Found problems: 451
Solve in real numbers the equation
$$\left(x^2-3x-2\right)^2-3\left(x^2-3x-2\right)-2-x=0.$$
Let $n\geq2$ be an integer. An $n$-tuple $(a_1,a_2,\dots,a_n)$ of not necessarily different positive integers is [i]expensive[/i] if there exists a positive integer $k$ such that $$(a_1+a_2)(a_2+a_3)\dots(a_{n-1}+a_n)(a_n+a_1)=2^{2k-1}.$$
a) Find all integers $n\geq2$ for which there exists an expensive $n$-tuple.
b) Prove that for every odd positive integer $m$ there exists an integer $n\geq2$ such that $m$ belongs to an expensive $n$-tuple.
[i]There are exactly $n$ factors in the product on the left hand side.[/i]
Four buddies bought a ball. First one paid half of the ball price. Second one gave one third of money that other three gave. Third one paid a quarter of sum paid by other three. Fourth paid $5\$$. How much did the ball cost?
A given natural number $N$ is being decomposed in a sum of some consecutive integers.
[b]a.)[/b] Find all such decompositions for $N=500.$
[b]b.)[/b] How many such decompositions does the number $N=2^{\alpha }3^{\beta }5^{\gamma }$ (where $\alpha ,$ $\beta $ and $\gamma $ are natural numbers) have? Which of these decompositions contain natural summands only?
[b]c.)[/b] Determine the number of such decompositions (= decompositions in a sum of consecutive integers; these integers are not necessarily natural) for an arbitrary natural $N.$
[b]Note by Darij:[/b] The $0$ is not considered as a natural number.
Show that there are infinitely many integers $m \ge 2$ such that $m \choose 2$ $= 3$ $n \choose 4$ holds for some integer $n \ge 4$. Give the general form of all such $m$.
Determine all pairs $(x, y)$ of positive integers such that \[\sqrt[3]{7x^2-13xy+7y^2}=|x-y|+1.\]
[i]Proposed by Titu Andreescu, USA[/i]
Positive integers $k, m, n$ satisfy the equation $m^2 + n = k^2 + k$. Show that $m \le n$.
Find all real triples $(a,b,c)$ satisfying
\[(2^{2a}+1)(2^{2b}+2)(2^{2c}+8)=2^{a+b+c+5}.\]
For a given positive real parameter $p$, solve the equation $\sqrt{p+x}+\sqrt{p-x }= x$.
Solve equation $x(x+1) = y(y+4)$ where $x$, $y$ are positive integers
For which real numbers $p$ does the equation $x^{2}+px+3p=0$ have integer solutions ?
Find the real roots of the equation $$(5-x)^4+ (x-2)^ 4 = 17$$ and the real roots of a more general equation $$(a - x) ^4+ (x - b)^4 = c$$
Let $a$ and $b$ be real numbers such that$$\frac{a}{a^2-5} = \frac{b}{5-b^2} = \frac{ab}{a^2b^2-5}$$where $a+b \neq 0$. $a^4 + b^4 =$ ?
Solve the equation $ \sin x\sin 2x\cdots\sin nx+\cos x\cos 2x\cdots\cos nx =1, $ for $ n\in\mathbb{N} $ and $ x\in\mathbb{R} . $
Let the positive integers $x_1$, $x_2$, $...$, $x_{100}$ satisfy the equation
\[\frac{1}{\sqrt{x_1}}+\frac{1}{\sqrt{x_2}}+...+\frac{1}{\sqrt{x_{100}}}=20.\]
Show that at least two of these integers are equal to each other.
Provided the equation $xyz = p^n(x + y + z)$ where $p \geq 3$ is a prime and $n \in \mathbb{N}$. Prove that the equation has at least $3n + 3$ different solutions $(x,y,z)$ with natural numbers $x,y,z$ and $x < y < z$. Prove the same for $p > 3$ being an odd integer.
Given that $a,b,c$ are distinct real numbers, show that the equation
\[\frac{1}{x-a}+\frac{1}{x-b}+\frac{1}{x-c}=0\]
has a real root.
Find the number of solutions to the equation:
$6\{x\}^3 + \{x\}^2 + \{x\} + 2x = 2018. $
With {x} we denote the fractional part of the number x.
Solve in $ \mathbb{C} $ the following equation: $ |z|+|z-5i|=|z-2i|+|z-3i|. $
Let $a, b, c$, and $d$ be prime numbers with $a \le b \le c \le d > 0$.
Suppose $a^2 + 2b^2 + c^2 + 2d^2 = 2(ab + bc - cd + da)$. Find $4a + 3b + 2c + d$.
Let $ABC$ be a right-angle triangle with $\angle B=90^{\circ}$. Let $D$ be a point on $AC$ such that the inradii of the triangles $ABD$ and $CBD$ are equal. If this common value is $r^{\prime}$ and if $r$ is the inradius of triangle $ABC$, prove that
\[ \cfrac{1}{r'}=\cfrac{1}{r}+\cfrac{1}{BD}. \]
The polynomials $P$, $Q$, $R$ with real coefficients, one of which is degree $2$ and two of degree $3$, satisfy the equality $P^2+Q^2=R^2$. Prove that one of the polynomials of degree $3$ has three real roots.
Find all real numbers $x$ which satisfied $|2x+1|+|x-1|=2-x$ .
Let $n\geq2$ be an integer. An $n$-tuple $(a_1,a_2,\dots,a_n)$ of not necessarily different positive integers is [i]expensive[/i] if there exists a positive integer $k$ such that $$(a_1+a_2)(a_2+a_3)\dots(a_{n-1}+a_n)(a_n+a_1)=2^{2k-1}.$$
a) Find all integers $n\geq2$ for which there exists an expensive $n$-tuple.
b) Prove that for every odd positive integer $m$ there exists an integer $n\geq2$ such that $m$ belongs to an expensive $n$-tuple.
[i]There are exactly $n$ factors in the product on the left hand side.[/i]
Solve the equation for real numbers $x,y,z$
$$(x-y+z)^2=x^2-y^2+z^2$$