Author Archives: kevin

AMC 10 Training – Number Theory – 1

Posted on February 25, 2024 by kevin

Let $a$, $b$, $c$ be real numbers with $|a-b|=3$, $|b-c|=4$, and $|c-d|=5$. What is the sum of all possible values of $|a-d|$? What is the greatest integer less than or equal to $$\dfrac{5^{200}+3^{200}}{5^{197}+3^{197}}$$ How many negative integers can be written … Continue reading →

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AMC 8/MathLeague Mock Test – 1

Posted on January 26, 2024 by kevin

If the hands on a circular clock start at midnight, what number will the hour hand point to 1000 hours later? $(A)\ 2 \ \ \ \ \ \ \ \ \ (B)\ 4 \ \ \ \ \ \ … Continue reading →

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Geometry Challenge – 14

Posted on November 30, 2023 by kevin

In the figure as shown, $ABCD$ is a unit square, $E$ is on $BC$, $F$ is on $CD$, $\angle{EAF}=45^\circ$, $\angle{AEB}=70^\circ$. A B C D E F 70° 45° ? A B C D E F G 70° 45° ? (1) … Continue reading →

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Number Theory Challenge – 2

Posted on November 8, 2023 by kevin

How many pairs of positive integers $(x,y)$ are there such that $x<y$ and $\dfrac{x^2+y^2}{x+y}$ is an integer which is a divisor of $2835$? BIMC 2018 Click for the solution Solution: Based on the result of Number Theory Challenge – 1, … Continue reading →

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Number Theory Challenge – 1

Posted on November 4, 2023 by kevin

Let $n=4k+3$ is a prime number and $a^2+b^2\equiv 0\pmod{n}$, prove that $a\equiv b \equiv 0\pmod{n}$. Click for the solution Proof: If $a\equiv 0\pmod{n}$, because $a^2+b^2\equiv 0\pmod{n}$, $b\equiv 0\pmod{n}$. If $a\not\equiv 0\pmod{n}$, because $n$ is a prime number, according to Fermat’s … Continue reading →

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