PIMSO 2023 - Mathematics - Grade 9 (Secondary 3) - National Round

Determine the largest possible value of $k$ such that $\frac{100-k}{k-1}$ is a positive integer.

Editorial note: For automatic assessment, assume k is an integer, as in the supplied key. The source omits this domain; see the solution for the unrestricted-real alternative.

Determine the sum of the reciprocals of the positive divisors of 54.

Evaluate $20-1+19-2+18-3+\ldots+11-10$

How many positive divisors (factors) does 1200 have?

If x and y are positive integers and $xy+x+y=54$, what is $x+y$?

What is the absolute difference between the largest and the smallest possible four-digit numbers that you can form using the digits 2, 3, 5, 7, and 9?

Editorial note: Assume no digit is repeated.

For Numbers 7-10:

A number of species faced extinction during the Mesozoic era. Species R, S, T, U, V, W, and X were species that all became extinct during this era, and scientists have determined that their extinctions occurred based on the following constraints:

W became extinct before X. U became extinct after T.

S became extinct before T. X became extinct before U. V became extinct after U.

S became extinct before W.

R became extinct before S.

Which were the last two species to go extinct?

Editorial note: Enter the two species in extinction order, separated by a comma, for example A, B.

For Numbers 7-10:

A number of species faced extinction during the Mesozoic era. Species R, S, T, U, V, W, and X were species that all became extinct during this era, and scientists have determined that their extinctions occurred based on the following constraints:

W became extinct before X. U became extinct after T.

S became extinct before T. X became extinct before U. V became extinct after U.

S became extinct before W.

R became extinct before S.

Which species could have been the fourth one to go extinct? [Hint: There is more than one answer]

Editorial note: List all possible species, in alphabetical order separated by commas, for example A, B, C.

For Numbers 7-10:

A number of species faced extinction during the Mesozoic era. Species R, S, T, U, V, W, and X were species that all became extinct during this era, and scientists have determined that their extinctions occurred based on the following constraints:

W became extinct before X. U became extinct after T.

S became extinct before T. X became extinct before U. V became extinct after U.

S became extinct before W.

R became extinct before S.

Which is the smallest number of species that could have gone extinct before T?

For Numbers 7-10:

A number of species faced extinction during the Mesozoic era. Species R, S, T, U, V, W, and X were species that all became extinct during this era, and scientists have determined that their extinctions occurred based on the following constraints:

W became extinct before X. U became extinct after T.

S became extinct before T. X became extinct before U. V became extinct after U.

S became extinct before W.

R became extinct before S.

How many possible orders are there for the species’ extinction pattern?

What is the next number in the series? $1,2,10,37,101,\_\_\_$

Editorial note: Continue the pattern in which successive differences are consecutive cubes.

A book has 820 pages. If Shane starts reading from page 217 and reads 12 pages forward, then 3 pages backward, and continues this pattern, how many times will she flip the page until she reaches the end of the book?

Editorial note: This source item is assessed manually because “flip” has an unclear counting unit. State your interpretation.

Suppose $x,y,z>0$ and $x+y+z=10$, what is the maximum value of $xyz$?

If a,b, and c are the roots of the polynomial $x^3+5x^2+7x+8$, what is $ab+bc+ca$?

Editorial note: Use all three polynomial roots, including complex roots and counting multiplicity.

Determine the largest possible value of x for which the following equation is true: $x^3-6x^2+11x-6=0$.

What is the coefficient of $x^3$ when the expression $(2x+3)^5$ is expanded?

Given that a and b are the roots of the quadratic expression $x^2+3x+1$, what is $1/a+1/b$?

If $x^2-7x+4=0$, what is the value of $x^2+16/x^2$?

Two chords AB and CD intersect inside a circle at point E. If AE = 3, BE = 6, and CE = 2, what is DE?

The angle measures of quadrilateral WXYZ are given as $\angle W=40^\circ$, $\angle X=93^\circ$, $\angle Y=62^\circ$. What is $\angle Z$?