MathDB

1

2024 Alg/NT Problem 10

There exists a unique pair of polynomials

(P(x),Q(x))

such that \begin{align*} P(Q(x))&= P(x)(x^2-6x+7) \\ Q(P(x))&= Q(x)(x^2-3x-2) \end{align*} Compute

P(10)+Q(-10)

.
Proposed by Connor Gordon

9

1

2024 Alg/NT Problem 9

Let

\mathbb Q_{\geq 0}

be the non-negative rational numbers,

f: \mathbb Q_{\geq 0} \to \mathbb Q_{\geq 0}

such that

f(z+1) = f(z)+1

,

f(1/z) = f(z)

for

z\neq 0

, and

f(0) = 0.

Define a sequence

P_n

of non-negative integers recursively via P_0 = 0, P_1 = 1, P_n = 2 P_{n-1}+P_{n-2} for every

n \geq 2

. Find

f\left(\frac{P_{20}}{P_{24}}\right).

Proposed by Robert Trosten

8

1

2024 Alg/NT Problem 8

Compute the number of non-negative integers

k < 2^{20}

such that

\binom{5k}{k}

is odd.
Proposed by David Tang

7

1

2024 Alg/NT Problem 7

Let

x_0

,

x_1

,

x_2

, and

x_3

be complex numbers forming a square centered at

0

in the complex plane with side length

2

. For each

0\leq k\leq 3

, there are four more complex numbers

z_{4k}, z_{4k+1}

,

z_{4k+2}

, and

z_{4k+3}

forming a square centered at

x_k

with side length

\sqrt 2

. Given that

\prod_{i=0}^{15} z_i

is a positive integer, how many possible values could it take?
Proposed by Hari Desikan

6

1

2024 Alg/NT Problem 6

Integers

a, b

satisfy the following property: the line

y = 2x + ab

passes through all intersection points of the two parabolas given by y = x^2 + 2x + a, y = 2x^2 +bx, which intersect at least once. How many such

(a, b)

satisfy

|ab| \leq 100

?
Proposed by Justin Hsieh

5

1

2024 Alg/NT Problem 5

Let

f(x)=(x+1)^{6}+(x-1)^{5}+(x+1)^{4}+(x-1)^3+(x+1)^2+(x-1)^1+1.

Find the remainder when

\sum_{j=-126}^{126}jf(j)

is divided by 1000.
Proposed by Hari Desikan

4

1

2024 Alg/NT Problem 4

For positive integer

n

, let

f(n)

be the largest integer

k

such that

k!\leq n

, let

g(n)=n-(f(n))!

, and for

j\geq 1

let

g^j(n)=\underbrace{g(\dots(g(n))\dots)}_{\text{$j$ times}}.

Find the smallest positive integer

n

such that

g^{j}(n)> 0

for all

j<30

and

g^{30}(n)=0

.
Proposed by Connor Gordon

3

1

2024 Alg/NT Problem 3

The positive integer

8833

has the property that

8833 = 88^2 + 33^2.

Find the (unique) other four-digit positive integer

\overline{abcd}

where

\overline{abcd} = (\overline{ab})^2 + (\overline{cd})^2.

Proposed by Allen Yang

2

1

2024 Alg/NT Problem 2

Suppose

P(x)=x^2+Ax+B

for real

A

and

B

. If the sum of the roots of

P(2x)

is

\tfrac 12

and the product of the roots of

P(3x)

is

\tfrac 13

, find

A+B

.
Proposed by Connor Gordon

1

2024 Alg/NT Problem 1

Connor is thinking of a two-digit number

n

, which satisfies the following properties:
[*] If

n>70

, then

n

is a perfect square. [*] If

n>40

, then

n

is prime. [*] If

n<80

, then the sum of the digits of

n

is

14

.
What is Connor's number?
Proposed by Connor Gordon

2024 CMIMC Algebra and Number Theory

Subcontests

2024 Alg/NT Problem 10

2024 Alg/NT Problem 9

2024 Alg/NT Problem 8

2024 Alg/NT Problem 7

2024 Alg/NT Problem 6

2024 Alg/NT Problem 5

2024 Alg/NT Problem 4

2024 Alg/NT Problem 3

2024 Alg/NT Problem 2

2024 Alg/NT Problem 1