PIMSO 2021 - Mathematics - Grade 5 - National Round

Find the ones digit of the product of the digits of the 2021-digit number $123412341234\ldots$.

A palindromic number is any number that reads the same forwards and backwards such as 12321, 2058502 or 100 272 001. What is the 30th 9-digit palindromic number?

Write the sum as a single fraction in simplest form: $\frac2{13}\times\frac{91}{47}+\frac2{13}\times\frac{156}{47}+\frac2{13}\times\frac{169}{47}+\frac2{13}\times\frac{195}{47}$.

Find the sum of all the numbers from 100 to 199 that are not divisible by 13.

Consider this phrase:
PIMSO MATH PHILIPPINES
Every second, the first letter of each word is moved to the end of the word. This continues until the statement reads the same as the beginning. In how many minutes will the original statement appear back again?

What is the units’ digit of the sum of $123^5+124^6+125^7+126^8+127^9$?

How many counting numbers between $2^4$ to $2^8$ are divisible by 2, 5 or 7?

Interpretation note (editorial): include both endpoints.

The natural numbers are arranged in a wave pattern as shown on the right. What is the sum of the numbers that appears on the 50th row?

The number 24682468…2468 has 476 digits. What is the tens digit of the product of all its digits?

The 8-digit number $\overline{ILOILOPH}$ is divisible by 88. If the digits PH are removed the number becomes 724724. What is the biggest possible value of $\overline{ILOILOPH}$?

The three athletes Gary, Ian and Kent reunited during the National Sports Finals. They were come from different regions in the country namely, Region III, Region VII and Region X. They were part of volleyball, baseball and basketball team of their regions.
Kent is not from Region VII.
Ian played baseball but he is not from Region III.
The athlete who played volleyball is not from Region X.
The athlete who played basketball is from Region VII.
From what region does Kent come from?

Find the remainder when the 2020th term of the number sequence below is divided by 7.
$2,3,5,8,12,17,23,30,38,47,\ldots$

Pattern note (editorial): continue the consecutive increasing differences.

The biggest number that can divide two numbers A and B is 17. The smallest number that can be divided by both A and B is 102. What is the value of B if A is 51?

Some years ago, there were 5 Sundays and 4 Mondays in the 11th month of the year. On which day of the week was the 20th in that month?

Alden, Bayani, Carlo, and Densio finished the race in a particular order. Each of them was asked and replied as follows:
Alden: I was not the first.
Bayani: Carlo was the first. I am not the last.
Carlo: I came to the finish line before Alden and Densio did.
Densio: I was the first.
If only one of them lied, who was the last to finish the race?

There is a village whose inhabitants are grouped into two tribes namely, Bayani and Kawal. Inhabitants of Bayani always tell the truth but those of Kawal always lie. While trying to look for the chieftain of the village, you meet a man who tells you that he is a liar. Does he live on the village or is he a visitor?

Four tourists, Oliver, Akbar, Arif, and Yuki, check-in to Boracay Mandarin Island Hotel. They came from Australia, Uzbekistan, Indonesia, and Japan.
Oliver is older than the tourist from Australia.
Akbar doesn’t love Indonesian food and hate to play with kangaroos.
Arif is older than the boy from Uzbekistan but younger than the Japanese.
Yuki loves kangaroo and kiwi fruit.
Who is the tourist that come from Japan?

In the circular sector below, PH and HL are straight lines that intersect at H on the circumference of circle O. Find $\angle PHL$.

A triangle has side lengths of HL = 27 cm, PH = 68cm, and PL = c cm, where the third side c is a whole number. How many values of c are possible?

The quadrilateral PHIL is divided into four regions as shown below. If the areas of $\triangle POH$, $\triangle POL$, and $\triangle LOI$ are $72\text{ cm}^2$, $48\text{ cm}^2$, and $56\text{ cm}^2$, respectively, find the area of $\triangle HOI$.