Practice (90/1000)
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Let $a_1, a_2, \cdots, a_{2n+1}$ be a set of integers such that, if any one of them is removed, the remaining ones can be divided into two sets of $n$ integers with equal sums. Prove $a_1 = a_2 = \cdots = a_{2n+1}$.
Prove that if positive integer $a$ and $b$ are such that $ab+1$ divides $a^2 + b^2$. then $$\frac{a^2+b^2}{ab+1}$$ is a square number.
Find the maximal value of $m^2+n^2$ if $m$ and $n$ are integers between $1$ and $1981$ satisfying $(n^2-mn-m^2)^2=1$.
Solve in positive integers $x^2 + y^2 + x+y+1 = xyz$
Solve in positive integers $x$, $y$, $u$, $v$ the system of equations
$$
\left\{
\begin{array}{ll}
x^2 +1 &= uy\\
y^2 + 1&= vx
\end{array}
\right.
$$
Show that if there is a triple $(x, y, z)$ of positive integers such that $$x^2 +y^2 +1 = xyz$$
then $z=3$, and find all such triples.
Find all solutions of $a^3 + b^3 = 2(s^2+t^2)$
Solve in integers $x^2 + y^2 +z^2 - 2xyz=0$.
Show that there exists an infinite sequence of positive integers $a_1, a_2, \cdots$ such that $$S_n=a_1^2 + a_2^2 + \cdots + a_n^2$$ is square for any positive integer $n$.
Show that the sides of a Pythagorean triangle in which the hypotenuse exceeds the larger leg by 1 are given by $\frac{n^2-1}{2}$, $n$ and $\frac{n^2+1}{2}$
Show that if the lengths of all the three sides in a right triangle are whole numbers, then radius of its incircle is always a whole number too.
Let $a$, $b$, $c$, $d$, and $e$ be five positive integers. If $ab+c=3115$, $c^2+d^2=e^2$, both $a$ and $c$ are prime numbers, $b$ is even and has $11$ divisors. Find these five numbers
25 boys and 8 girls sit in a circle. If there are at least two boys between any two girls, how many different sitting plans are there? (Two sitting plans will be considered as the same if they differ just by rotating.)
Solve in positive integers the equation $$m^2 - n^2 - 3n = 5$$
Let $ ABC$ be acute triangle. The circle with diameter $ AB$ intersects $ CA,\, CB$ at $ M,\, N,$ respectively. Draw $ CT\perp AB$ and intersects above circle at $ T$, where $ C$ and $ T$ lie on the same side of $ AB$. $ S$ is a point on $ AN$ such that $ BT = BS$. Prove that $ BS\perp SC$.
Let $ a,\, b,\, c$ be side lengths of a triangle and $ a+b+c = 3$. Find the minimum of \[ a^{2}+b^{2}+c^{2}+\frac{4abc}{3}\]
Sequence $ \{a_{n}\}$ is defined by $ a_{1}= 2007,\, a_{n+1}=\frac{a_{n}^{2}}{a_{n}+1}$ for $ n \ge 1.$ Prove that $ [a_{n}] =2007-n$ for $ 0 \le n \le 1004,$ where $ [x]$ denotes the largest integer no larger than $ x.$
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.
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 \[ \{n | f(n^{2}-214n-1998) \geq 0,\ n \in\mathbb{Z}\}.\]
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}$.
The inradius of triangle $ ABC$ is $ 1$ and the side lengths of $ ABC$ are all integers. Prove that triangle $ ABC$ is right-angled.
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.
As shown below, $ABCD$ is a unit square, $\angle{CBE} = 20^\circ$, and $\angle{FBA} = 25^\circ$. Find the circumstance of $\triangle{DEF}$.
Find all the Pythagorean triangles whose two sides are consecutive integers.
Find all the positive integer triplets $(m, n, k)$ that satisfy the equation $$1!+2!+3!+\cdots+m!=n^k$$
where $m, n , k > 1$