HMMT 二月 2024 · ALGNT 赛 · 第 13 题

HMMT February 2024 February 17, 2024 Algebra and Number Theory Round 2

1. Suppose r , s , and t are nonzero reals such that the polynomial x + rx + s has s and t as roots, and 2 the polynomial x + tx + r has 5 as a root. Compute s .
2. Suppose a and b are positive integers. Isabella and Vidur both fill up an a × b table. Isabella fills it up with numbers 1 , 2 , . . . , ab , putting the numbers 1 , 2 , . . . , b in the first row, b + 1 , b + 2 , . . . , 2 b in the second row, and so on. Vidur fills it up like a multiplication table, putting ij in the cell in row i and column j . (Examples are shown for a 3 × 4 table below.) 1 2 3 4 1 2 3 4 5 6 7 8 2 4 6 8 9 10 11 12 3 6 9 12 Isabella’s Grid Vidur’s Grid Isabella sums up the numbers in her grid, and Vidur sums up the numbers in his grid; the difference between these two quantities is 1200 . Compute a + b .
3. Compute the sum of all two-digit positive integers x such that for all three-digit (base 10 ) positive integers a b c , if a b c is a multiple of x , then the three-digit (base 10 ) number b c a is also a multiple of x . 3
4. Let f ( x ) be a quotient of two quadratic polynomials. Given that f ( n ) = n for all n ∈ { 1 , 2 , 3 , 4 , 5 } , compute f (0) .
5. Compute the unique ordered pair ( x, y ) of real numbers satisfying the system of equations x 1 y 1 √ − = 7 and √ + = 4 . 2 2 2 2 x y x + y x + y n !
6. Compute the sum of all positive integers n such that 50 ≤ n ≤ 100 and 2 n + 3 does not divide 2 − 1 . 3 3 3
7. Let P ( n ) = ( n − 1 )( n − 2 ) . . . ( n − 40 ) for positive integers n . Suppose that d is the largest positive integer that divides P ( n ) for every integer n > 2023 . If d is a product of m (not necessarily distinct) prime numbers, compute m . 2 π 2 π
8. Let ζ = cos + i sin . Suppose a > b > c > d are positive integers satisfying 13 13 √ a b c d | ζ + ζ + ζ + ζ | = 3 . Compute the smallest possible value of 1000 a + 100 b + 10 c + d .
9. Suppose a , b , and c are complex numbers satisfying 2 a = b − c, 2 b = c − a, and 2 c = a − b. Compute all possible values of a + b + c .
10. A polynomial f ∈ Z [ x ] is called splitty if and only if for every prime p , there exist polynomials g , h ∈ Z [ x ] with deg g , deg h < deg f and all coefficients of f − g h are divisible by p . Compute p p p p p p 4 2 the sum of all positive integers n ≤ 100 such that the polynomial x + 16 x + n is splitty.