Practice (97)

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Find all the ordered integers $(a, b, c)$ which satisfy $a+b+c=450$ and $\sqrt{a+\sqrt{b}}+\sqrt{a-\sqrt{b}}=2c$.

Find all pairs of integers $(x,y)$ such that $$x^3+y^3=(x+y)^2$$.

Find $a$, $b$, so that $(x-1)^2$ divides $ax^4+bx^3+1$.

Show that if there exist integer $x$, $y$, and $z$ such that $3^x + 4^y=5^z$, then both $x$ and $z$ must be even.


Find all ordered integer pairs $(x, y)$ such that $x^3 + y^3=2019$.

Find all the integer pairs $(x, y)$ such that $x^3 = 2^y + 15$.

Determine all positive integer $n$ such that the following equation is solvable in integers: $$x^n + (2+x)^n + (2-x)^n = 0$$


$\textbf{Make Four Liters}$

If you have an infinite supply of water, a $5$-liter bucket, and a $3$-liter bucket, how would you measure exactly $4$ liters of water? The buckets do not have any intermediate scales.


Let $T$ be the triangle in the coordinate plane with vertices $\left(0,0\right)$, $\left(4,0\right)$, and $\left(0,3\right)$. Consider the following five isometries (rigid transformations) of the plane: rotations of $90^{\circ}$, $180^{\circ}$, and $270^{\circ}$ counterclockwise around the origin, reflection across the $x$-axis, and reflection across the $y$-axis. How many of the $125$ sequences of three of these transformations (not necessarily distinct) will return $T$ to its original position? (For example, a $180^{\circ}$ rotation, followed by a reflection across the $x$-axis, followed by a reflection across the $y$-axis will return $T$ to its original position, but a $90^{\circ}$ rotation, followed by a reflection across the $x$-axis, followed by another reflection across the $x$-axis will not return $T$ to its original position.)


Solve $17^x-15^y=2$ in positive integers.