PIMSO 2023 - Mathematics - Secondary 4 / Grade 10 - National Round

Determine the largest possible value of $k$ such that $\frac{100-k}{2k+1}$ is a positive integer.

Determine the sum of the reciprocals of the positive divisors of 100.

Evaluate $30-11+29-12+28-13+\ldots+21-20$

How many positive divisors (factors) does 2400 have?

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

What is the absolute difference between the largest and the smallest possible four-digit numbers that you can form using the digits 1, 4, 6, 7, and 8?

For Questions 7-9

A fitness trainer will schedule exactly one session for each of six people—Len, May, Nan, Richie, Sam, and Tim—one session per day for six consecutive days. The schedule must conform to the following conditions:

- May's session is later in the schedule than Nan's session.
- Tim's session is later in the schedule than Richie's session.
- Richie's session is exactly three days after Len's session.
- Nan's session is on the first day or else the third day.

If Nan's session is scheduled for the first day, then the session for which of the following people must be scheduled for the sixth day?

For Questions 7-9

A fitness trainer will schedule exactly one session for each of six people—Len, May, Nan, Richie, Sam, and Tim—one session per day for six consecutive days. The schedule must conform to the following conditions:

- May's session is later in the schedule than Nan's session.
- Tim's session is later in the schedule than Richie's session.
- Richie's session is exactly three days after Len's session.
- Nan's session is on the first day or else the third day.

Who among the 6 people would find this schedule acceptable: the person's session is on the third day and another acceptable schedule in which the person's session is on the fifth day?

For Questions 7-9

A fitness trainer will schedule exactly one session for each of six people—Len, May, Nan, Richie, Sam, and Tim—one session per day for six consecutive days. The schedule must conform to the following conditions:

- May's session is later in the schedule than Nan's session.
- Tim's session is later in the schedule than Richie's session.
- Richie's session is exactly three days after Len's session.
- Nan's session is on the first day or else the third day.

If May’s session is scheduled for a day either immediately before or immediately after Sam's session, then Len's session must be scheduled for the ___ day.

Find the sum of the infinite geometric series: $3+1.5+0.75+0.375+\ldots$

A water tank can be filled by an inlet pipe in 8 hours. It takes twice that long for the outlet pipe to empty the tank. How long would it take to fill the tank if both pipes were open?

Mom had just filled the cookie jar when the 3 children went to bed. That night one child woke up, ate half of the cookies and went back to bed. Later, the second child woke up, ate half of the remaining cookies, and went back to bed. Still later, the third child woke up, ate half of the remaining cookies, leaving 3 cookies in the jar. How many cookies were in the jar to begin with?

Suppose $x,y,z,w>0$ and $x+y+z+w=12$, what is the maximum value of $xyzw$?

If $a,b,$ and $c$ are the roots of the polynomial $3x^3+12x^2+9x+8$, what is $ab+bc+ca$?

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

What is the coefficient of $x^4$ when the expression $(2x+3)^5$ is expanded?

Given that $a$ and $b$ are the roots of the quadratic expression $x^2+14x+2$, what is $1/a+1/b$?

If $x^2-3x+1=0$, what is the value of $x^2+1/x^2$?

Two chords AB and CD intersect inside a circle at point E. If $AE=4$, $BE=6$, and $CE=3$, what is DE?

The angle measures of quadrilateral WXYZ are given as $\angle W=30^\circ$, $\angle X=75^\circ$, $\angle Y=87^\circ$. What is $\angle Z$?