18))?

(b) Explain the statement of Theorem 1 in your own words.

(c) Show that #{n  x : (n) is odd}  ↵ ·

log log x log x^1 /x^

for some ↵ > 0 and x > 9.

(d) Prove that ((n)) < k ·

(n) log n if^

Q

p pkn

p|(n).

(e) On page 5, explain how the authors arrive at Equation (1) from the preceding lines.

(f*) Bonus Problem: Use a computer program (e.g., Python) to check how many integers n  x

satisfy the condition ((n))  cn for varying values of c and x.