Vasilije in Cacak (Codeforces 1878C)
Tags: Math, math

Aca and Milovan, two fellow competitive programmers, decided to give Vasilije a problem to test his skills.Vasilije is given three positive integers: $$$n$$$, $$$k$$$, and $$$x$$$, and he has to determine if he can choose $$$k$$$ distinct integers between $$$1$$$ and $$$n$$$, such that their sum is equal to $$$x$$$.Since Vasilije is now in the weirdest city in Serbia where Aca and Milovan live, Cacak, the problem seems weird to him. So he needs your help with this problem.

Input: The first line contains a single integer $$$t$$$ ($$$1 \le t \le 10^4$$$) — the number of test cases.The only line of each test case contains three integers $$$n$$$, $$$k$$$ and $$$x$$$ ($$$1 \le n \le 2 \cdot 10^5$$$, $$$1 \le k \le n$$$, $$$1 \le x \le 4 \cdot 10^{10}$$$) — the maximum element he can choose, the number of elements he can choose and the sum he has to reach.Note that the sum of $$$n$$$ over all test cases may exceed $$$2 \cdot 10^5$$$.

Output: For each test case output one line: "YES", if it is possible to choose $$$k$$$ distinct integers between $$$1$$$ and $$$n$$$, such that their sum is equal to $$$x$$$, and "NO", if it isn't. You can output the answer in any case (for example, the strings "yEs", "yes", "Yes", and "YES" will be recognized as a positive answer).

Note: In the first test case $$$n = 5,\ k=3,\ x=10$$$, so we can choose the numbers: $$$2$$$, $$$3$$$, $$$5$$$, whose sum is $$$10$$$, so the answer is "YES".In the second test case $$$n = 5, \ k=3, \ x=3$$$, there is no three numbers which satisfies the condition, so the answer is "NO". It can be shown that there are no three numbers whose sum is $$$3$$$.

Sample Inputs:
12
5 3 10
5 3 3
10 10 55
6 5 20
2 1 26
187856 87856 2609202300
200000 190000 19000000000
28 5 2004
2 2 2006
9 6 40
47202 32455 613407217
185977 145541 15770805980

Sample Outputs:
YES
NO
YES
YES
NO
NO
YES
NO
NO
NO
YES
YES

Solution (Python):
from sys import stdin
input = stdin.readline

def ri():
    return int(input())

def rl(v_type=int, output=list):
    return output(map(v_type, input().split()))

def rs():
    return input()[:-1]

def out(x, sep='\n'):
    print(sep.join(x))

def solve():
    (n, k, x) = rl()
    t = k * (k + 1) // 2
    if k * (n + 1) - t >= x >= t:
        return 'Yes'
    return 'NO'
outp = []
for TC in range(ri()):
    outp.append(str(solve()))
out(outp)

Solution (Python):
from sys import stdin
input = stdin.readline

def ri():
    return int(input())

def rl(v_type=int, output=list):
    return output(map(v_type, input().split()))

def rs():
    return input()[:-1]

def out(x, sep='\n'):
    print(sep.join(x))

def solve():
    (n, k, s) = rl(output=tuple)
    if s <= k * n - (k - 1) * k / 2 and s >= k * (k + 1) / 2:
        return 'YES'
    return 'NO'
output = []
for _ in range(ri()):
    output.append(solve())
out(output)

Solution (Python):
for s in [*open(0)][1:]:
    (n, k, x) = map(int, s.split())
    print('NYOE S'[k * k + k <= 2 * x <= (2 * n - k + 1) * k::2])