question:Note: This question carries 200 points In the mystical land of Zephyria, there exist ancient portals that connect various cities in a hidden pattern. To unlock these portals, a unique key needs to be formed by combining the attributes of two cities. Given the attributes of city A and city B, your task is to find how many unique ways the portals can be unlocked following a specific rule. Given two integers P and Q representing the attributes of city A and city B respectively, you have to determine the number of pairs (a, b) such that 1 leq a leq P and 1 leq b leq Q, for which the condition a^2 + b^3 = a oplus b holds true, where oplus denotes the bitwise XOR operation. Also, find the number of distinct values of a that can participate in valid pairs. -----Input :----- - The first line contains T denoting the number of test cases. - Each test case contains two integers P and Q. -----Output :----- - For each test case, print two integers: the number of pairs (a, b) and the number of distinct values of a. -----Constraints :----- - 1 ≤ T ≤ 3000 - 1 ≤ P, Q ≤ 10^8 -----Sample Input :----- 1 2 5 -----Sample Output :----- 1 1 -----Explanation :----- For the given test case with P=2 and Q=5, there is only one pair (2,1) that satisfies the condition: 2^2 + 1^3 = 4 + 1 = 5 and 2 oplus 1 = 3, so the condition does not hold, implying no valid pairs. The sample incorrect output is just an illustrative example.

question:Polycarp enjoys creating string patterns based on mathematical operations. He starts with a string consisting of digits '1' to '9' and performs a series of operations until he reaches a desired length. He can perform two types of operations on the string: - Append the string to itself. - Remove the first character of the string. Given an integer n signifying the desired length, your task is to determine the minimum initial string length required so that by performing the above operations, Polycarp can obtain a string of length exactly n. -----Input:----- - A single integer n indicating the desired length of the string. -----Output:----- Print a single integer which is the minimum initial string length required. -----Constraints----- - 1 leq n leq 10^6 -----Sample Input:----- 5 -----Sample Output:----- 3 -----EXPLANATION:----- In the example, Polycarp can start with the initial string "123". He performs the following operations: 1. Initial string "123" 2. Append string to itself: "123123" 3. Remove the first character: "23123" The final string "23123" has the desired length of 5. The minimal initial string length Polycarp needed was 3.

question:In a faraway forest, there are several magical creatures called Elphins who like to cast spells on each other. Each Elphin is positioned on a unique magical coordinate along a 1-dimensional axis and can cast spells either to the right or to the left. The distance of their spell cast is fixed for each Elphin and does not vary. The head wizard wants to know if there is any pair of Elphins that can cast spells and hit each other simultaneously. Help the head wizard determine if such a pair exists. Each Elphin has a specific fixed position and a spell range, which indicates how far their spell can reach in either direction. -----Input:----- - The first line contains an integer n (1 <= n <= 100) - the number of Elphins in the forest. - Each of the next n lines contains two integers p(i) and r(i) (-10^4 <= p(i) <= 10^4, 1 <= |r(i)| <= 2 * 10^4) - where p(i) is the position of the i-th Elphin, and r(i) is the distance it can cast spells. Positive values of r(i) indicate the spell is cast to the right, and negative values indicate the spell is cast to the left. No two Elphins stand at the same position. -----Output:----- If there is any pair of Elphins that can cast spells and hit each other simultaneously, output "TRUE", otherwise output "FALSE". -----Sample Input:----- 3 2 3 5 -3 6 4 -----Sample Output:----- TRUE

question:In a small seaside town, there is a bustling fish market where the daily business revolves around fishermen bringing their catches of fresh fish. Each day, fishermen sell their catches, and the fish market keeps a record of each sale. The fish market wants to analyze the sales data to identify which day had the highest aggregate weight of the fish sold. Each sale record includes the day identifier and the weight of the fish sold on that day. Your task is to write a program that processes the records of these sales and determines the day with the highest aggregate weight of the fish sold. If there are multiple days with the same maximum aggregate weight, the earliest day identifier should be returned. -----Input----- The first line of input contains a single integer T - the number of test cases. Each test case is structured as follows: - The first line contains a single integer N — the number of sales records. - The next N lines contain the sales records in the format "day weight", where - `day` is a string representing the unique identifier of the day. - `weight` is an integer representing the weight of the fish sold on that day. -----Output----- For each test case, output a single line with the identifier of the day that has the highest aggregate weight of the fish sold. -----Constraints----- - 1 ≤ T ≤ 100 - 1 ≤ N ≤ 1000 - 1 ≤ `weight` ≤ 1000 - The length of `day` will not exceed 10 characters. - All day identifiers in a single test case are unique. -----Example----- Input: 2 5 Monday 100 Tuesday 200 Monday 150 Wednesday 300 Tuesday 100 3 Friday 100 Saturday 400 Sunday 400 Output: Monday Saturday Explanation: In the first test case, Monday has a total weight of 250, Tuesday has a total weight of 300, and Wednesday has 300. Since Tuesday and Wednesday have the same aggregate weight, the earliest day identifier, Tuesday, is selected. In the second test case, both Saturday and Sunday have the same weight of 400, and since Saturday comes earlier, it is selected.