Found problems: 85335
Prove that if $n$ is a positive integer then $2 (n^2 + 1) - n$ is not a square of an integer.
Determine all triplets of real numbers $ x,y,z$ satisfying \[1\plus{}x^4\leq 2(y\minus{}z)^2,\quad 1\plus{}y^4\leq 2(x\minus{}z)^2,\quad 1\plus{}z^4\leq 2(x\minus{}y)^2.\]
A cube has been divided into $27$ equal-sized sub-cubes. We take a line that passes through the interiors of as many sub-cubes as possible. How many does it pass through?
Let $a_1,a_2,\cdots, a_n$ and $b_1,b_2,\cdots, b_n$ be two permutations of the numbers $1,2,\cdots, n$. Show that
\[\sum_{i=1}^n i(n+1-i) \le \sum_{i=1}^n a_ib_i \le \sum_{i=1}^n i^2\]
Does there exist an equilateral triangle
(a) on a plane; (b) in a 3-dimensional space;
such that all its three vertices have integral coordinates?
Let $N$ be a given positive integer. Consider a permutation of $1,2,3,\cdots,N$, denoted as $p_1,p_2,\cdots,p_N$. For a section $p_l, p_{l+1},\cdots, p_r$, we call it "extreme" if $p_l$ and $p_r$ are the maximum and minimum value of that section. We say a permutation $p_1,p_2,\cdots,p_N$ is "super balanced" if there isn't an "extreme" section with a length at least $3$. For example, $1,4,2,3$ is "super balanced", but $3,1,2,4$ isn't. Please answer the following questions:
1. How many "super balanced" permutations are there?
2. For each integer $M\leq N$. How many "super balanced" permutations are there such that $p_1=M$?
[i]Proposed by ltf0501[/i]
Find all polynomials P(x) such that
P(x)+P(1/x)=x+1/x
Determine all the functions $f : \mathbb{R}\rightarrow\mathbb{R}$ that satisfies the following.
$f(xf(x)+f(x)f(y)+y-1)=f(xf(x)+xy)+y-1$
Find all real values of $u$ such that the curves $y=x^2+u$ and $y=\sqrt{x-u}$ intersect in exactly one point.
$p$ is a prime. Find all relatively prime positive integers $m$, $n$ such that
\[
\frac{m}{n}+\frac{1}{p^2}=\frac{m+p}{n+p}
\]
[b]10.[/b] A car is used by $n$ drivers. Every morning the drivers choose by drawing that one of them who will drive the car that day. Let us define the random variable $\mu (n)$ as the least positive integer such that each driver drives at least one day during the first $\mu (n)$ days. Find the limit distribution of the random variable
$\frac {\mu (n) -n \log n}{n}$
as $n \to \infty$. [b](P. 9)[/b]
Show that there exists an integer polynomial $P$ such that $P(1) = 2024$ and the set of prime divisors of {$P(2^k)$},$k=0,1,2,.....$ is an infinite set.
Given $P(z)= z^2 +az +b,$ where $a,b \in \mathbb{C}.$ Suppose that $|P(z)|=1$ for every complex number $z$ with $|z|=1.$ Prove that $a=b=0.$
Eight students were asked to solve $8$ problems (the same set of problems for each of the students).
(a) Each problem was solved by $5$ students. Prove that one canfind two students so that each of the problems was solved by at least one of them.
(b) If each problem was solved by $4$ students, then it is possible that no such pair of students exists. Prove this.
(S Tokarev)
Let $n \ge 50 $ be a natural number. Prove that $n$ is expressible as sum of two natural numbers $n=x+y$, so that for every prime number $p$ such that $ p\mid x$ or $p\mid y $ we have $ \sqrt{n} \ge p $. For example for $n=94$ we have $x=80, y=14$.
Show that the equation $\{x^3\}+\{y^3\}=\{z^3\}$ has infinitely many rational non-integer solutions.
What is the maximum value of the function $$\frac{1}{\left|x+1\right|+\left|x+2\right|+\left|x-3\right|}?$$
$\text{(A) }\frac{1}{3}\qquad\text{(B) }\frac{1}{4}\qquad\text{(C) }\frac{1}{5}\qquad\text{(D) }\frac{1}{6}\qquad\text{(E) }\frac{1}{7}$
We consider a fixed point $P$ in the interior of a fixed sphere$.$ We construct three segments $PA, PB,PC$, perpendicular two by two$,$ with the vertexes $A, B, C$ on the sphere$.$ We consider the vertex $Q$ which is opposite to $P$ in the parallelepiped (with right angles) with $PA, PB, PC$ as edges$.$ Find the locus of the point $Q$ when $A, B, C$ take all the positions compatible with our problem.
Let $a,b,c$ be positive reals. Prove that
\[ \frac{(a-b)(a-c)}{2a^2 + (b+c)^2} + \frac{(b-c)(b-a)}{2b^2 + (c+a)^2} + \frac{(c-a)(c-b)}{2c^2 + (a+b)^2} \geq 0. \]
[i]Calvin Deng.[/i]
Let the sum of positive numbers $x_1, x_2, ... , x_n$ be $1$. Let $s$ be the greatest of the numbers $$\left\{\frac{x_1}{1+x_1}, \frac{x_2}{1+x_1+x_2}, ..., \frac{x_n}{1+x_1+...+x_n}\right\}$$
What is the minimal possible $s$? What $x_i $correspond it?
How many positive integer solutions does the equation have $$\left\lfloor\frac{x}{10}\right\rfloor= \left\lfloor\frac{x}{11}\right\rfloor + 1?$$
($\lfloor x \rfloor$ denotes the integer part of $x$, for example $\lfloor 2\rfloor = 2$, $\lfloor \pi\rfloor = 3$, $\lfloor \sqrt2 \rfloor =1$)
Let $x_1, x_2, x_3$ be positive real numbers. Prove that
$$\frac{(x_1^2+x_2^2+x_3^2)^3}{(x_1^3+x_2^3+x_3^3)^2}\le 3$$
Suppose that $x, y, z$ are non-zero real numbers such that $$\begin{cases}x = 2 - \dfrac{y}{z} \\ \\ y = 2 -\dfrac{z}{x} \\ \\ z = 2 -\dfrac{x}{y}.\end{cases}$$
Find all possible values of $T = x + y + z$
Let $n$ be a positive integer. $S_1, S_2, \ldots, S_n$ are pairwise non-intersecting sets, and $S_k $ has exactly $k$ elements $(k = 1, 2, \ldots, n)$. Define $S = S_1\cup S_2\cup\cdots \cup S_n$. The function $f: S \to S $ maps all elements in $S_k$ to a fixed element of $S_k$, $k = 1, 2, \ldots, n$. Find the number of functions $g: S \to S$ satisfying $f(g(f(x))) = f(x).$
The natural number $N$ is a multiple of $2009$ and the sum of its (decimal) digits equals $2009$.
(a) Find one such number.
(b) Find the smallest such number.