PIMSO 2021 - Mathematics - Grade 6 - National Round

Find the value of $815\times(640000-12000+225)$.

What is the highest power of 2 in the standard form of $\dfrac{20!}{10!}$?

The operation # is defined as $a\mathbin{\#}b=a^2-ab+b^2$. What is the value of $(5\mathbin{\#}4)\mathbin{\#}2$?

If you will arrange A, B, and C in descending order, which is the largest?

$A=4^8$, $B=3^{32}$, and $C=6^{16}$

What is the smallest number that should be multiplied to 750 to make it a perfect square?

Find the sum of $\dfrac{3}{6\times9}+\dfrac{3}{9\times12}+\dfrac{3}{12\times15}+\dfrac{3}{15\times18}+\dfrac{3}{18\times21}+\cdots+\dfrac{3}{96\times99}+\dfrac{3}{99\times102}$.

How many counting numbers between $2^4$ and $2^8$ are not divisible by 2, 5 or 7?

The Least Common Multiple of $2^{16}$, $6^7$, and $m$ is $36^4$. How many possible values of $m$ are there?

A, B, and C are prime numbers and $A\lt B\lt C$. What is the value of C in $3A+2B+5C=143$?

What is the unit’s digit of the sum of the addition sentence below?

$622^4+623^5+624^6+625^7+626^8+627^9+628^{10}$

What is the value of A in $\dfrac{A}{1\times3}+\dfrac{A}{3\times5}+\dfrac{A}{5\times7}+\cdots+\dfrac{A}{97\times99}=1$?

Mother bought a watch, a cellphone and a necklace for her 3 daughters Anita, Benita, and Catriona. At any situation, one of the following statements is true and the other three are false. Which statement is true?

i. Anita received the watch.
ii. Benita received the cellphone.
iii. Catriona did not receive the necklace.
iv. Neither Catriona nor Anita received the cellphone.

Three friends draw 3 number cards from a tray of cards numbered 1, 2, 3 … 8, and 9.
Alma: The product of my numbers is 21.
Bella: The sum of my 2 numbers subtracted from my largest number gives a difference of zero.
Carmen: The sum of my numbers is 18.
Who drew the card numbered 9?

Pedro has a tetrahedron. His friends ask him questions which he answered as follows:
Mario: Does your polyhedron have more than 6 faces.
Pedro: Yes. It has more than 6 faces.
Nardo: Does it have two bases and more than 12 edges?
Pedro: Yes. It has two bases and more than 12 edges.
Lando: Does it have 16 vertices?
Pedro: You are correct! It has 16 vertices.
How many edges does Pedro’s polyhedron have?

In the circular sector below, PH, HL and LS are straight lines. Points P, H, L, and S lie on the circumference of circle O. $PH=HL$ and $\angle LSO=35^\circ$. Find $\angle HLS$.

The isosceles trapezoid TUVW overlaps triangle QRS as shown below. QR = 21 cm, RS = 29 cm and SQ = 20 cm. TU = 15 cm, VW = 31 cm, and the height of the trapezoid UP = 12 cm. Find the difference between the shaded areas X and Y. (The dimensions are not drawn to scale.)

A triangle has sides HL = 32 in, PH = 69 in, and PL = c in, where the third side c is a whole number. How many possible values of c are there?

The quadrilateral PHIL is divided into four regions as shown below. If the areas of $\triangle POH$, $\triangle POL$, and $\triangle LOI$ are 110 cm², 90 cm², and 126 cm², respectively, find the area of $\triangle HOI$.

In the triangle PHL below, how many more small triangles must be shaded so that $\dfrac{5}{16}$ of the biggest triangle is shaded?

Two circles P and H overlap as shown below. $\dfrac16$ of circle P and $\dfrac18$ of circle H are shaded. If the shaded region is 5 cm², find the area of the unshaded portion of the figure.