Practice (TheColoringMethod)
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Find the largest 7-digit integer such that all its 3-digit subpart is either a multiple of 11 or multiple of 13.
Find a $4$-digit square number $x$ such that if every digit of $x$ is increased by 1, the new number is still a perfect square.
If $a+b=\sqrt{5}$, compute $\frac{a^2 -a^2b^2 + b^2 +2ab}{a+ab+b}$.
What is Heron's formula to calculate a triangle's area given the lengths of three sides?
(Stewart's Theorem) Show that $$b^2m + c^2n = a(d^2 +mn)$$
What is the angle bisector's theorem?
Show that the next integer bigger than $(\sqrt{2}+1)^{2n}$ is divisible by $2^{n+1}$.
Let $m$ be an odd positive integer, and not a multiple of 3. Show that the integer part of $4^m - (2+\sqrt{2})^m$ is a multiple of 112.
As shown in the figure, in a regular triangular house, all the rooms are in the shape of regular triangles. There are doors between adjacent rooms. Starting from one of the rooms, go through the doors to visit other rooms, without repeating rooms or leaving the house. Including the starting room, how many rooms can be visited?
Solve $4x^2+27x-9\equiv 0\pmod{15}$
Solve $5x^3 -3x^2 +3x-1\equiv 0\pmod{11}$
Solve $3x^{15}-x^{13}-x^{12} -x^{11} -3x^5 +6x^3 -2x^2 +2x-1\equiv 0 \pmod{11}$
Solve $14x\equiv 30 \pmod{21}$
Solve $17x\equiv 229\pmod{1540}$.
Solve $$\left\{ \begin{array}{rcl} x &\equiv 2 &\pmod{3}\\ x &\equiv 2 &\pmod{5}\\ x &\equiv -3 &\pmod{7}\\x &\equiv -2 &\pmod{13} \end{array}\right.$$
How many digits are there if the numbers $2^{2015}$ and $5^{2015}$ are written one after another?
What value of $a$ satisfies $27x^3 - 16\sqrt{2}=(3x-2\sqrt{2})(9x^2 + 12x\sqrt{2}+a)$?
If the first $25$ positive integers are multiplied together, in how many zeros does the product terminate?
What is the smallest positive number $x$ for which $\left(16^\sqrt{2}\right)^x$ represents a positive integer?
Of the pairs of positive integers $(x, y)$ that satisfies $3x+7y=188$, which ordered pair has the least positive difference $x-y$?
What is the smallest positive integer greater than $5$ which leaves a remainder of $5$ when divided by each of $6$, $7$, $8$, and $9$?
What are all the ordered pairs of positive numbers $(x, y)$ for which $x=\sqrt{2y}$ and $y=\sqrt{x}$?
How many minutes past $4$ o'clock are the hands of a standard $12$-hour clock first perpendicular to each other?
I first drove 16 km at 48 km/hr, then I drove 20 km at 40 km/hr, and finally I drove 24 km at 36 km/hr. What was my average speed in km/hr, for the entire trip?
How many different triangles have vertices selected from the seven points (-4, 0), (-2, 0), (0,0), (2,0), (4,0), (0,2), and (0,4)?