PIMSO 2025 - Mathematics - Senior Secondary - National Round

The remainder of $\frac{3^{677}}{9^{45}}$ when divided by 80 is n. What is n?

The highest power of 200 in $90!$ is $n$. The highest power of 9 in $30!$ is $m$. What is $m\times n$?

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

The remainder obtained when $1!+2!+3!+\cdots+2023!+2024!+2025!$ Is divided by 14, is

There are n ordered pairs (a, b) that satisfy the equation $ab=2^4\times3^2\times442$. If a and b are integers, what is n?

How many numbers that divide $(2.4\times10^3)\div(3\times10^{-2})$ are perfect squares?

If $A=\{a_1,a_2,a_3,\ldots,a_m\}$, $B=\{b_1,b_2,b_3,\ldots,b_n\}$ and $C=\{c_1,c_2,c_3,\ldots,c_l\}$, then $A\times B\times C=\{a_1,a_2,a_3,\ldots,a_m\}\times\{b_1,b_2,b_3,\ldots,b_n\}\times\{c_1,c_2,c_3,\ldots,c_l\}$. Find the cardinality of the power set of $A\times B\times C$.

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

Original choices: A. -1; B. 0; C. 1; D. 2.

Editorial note: The printed conditions do not certify a unique valid answer. Explain the source issue rather than choose an automatic key.

What is the square root of the sum of the infinite series $\{\frac52,\frac56,\frac5{12},\frac5{20},\ldots\}$?

The There are 16 different books to be placed on 3 labeled shelves: Shelf X, Shelf Y, and Shelf Z. The books must be arranged so that: Shelf Y holds 1 more book than Shelf X, Shelf Z holds 2 more books than Shelf Y, and each book is placed on exactly one shelf. In how many distinct ways can the books be distributed across the 3 shelves under these conditions?

What is the remainder when $(x-2)^{2019}+(x-3)^{2021}$ is divided by $(x^2-5x+6)$? Hint! Use Remainder Theorem

If $(189x+567y)$ is a square number and both x and y are positive integers, what is the minimum value of $x^2+y^2$?

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

The sides of a cyclic quadrilateral are 6,8,10, and 12 cm. If the area is in the form $a\sqrt b$, what is a+b? Hint! A cyclic quadrilateral is a four-sided figure (quadrilateral) in which all four vertices lie on the circumference of a single circle.

If $(3+x)(3+x^2)(3+x^3)\cdots(3+x^{100})=\sum_{r=0}^n a_nx^r$, then n equals

If What is the value of a+b in the binomial expansion $(ax+by)^{10}$ if the coefficient of the 3rd term of the binomial expansion of $(ax+by)^9$ is 36 and if the coefficient of the 4th term of the binomial expansion of $(ax+by)^9$ is equal to 84?

One unit in the grid shown below is 1 cm long. Find the area of the shaded figure.

If $x\in\{1,2,3,\ldots,9\}$ and $f_n(x)=xxx\ldots x$ (n digits), then $f_n^2(3)+f_n(2)=$?

Three 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: The printed conditions do not certify a unique valid answer. Explain the source issue rather than choose an automatic key.

Debby walked 10m towards East, then turns to her left and walked 4m and then turns to her right and walked 3m. Then, she turned to his right again and walked 2m. Then she turned to her left and she walked 6m. Again, she turned to her left and walked 8m. Finally, she turned to her left again and walked 9m. What is the shortest distance between her final and initial position?