SEAMO X 2024 - Grade 7 & 8

Admin dot · 2026-05-20T04:34:20Z

Find the remainder when $2024^{2024}$ is divided by 7. In triangle 𝐴𝐵𝐶, 𝐴𝐵 = 3, 𝐵𝐶 = 4 and 𝐴𝐶 = 5. Given that sin 2𝐴 = $frac{𝑥}{𝑦}$ where 𝑥 and 𝑦 are coprime positive integers, find the value of 𝑥 + 𝑦. Find the remainder when $𝑥^3 + 3𝑥^2 + 2𝑥 + 1$ is divided by (𝑥 − 2). Evaluate $$sqrt[256]{(2+1)(2^2+1)(2^4+1)(2^8+1)...(2^{256}+1)+1}$$ Find the least value of $2 log_{100} 𝑎 − log_𝑎(0.0001)$, for 𝑎 > 1. It is known that $overline{𝑎𝑏𝑐𝑑}$ is a multiple of 11 and 𝑏 + 𝑐 = 𝑎, $overline{𝑏𝑐}$ is a perfect square. Find the smallest such number, given that none of its digits is 0. It is given that $$y=sqrt{frac{x^2-2}{5x-4}}-sqrt{frac{x^2-2}{4-5x}}+2$$ Find the value of $x^2+y^2$. For each positive integer 𝑛, define $$A_n=frac{20^n+24^n}{n!},text{ where }n!=1times 2times 3times cdots times n$$ Find the value of 𝑛 that maximises $𝐴_𝑛$. How many ways are there to put 7 different-coloured beads into 4 identical baskets so that each basket has at least ONE bead? Evaluate $$480(frac{1}{2^2-1}+frac{1}{3^2-1}+cdots +frac{1}{15^2-1})$$ 𝛼 and 𝛽 are 2 distinct real roots of $𝑥^2 + 2(𝑘 + 3)𝑥 + 𝑘^2 + 3 = 0$, where 𝑘 is an integer. Find the minimum value of $(𝛼 − 1)^2 + (𝛽 − 1)^2$. If 𝑎, 𝑏 and 𝑐 are real numbers such that 𝑎 + 2𝑏 + 𝑐 = 4, find the maximum value of 𝑎𝑏 + 𝑏𝑐 + 𝑐𝑎. 𝐴𝐵𝐶𝐷 is a square with side of length 2 cm. 𝐵𝑃𝐶 is an equilateral triangle. If the area of ∆𝐵𝑃𝐷 is 𝑘 $text{cm}^2$, find the value of $sqrt{3} − 𝑘$. The figure below shows a solid cube of volume 1 $text{cm}^3$. Let 𝑀 be the midpoint of the edge 𝐺𝐶. If the shortest path for an ant to crawl from the vertex 𝐴 to 𝑀 is $frac{sqrt{𝑎}}{𝑏}$ cm, where 𝑎, 𝑏 are integers and 𝑎 has no squared factor, find 𝑎 + 𝑏. The point 𝐼 below is the in-center of ∆𝐴𝐵𝐶. The line 𝐴𝐼 produced meets the circumcircle at 𝐷. It is known that 𝐴𝐵 = 3, 𝐴𝐶 = 4, and $𝑆_{Δ𝐼𝐵𝐶} = 𝑆_{Δ𝐷𝐵𝐶}$. Find the value of 4𝐵𝐶.

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Find the remainder when $2024^{2024}$ is divided by 7.

In triangle 𝐴𝐵𝐶, 𝐴𝐵 = 3, 𝐵𝐶 = 4 and 𝐴𝐶 = 5. Given that sin 2𝐴 = $\frac{𝑥}{𝑦}$ where 𝑥 and 𝑦 are coprime positive integers, find the value of 𝑥 + 𝑦.

Find the remainder when $𝑥^3 + 3𝑥^2 + 2𝑥 + 1$ is divided by (𝑥 − 2).

Evaluate $$\sqrt[256]{(2+1)(2^2+1)(2^4+1)(2^8+1)...(2^{256}+1)+1}$$

Find the least value of $2 \log_{100} 𝑎 − \log_𝑎(0.0001)$, for 𝑎 > 1.

It is known that $\overline{𝑎𝑏𝑐𝑑}$ is a multiple of 11 and 𝑏 + 𝑐 = 𝑎, $\overline{𝑏𝑐}$ is a perfect square. Find the smallest such number, given that none of its digits is 0.

It is given that $$y=\sqrt{\frac{x^2-2}{5x-4}}-\sqrt{\frac{x^2-2}{4-5x}}+2$$ Find the value of $x^2+y^2$.

For each positive integer 𝑛, define $$A_n=\frac{20^n+24^n}{n!},\text{ where }n!=1\times 2\times 3\times \cdots \times n$$ Find the value of 𝑛 that maximises $𝐴_𝑛$.

How many ways are there to put 7 different-coloured beads into 4 identical baskets so that each basket has at least ONE bead?

Evaluate $$480(\frac{1}{2^2-1}+\frac{1}{3^2-1}+\cdots +\frac{1}{15^2-1})$$

𝛼 and 𝛽 are 2 distinct real roots of $𝑥^2 + 2(𝑘 + 3)𝑥 + 𝑘^2 + 3 = 0$, where 𝑘 is an integer. Find the minimum value of $(𝛼 − 1)^2 + (𝛽 − 1)^2$.

If 𝑎, 𝑏 and 𝑐 are real numbers such that 𝑎 + 2𝑏 + 𝑐 = 4, find the maximum value of 𝑎𝑏 + 𝑏𝑐 + 𝑐𝑎.

𝐴𝐵𝐶𝐷 is a square with side of length 2 cm. 𝐵𝑃𝐶 is an equilateral triangle. If the area of ∆𝐵𝑃𝐷 is 𝑘 $\text{cm}^2$, find the value of $\sqrt{3} − 𝑘$.

The figure below shows a solid cube of volume 1 $\text{cm}^3$. Let 𝑀 be the midpoint of the edge 𝐺𝐶. If the shortest path for an ant to crawl from the vertex 𝐴 to 𝑀 is $\frac{\sqrt{𝑎}}{𝑏}$ cm, where 𝑎, 𝑏 are integers and 𝑎 has no squared factor, find 𝑎 + 𝑏.

The point 𝐼 below is the in-center of ∆𝐴𝐵𝐶. The line 𝐴𝐼 produced meets the circumcircle at 𝐷. It is known that 𝐴𝐵 = 3, 𝐴𝐶 = 4, and $𝑆_{Δ𝐼𝐵𝐶} = 𝑆_{Δ𝐷𝐵𝐶}$. Find the value of 4𝐵𝐶.

This topic was modified 7 hari ago 2 times by Admin dot

Posted : 20/05/2026 4:34 am

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