Counting Numbers

by steven_zhang123, Mar 30, 2025, 12:12 AM

Let the decimal representations of numbers $A$ and $B$ be given as: $A = 0.a_1a_2\cdots a_k > 0$ , $B = 0.b_1b_2\cdots b_k > 0$ (where $a_k, b_k$ can be 0), and let $S$ be the count of numbers $0.c_1c_2\cdots c_k$ such that $0.c_1c_2\cdots c_k < A$ and $0.c_kc_{k-1}\cdots c_1 < B$ ( $c_k, c_1$ can also be 0). (Here, $0.c_1c_2\cdots c_r (c_r \neq 0)$ is considered the same as $0.c_1c_2\cdots c_r0\cdots0$ ).

Prove: $\left| S - 10^k AB \right| \leq 9k.$

China TST

combinatorics

Perfect Numbers

by steven_zhang123, Mar 30, 2025, 12:09 AM

If the sum of all positive divisors (including itself) of a positive integer $n$ is $2n$ , then $n$ is called a perfect number. For example, the sum of the positive divisors of 6 is $1 + 2 + 3 + 6 = 2 \times 6$ , hence 6 is a perfect number.
Prove: There does not exist a perfect number of the form $p^a q^b r^c$ , where $a, b, c$ are positive integers, and $p, q, r$ are odd primes.

China TST

number theory

Roots of unity

by steven_zhang123, Mar 30, 2025, 12:08 AM

Let $k, n$ be positive integers, and let $\alpha_1, \alpha_2, \ldots, \alpha_n$ all be $k$ -th roots of unity, satisfying:
$\alpha_1^j + \alpha_2^j + \cdots + \alpha_n^j = 0 \quad \text{for any } j (0 < j < k).$ Prove that among $\alpha_1, \alpha_2, \ldots, \alpha_n$ , each $k$ -th root of unity appears the same number of times.

algebra

China TST

Graph Theory Test in China TST (space stations again)

by steven_zhang123, Mar 30, 2025, 12:05 AM

MO Space City plans to construct $n$ space stations, with a unidirectional pipeline connecting every pair of stations. A station directly reachable from station P without passing through any other station is called a directly reachable station of P. The number of stations jointly directly reachable by the station pair $\{P, Q\}$ is to be examined. The plan requires that all station pairs have the same number of jointly directly reachable stations.

(1) Calculate the number of unidirectional cyclic triangles in the space city constructed according to this requirement. (If there are unidirectional pipelines among three space stations A, B, C forming $A \rightarrow B \rightarrow C \rightarrow A$ , then triangle ABC is called a unidirectional cyclic triangle.)

(2) Can a space city with $n$ stations meeting the above planning requirements be constructed for infinitely many integers $n \geq 3$ ?

graph theory

combinatorics

China TST

Graph again

by steven_zhang123, Mar 29, 2025, 11:53 PM

Let $L_3 = \{3\}$ , $L_n = \{3, 4, \ldots, h\}$ (where $h > 3$ ). For any given integer $n \geq 3$ , consider a graph $G$ with $n$ vertices that contains a Hamiltonian cycle $C$ and has more than $\frac{n^2}{4}$ edges. For which lengths $l \in L_n$ must the graph $G$ necessarily contain a cycle of length $l$ ?

combinatorics

graph theory

China TST

Why are there so many Graphs in China TST 2001?

by steven_zhang123, Mar 29, 2025, 11:44 PM

Let $k$ be a given integer, $3 < k \leq n$ . Consider a graph $G$ with $n$ vertices satisfying the condition: for any two non-adjacent vertices $x$ and $y$ in graph $G$ , the sum of their degrees must satisfy $d(x) + d(y) \geq k$ . Please answer the following questions and prove your conclusions.

(1) Suppose the length of the longest path in graph $G$ is $l$ satisfying the inequality $3 \leq l < k$ , does graph $G$ necessarily contain a cycle of length $l+1$ ? (The length of a path or cycle refers to the number of edges that make up the path or cycle.)

(2) For the case where $3 < k \leq n-1$ and graph $G$ is connected, can we determine that the length of the longest path in graph $G$ , $l \geq k$ ?

(3) For the case where $3 < k = n-1$ , is it necessary for graph $G$ to have a path of length $n-1$ (i.e., a Hamiltonian path)?

graph theory

combinatorics

China TST

The Quest for Remainder

by steven_zhang123, Mar 29, 2025, 11:42 PM

Given sets $A = \{1, 4, 5, 6, 7, 9, 11, 16, 17\}$ , $B = \{2, 3, 8, 10, 12, 13, 14, 15, 18\}$ , if a positive integer leaves a remainder (the smallest non-negative remainder) that belongs to $A$ when divided by 19, then that positive integer is called an $\alpha$ number. If a positive integer leaves a remainder that belongs to $B$ when divided by 19, then that positive integer is called a $\beta$ number.
(1) For what positive integer $n$ , among all its positive divisors, are the numbers of $\alpha$ divisors and $\beta$ divisors equal?
(2) For which positive integers $k$ , are the numbers of $\alpha$ divisors less than the numbers of $\beta$ divisors? For which positive integers $l$ , are the numbers of $\alpha$ divisors greater than the numbers of $\beta$ divisors?

number theory

China TST

2025 TST 22

by EthanWYX2009, Mar 29, 2025, 2:50 PM

Let $A$ be a set of 2025 positive real numbers. For a subset $T \subseteq A$ , define $M_T$ as the median of $T$ when all elements of $T$ are arranged in increasing order, with the convention that $M_\emptyset = 0$ . Define
$P(A) = \sum_{\substack{T \subseteq A \\ |T| \text{ odd}}} M_T, \quad Q(A) = \sum_{\substack{T \subseteq A \\ |T| \text{ even}}} M_T.$ Find the smallest real number $C$ such that for any set $A$ of 2025 positive real numbers, the following inequality holds:
$P(A) - Q(A) \leq C \cdot \max(A),$ where $\max(A)$ denotes the largest element in $A$ .

combinatorics

A and B play a game

by EthanWYX2009, Mar 29, 2025, 2:49 PM

Let $n \geq 2$ be an integer. Two players, Alice and Bob, play the following game on the complete graph $K_n$ : They take turns to perform operations, where each operation consists of coloring one or two edges that have not been colored yet. The game terminates if at any point there exists a triangle whose three edges are all colored.

Prove that there exists a positive number $\varepsilon$ , Alice has a strategy such that, no matter how Bob colors the edges, the game terminates with the number of colored edges not exceeding
$\left( \frac{1}{4} - \varepsilon \right) n^2 + n.$

This post has been edited 1 time. Last edited by EthanWYX2009, Yesterday at 2:56 PM

combinatorics

China TST

How many cases did you check?

by avisioner, Jul 17, 2024, 12:01 PM

Determine all ordered pairs $(a,p)$ of positive integers, with $p$ prime, such that $p^a+a^4$ is a perfect square.

Proposed by Tahjib Hossain Khan, Bangladesh

This post has been edited 1 time. Last edited by avisioner, Jul 20, 2024, 4:57 PM
Reason: Proposer name added

number theory

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Pi in the Sky

by steven_zhang123, Mar 30, 2025, 12:12 AM

by steven_zhang123, Mar 30, 2025, 12:09 AM

by steven_zhang123, Mar 30, 2025, 12:08 AM

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by avisioner, Jul 17, 2024, 12:01 PM