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Evaluate $$\frac{1}{\sqrt{8}+\sqrt{11}}+\frac{1}{\sqrt{11}+\sqrt{14}}+\frac{1}{\sqrt{14}+\sqrt{17}}+\cdot\cdot\cdot+\frac{1}{\sqrt{6038}+\sqrt{6041}}$$
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Yellow, Red and Blue are used to colour the rings shown below, such that no adjacent rings have the same colours; each colour is used exactly twice. How many ways are there to do so?
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How many real numbers satisfy the equation below? $$\frac{a^{5}+a^{4}-2a^{3}-a^{2}-a+2}{a^{3}-a}$$
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Let $x, y, z$ be three numbers all larger than 1 and let $w$ be a positive number such that $\log_{x}w=24$, $\log_{y}w=40$, and $\log_{xyz}w=12$. Find the value of $\log_{z}w.$
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Given that $a=\frac{\sqrt{5}-1}{2}$ , evaluate: $$\sin^{2}1^{\circ}+\sin^{2}2^{\circ}+\sin^{2}3^{\circ}+\cdot\cdot\cdot+\sin^{2}360^{\circ}$$
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How many numbers at the most can you select from $\{1, 2, 3, ..., 2024\}$ so the sum of any 3 of the numbers is divisible by 33?
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$E, F$ are points on $BC, CD$ respectively in rectangle $ABCD$. It is given $AE=4$, $EF=3$, and $AF=5.$ Find the area of $ABCD$ if $S_{\triangle ABE}=S_{\triangle ECF}+S_{\triangle ADF}.$
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$PA, PB$ are tangents to the circle. $PQR$ is a straight line cutting through the circle and intersecting $AB$ at $S$. $F$ is a point on $AR$ such that $QF \parallel PA$. $QF$ intersects $AB$ at $E$. Given that $QE=3,$ find $EF$.
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Find the value of: $$f(1)+f(2) + f (3) +\cdot\cdot\cdot+f(2024)$$
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Find the remainder when $1^{5}+2^{5}+3^{5}+\cdot\cdot\cdot+99^{5}+100^{5}$ is divided by 4.
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$DN$ and $AD$ are lines perpendicular to $AB$ and $BC$ respectively in $\triangle ABC$. $DM$ is perpendicular to $AC$. $NM$ and $BC$ produced meet at $E$. It is known $AD=DE=1$. Find: $$\cot \angle CAD \cdot \cot \angle BAD$$
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Suppose $a$ and $b$ are the roots of the quadratic equation $x^{2}+(\sin 10^{\circ})x+1=0$, and $c$ and $d$ are the roots of the quadratic equation $x^{2}+(\cos 10^{\circ})x-1=0$. Find the value of: $$\frac{1}{a^{2}}+\frac{1}{b^{2}}+\frac{1}{c^{2}}+\frac{1}{d^{2}}$$
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A point $(x, y)$ in the $xy$-plane is called a lattice point if both $x$ and $y$ are integers. For any integer $n$, let $f(n)$ be the number of lattice points on the line segment joining $(0,0)$ and $(n,n+5)$. For instance, we have $f(0)=6$ and $f(1)=2$. Find: $$f(1)+f(2) + f (3) +\cdot\cdot\cdot+f(2024)$$
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In the figure shown (in the file), $E$ is the midpoint of $BC$, $M$ is the midpoint of $AC$. Suppose that $D$ is a point on $AE$ such that $AD=BD=CD=169\text{ cm}$, and $EM=156\text{ cm}$. Find the length of $DE$.
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Let $\triangle ABC$ be an equilateral triangle inscribed in a circle, and let $M$ be a point on the arc $BC$ as shown. Find $AM$, given that $BM = 3$ and $MC = 5$.
Posted : 20/05/2026 6:20 am
