Practice (TheColoringMethod)

2340

For every point on the plane, one of $ n$ colors are colored to it such that: $ (1)$ Every color is used infinitely many times. $ (2)$ There exists one line such that all points on this lines are colored exactly by one of two colors. Find the least value of $ n$ such that there exist four concyclic points with pairwise distinct colors.

2342

Let $ f$ be a function given by $ f(x) = \lg(x+1)-\frac{1}{2}\cdot\log_{3}x$. a) Solve the equation $ f(x) = 0$. b) Find the number of the subsets of the set \[ \{n | f(n^{2}-214n-1998) \geq 0,\ n \in\mathbb{Z}\}.\]

2343

Let $ n$ be a positive integer and $ [ \ n ] = a.$ Find the largest integer $ n$ such that the following two conditions are satisfied: $ (1)$ $ n$ is not a perfect square; $ (2)$ $ a^{3}$ divides $ n^{2}$.

2344

The inradius of triangle $ ABC$ is $ 1$ and the side lengths of $ ABC$ are all integers. Prove that triangle $ ABC$ is right-angled.

2345

Joe is playing with a set of $6$ masses: $1$g, $2$g, $4$g, $8$g, $16$g, and $32$g. He found that some weights can be measured in more than one way. For example, $7$g can be measure by putting $1$g, $2$g, and $4$g on one side of a balance. It can also be achieved by putting $1$g and $8$g on different sides of a balance. He therefore wonder which weight can be measured using these masses in the most number of different ways? Can you help him to find it out? Describe how will you approach this problem. The final answer is optional.

2346

As shown below, $ABCD$ is a unit square, $\angle{CBE} = 20^\circ$, and $\angle{FBA} = 25^\circ$. Find the circumstance of $\triangle{DEF}$.

2347

Find all the Pythagorean triangles whose two sides are consecutive integers.

2348

Find all the positive integer triplets $(m, n, k)$ that satisfy the equation $$1!+2!+3!+\cdots+m!=n^k$$ where $m, n , k > 1$

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: $3\times (5x + 1)=y^2$

Practice (TheColoringMethod)

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