PIMSO 2025 - Mathematics - Secondary 3 - National Round

What is the remainder when $\frac{32^{405}}{2^{25}}$ is divided by 9?

The highest power of 8 in 90! is n. The highest power of 25 in 90! is m. What is m + n?

The prime factorization of the product of the factors of 15 is $3^2\times5^2$. What is the highest exponent on the prime factorization of the product of the factors of 140? Hint! For any positive integer $n$, the product of all its factors (divisors) is $n^{d(n)/2}$, where $d(n)$ is the number of positive divisors of $n$.

How many numbers from 1 to 1500 are relatively prime to both 8 and 15? Hint! Two integers are relatively prime (or coprime) if their only common positive divisor is 1. This means their greatest common divisor (GCD) is 1.

How many n ordered pairs (a, b) that satisfy the equation ab=7657. If a and b are integers?

What is the sum of all perfect cube divisors of 10,125?

How many different measures of sides are there in a square whose area is between $4^8$ and $7^{12}$?

Original choices: A. 117649; B. 117393; C. 117392; D. 117648.

Editorial note: Explain the source limitation and any conditional interpretation; this item is manually assessed.

If $A=\{p,i,m,s,o\}$ and $B=\{1,2,3,4,5,6,7,8,9\}$ then $D=A\times B=\{p,i,m,s,o\}\times\{1,2,3,4,5,6,7,8,9\}$. How many subsets does D have? Hint! $M=\{1,2\}$, $N=\{3,4,5\}$, $M\times N\;\{(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)\}$. Subset: A set whose elements are all members of another set.

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

If (475x + 1900y) is a square number and both x and y are positive integers, what is the minimum value of (x + y)?

In the equation 15x + 16y + 17z = 208, where x, y, and z are integers, find the value of 4(x + y + z).

Original choices: A. 52; B. 50; C. 48; D. 46.

Editorial note: Explain the source limitation and any conditional interpretation; this item is manually assessed.

Find the sum of the coefficients of the polynomial $(7x^3-2x+14)(11+2x-8x^2)^2(2x-8)^3$.

What is the remainder when $(x+5)^{2019}+(x+4)^{2021}$ is divided by $(x^2+9x+20)$?

At a company picnic, the organizers prepared food and drinks for the employees. Every 12 employees share 5 pizzas. Every 6 employees share 3 bowls of salad. Every 9 employees share 2 bottles of juice. The coordinator noticed that the total number of sandwiches, salads, and juice bottles exceeds the number of employees by 15. How many employees attended the picnic?

The quadrilateral PHIL below is a rectangle. B is the midpoint of side PH. The areas of $\triangle JAG$ and $\triangle CID$ are 60 cm² and 20 cm², respectively. The areas of quadrilaterals FECH and JEDL are 420 cm² and 480 cm², respectively. Find the area of $\triangle BFG$.



Original choices: A. 20 cm²; B. 25 cm²; C. 30 cm²; D. 35 cm².

Editorial note: Explain the source limitation and any conditional interpretation; this item is manually assessed.

In the figure below, the arcs PMS and HIR are portions of concentric circles with center at O. Central angles HIO and IOR are 15° and 30°, respectively. Arc HI is $\frac{3\pi}{4}$ m. The ratio of radius OS to OR is 4:3. Find the area of the shaded region. Use $\pi=\frac{22}{7}$.

In the Park, there is a garden 72 m long and 52 m wide. It is divided into four portions as shown in the figure. Given that AB = KY = 48 m, MN = HU = 38 m, and the area of quadrilateral MAON is 816 m², determine the area of BAOU.



Original choices: A. 684 ²; B. 792 m²; C. 936 m²; D. 1056 m².

Editorial note: Explain the source limitation and any conditional interpretation; this item is manually assessed.

The diagonal length AB of a cube is $\sqrt{432}$ cm. How many liters is the volume of the cube?

The perimeter of a triangle formed by points A(1,2), B(3,-4), and C(-2,-3) is in the form $x\sqrt{5x}+\sqrt{2y}+\sqrt{2z}$. What is $x+y+z$?

Angle TOA measures 120°. What is $(\sin\theta+\cos\theta)^2$?