Practice (90/1000)

2349

Solve in positive integers the equation $x^2 + y^2 = z^4$, where $\gcd(x,y)=1$ and $x$ is even.

2350

Show that the sum and difference of two squares cannot be both squares themselves.

2351

If for a given positive integer $n$, the equation $x^n + y^n = z^n$ is not solvable in positive integer. Show that the equation $$x^{2n} + y^{2n} = z^{2n}$$ is not solvable in positive integers either.

2352

Show that the equation $$x^2 + y^2 -19xy - 19 =0$$ is not solvable in integers.

2353

Solve in positive integers $$x^3 + y^3 + z^3 = 3xyz$$

2354

What is the minimal number of masses required in order to measure any weight between 1 and $n$ grams. Note that a mass can be put on either sides of the balance.

2355

Compute the value of $\sin{18^\circ}$ using regular geometry.

2356

Let $F$ be a point inside $\triangle{ABC}$ such that $\angle{CAF} = \angle{FAB} = \angle{FBC} = \angle{FCA}$, show that the lengths of three sides form a geometric sequence.

2357

Suppose the point $F$ is inside a square $ABCD$ such that $BF=1$, $FA=2$, and $FD=3$, as shown. Find the measurement of $\angle{BFA}$.

2358

Solve this equation in positive integers $$x^3 - y^3 = xy + 61$$

2359

Solve in integers the equation $$(x+y)^2 = x^3 + y^3$$

2360

Let $k$ be a positive integer, show that $(4k+3)$ cannot be a square number.

2361

How many numbers in this series are squares? $$1, 14, 144, 1444, 14444, \cdots$$

2362

Find all positive integer $n$ such that $n$ is a square and its last four digits are the same.

2363

Solve the following equation in positive integers: $15x - 35y + 3 = z^2$

2364

Find a four-digit square number whose first two digits are the same and the last two digits are the same too.

2365

Solve the following equation in positive integers: $3\times (5x + 1)=y^2$

2366

Find all pairs of integers $(x, y)$ such that $5\times (x^2 + 3)= y^2$.

2368

If we arrange all the square numbers ascendingly as a queue: $1491625364964\cdots$ What is the $612^{th}$ digit?

2369

Mary typed a six-digit number, but the two 1s she typed didn't show. What appeared was 2002. How many different six-digit numbers could she have typed?

2370

How many ordered triples of positive integers $(x, y, z)$ satisfy $(x^y)^z = 64$?

2371

Let $P(x) = kx^3 + 2k^2x^2 + k^3$. Find the sum of all real numbers $k$ for which $x - 2$ is a factor of $P(x)$.

2372

The vertex $E$ of square $EFGH$ is at the center of square $ABC$D. The length of a side of $ABCD$ is 1 and the length of a side of $EFGH$ is 2. Side $EF$ intersects $CD$ at $I$ and $EH$ intersects $AD$ at $J$. If angle $EID = 60^\circ$, what is the area of quadrilateral $EIDJ$?

2373

What is the smallest integer n for which any subset of {1, 2, 3, . . . , 20} of size $n$ must contain two numbers that differ by 8?

2374

Let $f$ be a real-valued function such that $f(x) + 2f(\frac{2002}{x}) = 3x$ for all $x > 0$. Find $f(2)$.

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