PIMSO 2025 - Mathematics - Secondary 1 - National Round

$\overline{XYYX}$ is a cube number. What is X + Y?

What is the unit digit of the simplest form of $\frac{20^{50}}{25^{20}}$?

How many times can you multiply 20 by itself in 30!?

$n!=n\times(n-1)\times(n-2)\times\cdots\times1$

If P, I, M, S and O are whole numbers, How many factors of $P\times I\times M\times S\times O$.

$7.65=P+\cfrac{1}{I+\cfrac{1}{M+\cfrac{1}{S+\cfrac{1}{O}}}}$

Three pipes can fill a tank individually in 6 hours, 8 hours, and 12 hours. If all three pipes are open at the same time, how long will it take to fill the tank?

Simplify the following sum into $\frac{x}{y}$ such that such that x and y have no common factors. Find the value of xy.

$\frac{2025}{100\times102}+\frac{2025}{102\times104}\times\frac{2025}{106\times108}\times\cdots\times\frac{2025}{288\times300}$



Original choices: A. 135; B. 108; C. 405; D. 408.

Editorial note: The printed conditions need clarification. Discuss the issue and justify any conditional interpretation; no automatic answer is assigned.

There are n ordered pairs (a, b) that satisfy the equation ab=442. If a and b are integers, what is n?

If $b\lt a\lt0$, $c>0$, and $|c|>|b|>|a|$, simplify $2|a+b|+|b+c|+|c+a|$?

How many possible divisors does 15! or 15 × 14 × 13 × 12 × … × 3 × 2 × 1 have?

Find the sum of the coefficients of the polynomial $(4x^3-7x+12)(11+5x-10x^2)^2$.

Suppose that in a set of 9 positive integers 18 is added. When 18 is added the average of all the integers in the set increases by 1.5. Find the sum of the 9 positive integers.

Given that $(A+B)=12$, $A\ne B$ and $(\overline{AB}-\overline{BA})$ is a square number, find $A^2-B$.

Note! two-digit numbers: $\overline{AB}=10A+B$

If $(207x+1035y)$ is a square number and both x and y are positive integers, what is the minimum value of $(x+y)$?

Two squares overlap as shown below. The side of the bigger square is x cm while the smaller square is y cm. Both x and y are integers. The shaded region between the squares is 32 cm². Find the sum of all possible values (x - y) that satisfy the condition.

Source-label note: The printed labels are A, B, C, B; the fourth choice is indexed D here, without changing its value.

The sequence is defined as

b, 2b+3, 3b+6, 4b+9, …

For a particular value of b, the first term is equal to b, and each term after the first is 10 greater than the previous term. What is the value of the 2025th term in the sequence?

In the composite figure below, the total shaded area is $\frac{25\pi}{2}$ cm². Given that $PH\perp HI$, $PI=LI$ and $PI\perp LI$, calculate the area of $\triangle PIL$.

Given the three sides of $\triangle PHL$ as shown below, find the area of the inscribed circle in the triangle.

In the figure shown below, the circle O has points D, E, F on the circumference. B is a point on a tangent line AC that touches the circle at B. $\angle ABD=77^\circ$ and $\angle DEF=64^\circ$ Find $\angle BDF$.

How many trailing zeroes are there in 2025!?

Note! Trailing zeros are the zeros at the end of a number when written in decimal form.

If $(1-\frac{1}{2^2})(1-\frac{1}{3^2})(1-\frac{1}{4^2})(1-\frac{1}{5^{20}})\cdots(1-\frac{1}{2024^2})(1-\frac{1}{2025^2})=\frac{A}{B}$, $\gcd(A,B)=1)$ Find $A+B$.



Original choices: A. 2026; B. 3038; C. 3039; D. 4051.

Editorial note: The printed conditions need clarification. Discuss the issue and justify any conditional interpretation; no automatic answer is assigned.