PIMSO 2022 - Mathematics - Grade 9 / Secondary 3 - National Round

What will be the remainder when $1!+2!+3!+4!+5!+\cdots+1000!$ is divided by 10?

What is the sum of the series below up to its 1000th term? $$\frac13+\frac16+\frac16+\frac1{12}+\frac1{10}+\frac1{20}+\frac1{15}+\cdots$$

Pattern convention: alternate reciprocal triangular numbers and half of each preceding term.

How many positive integer divisors does 25000 have?

Drake partially forgot his 6-character passcode. He correctly remembers that four of the characters are A, B, C, and D, in that specific chronological order, but forgets their respective places in the passcode. How many ways can Henry enter 1, 2, 3, 4?

If you’re going to write down all integers from 100 to 1000 that are NOT divisible by 2, 5, or 7, how many will there be on the list?

What is the highest possible value of integer $k$ that will make $100000!$ divisible by $661^k$?

[Ian] The radius of circle A is 1/3 the radius of circle B. Circle A rolls around circle B one trip back to its starting point. How many times will circle A rotate in total?

Assume external rolling without slipping around a fixed circle B.

Complete the series: 13, 25, 1, 37, –11, 49, ___.

Pattern convention: successive differences alternate signs and are consecutive positive multiples of 12.

If the 9th day of January 202X is three days earlier than Friday, what will be the 365th day of the year 202X?

Families living in New Clark City love biking. There are a total of 2,022 families in the city each with two or four bicycles. Given that there are as many families owning either four bicycles as many as families owning two bicycles, how many bicycles are there in New Clark City?

What is the value of CVCC?



Editorial note: the sentence and addition figure use different four-letter strings; preserve the sentence as printed. Use distinct decimal digits for distinct letters, with no leading zero.

Vladimir’s wallet contains 20, 50, and 100-peso bills. If the total value of the bills is Php1,650 and there are 30 bills, how many combinations of bills are possible?

Interpretation: at least one bill of each named denomination is present.

Let $R$ and $F$ ($R,F\ne0$) be the roots of the equation $x^2-3x+2=0$. If $RF$ and $R-F$ are the roots of the equation $(c+1)x^2-sx+m=0$, find $c+1+s+m$.

Let $R$ and $F$ ($R,F\ne0$) be the roots of the equation $x^2-3x-m=0$. If $RF$ and $R+F$ are the roots of the equation $(c+1)x^2-sx+m=0$ and $c+1+s+m=0$, then find the value of $c+s-m$.

Simplify $S=\frac1{10}+\frac2{10^2}+\frac3{10^3}+\frac4{10^4}+\frac5{10^5}+\frac6{10^6}+\frac7{10^7}+\frac8{10^8}+\frac9{10^9}$.

If $S=\frac{10}{100}+\frac{200}{100^2}+\frac{3000}{100^3}+\frac{40000}{100^4}+\cdots+\frac{9000000000}{100^9}+\frac{100000000000}{100^{10}}$, what is $S+\frac1{10^9}$?

Simplify $\frac{22}{70}\left(\frac74+\frac7{12}+\frac7{24}+\frac7{40}+\frac7{60}+\frac7{84}+\cdots+\frac7{180}+\frac7{220}\right)$.

Let $f(x)=2x+1$ and $h(x)=x^3$. Find $f(h(0))h(f(0))$.

A rectangle lies on 2 intersecting identical circles such that the points of intersection of the 2 circles are also intersected by the rectangle. The radii of the circles are 85 cm each while the width of the rectangle is 168 cm. What is the distance between the centers of the 2 circles?

Given the figure below, if the distance between the center of the 2 identical bigger circles is 16 cm. while their radii are 6 cm each. If the radius of the smallest circle at the top is 3 cm, what is the shortest distance from the center of the topmost circle to line m? (Leave your answer in surd form.)



Input format: use sqrt(...) for square roots, pi for π, * for multiplication and / for division.