PIMSO 2023 - Mathematics - Senior Secondary - National Round

In the sequence $1,4,7,10,13,\ldots,2023$, how many are divisible by 5?

How many positive integer solutions $(a,b,c)$ where $a,b,$ and $c$ are distinct does $1/a+1/b+1/c=1/2$ have?

If $a,b,$ and $c$ are positive integers, then how many solutions $(a,b,c)$ does $a+b+c=10$ have?

Let $\{a_n\}$ be a sequence of numbers such that $a_n=a_{n-1}-a_{n-2}$, $a_1=1$, and $a_2=2023$. What is $a_{2023}$?

If $1+3+5+\ldots+2023=k^2$, what is $k$?

Editorial note: For automatic marking, use the positive value of k, as intended by the supplied key; the source omits this restriction.

Evaluate: $1/3+1/15+1/35+1/63+\ldots+1/399$

For questions 7-9
A professor is setting up advising meetings with her students John, Dan, Emily, Ashley, Ben, and Eli. She can meet with exactly one student per day at lunch, except Fridays, when she can meet with two students after school instead. The following conditions apply:
- Ashley's appointment is before Ben's but after Eli's
- The professor will meet with a male student on Wednesday
- John is one of the students who meets on Friday
- Dan cannot meet on Fridays
- Emily's appointment is after the student who meets on Wednesday

How many of the students can be assigned to Monday?

Editorial note: Assume the intended Monday-Friday week, each student met once, and "before/after" means an earlier/later day. Following the intended name-based convention, John, Dan, Ben and Eli are male; Emily and Ashley are female. The source does not explicitly supply genders.

For questions 7-9
A professor is setting up advising meetings with her students John, Dan, Emily, Ashley, Ben, and Eli. She can meet with exactly one student per day at lunch, except Fridays, when she can meet with two students after school instead. The following conditions apply:
- Ashley's appointment is before Ben's but after Eli's
- The professor will meet with a male student on Wednesday
- John is one of the students who meets on Friday
- Dan cannot meet on Fridays
- Emily's appointment is after the student who meets on Wednesday

Ashley's appointment could be on which of the following weekdays?

Editorial note: Assume the intended Monday-Friday week, each student met once, and "before/after" means an earlier/later day. Following the intended name-based convention, John, Dan, Ben and Eli are male; Emily and Ashley are female. The source does not explicitly supply genders. No choices are printed; answer with the weekday name.

For questions 7-9
A professor is setting up advising meetings with her students John, Dan, Emily, Ashley, Ben, and Eli. She can meet with exactly one student per day at lunch, except Fridays, when she can meet with two students after school instead. The following conditions apply:
- Ashley's appointment is before Ben's but after Eli's
- The professor will meet with a male student on Wednesday
- John is one of the students who meets on Friday
- Dan cannot meet on Fridays
- Emily's appointment is after the student who meets on Wednesday

If Emily is NOT on Thursday, who could be?

Editorial note: Assume the intended Monday-Friday week, each student met once, and "before/after" means an earlier/later day. Following the intended name-based convention, John, Dan, Ben and Eli are male; Emily and Ashley are female. The source does not explicitly supply genders. Give both possible names.

Find the sum of the infinite geometric series: $2+1+1/2+1/4+\ldots$

Ana, Ben, Carla, and Dan are seated in a row of four seats numbered #1 to #4. Jim looks at them and says:
"Ben is next to Carla."
"Anna is between Ben and Carla."
However each one of Jim's statements is false. Ben is actually sitting in seat #3. Who is sitting in seat #2?

Editorial note: Treat "Anna" in the quoted statement as the same person as "Ana" in the list.

It takes 10 hours to fill a pool with the inlet pipe. It can be emptied in 15 hours with the outlet pipe. If the pool is half full to begin with, how long will it take to fill it from there if both pipes are open?

Suppose that $f$ is a polynomial with degree 5 and leading coefficient 1. If $f(k)=2k$ for $k=1,2,\ldots,5$, then what is $f(6)$?

If $x^2-5x+1=0$, what is the value of $|x^2-1/x^2|$?

Editorial note: Enter an exact expression using sqrt(...) for a square root.

Factor completely: $x^4+4y^4$.

Editorial note: Factor over the integers (equivalently here, into real polynomial factors). Use ^ for powers and parentheses for factors.

If $a,b,$ and $c$ are the solutions of the equation $x^3+2x^2+3x+4=0$, what is the value of $1/(a+2)+1/(b+2)+1/(c+2)$?

If $x^2+y^2+z^2+w^2=100$, what is the maximum value of $x+y+z+w$?

Editorial note: The variables are taken to be real, the standard interpretation of this maximum problem.

If $a,b,$ and $c$ are real numbers such that $a+b+c=12$, what is the maximum value of $(a-1)(2b-4)(3c-9)$ if $a>1$, $b>2$, and $c>3$?

Find the radius of the shaded circle below: (Answer: 2/3)

Editorial note: The source includes its answer in the question; it is preserved. Read the figure as two equal white circles of diameter 2 tangent to each other and to the outer circle of diameter 4, with the shaded circle tangent to all three. These geometric relations are indicated by the drawing rather than stated in words.

What is the area of a triangle with side lengths 13, 14, and 15?