Found problems: 1148
Find all real $x$ that satisfy $\sqrt[3]{20x+\sqrt[3]{20x+13}}=13$.
Let $d$ be any positive integer not equal to $2, 5$ or $13$. Show that one can find distinct $a,b$ in the set $\{2,5,13,d\}$ such that $ab-1$ is not a perfect square.
A positive number $\dfrac{m}{n}$ has the property that it is equal to the ratio of $7$ plus the number’s reciprocal and $65$ minus the number’s reciprocal. Given that $m$ and $n$ are relatively prime positive integers, find $2m + n$.
Find all pairs of integers $(x, y)$ such that $5x^{2}-6xy+7y^{2}=383$.
Scientist have succeeded to find new numbers between real numbers with strong microscopes. Now real numbers are extended in a new larger system we have an order on it (which if induces normal order on $ \mathbb R$), and also 4 operations addition, multiplication,... and these operation have all properties the same as $ \mathbb R$.
[img]http://i14.tinypic.com/4tk6mnr.png[/img]
a) Prove that in this larger system there is a number which is smaller than each positive integer and is larger than zero.
b) Prove that none of these numbers are root of a polynomial in $ \mathbb R[x]$.
The figures $ F_1$, $ F_2$, $ F_3$, and $ F_4$ shown are the first in a sequence of figures. For $ n\ge3$, $ F_n$ is constructed from $ F_{n \minus{} 1}$ by surrounding it with a square and placing one more diamond on each side of the new square than $ F_{n \minus{} 1}$ had on each side of its outside square. For example, figure $ F_3$ has $ 13$ diamonds. How many diamonds are there in figure $ F_{20}$?
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label("$F_4$",(21,-4));[/asy]$ \textbf{(A)}\ 401 \qquad \textbf{(B)}\ 485 \qquad \textbf{(C)}\ 585 \qquad \textbf{(D)}\ 626 \qquad \textbf{(E)}\ 761$
In circle $\theta_1$ with radius $1$, circles $\phi_1, \phi_2, \dots, \phi_8$, with equal radii, are drawn such that for $1 \le i \le 8$, $\phi_i$ is tangent to $\omega_1$, $\phi_{i-1}$, and $\phi_{i+1}$, where $\phi_0 = \phi_8$ and $\phi_1 = \phi_9$. There exists a circle $\omega_2$ such that $\omega_1 \neq \omega_2$ and $\omega_2$ is tangent to $\phi_i$ for $1 \le i \le 8$. The radius of $\omega_2$ can be expressed in the form $a - b\sqrt{c} -d\sqrt{e - \sqrt{f}} + g \sqrt{h - j \sqrt{k}}$ such that $a, b, \dots, k$ are positive integers and the numbers $e, f, k, \gcd(h, j)$ are squarefree. What is $a+b+c+d+e+f+g+h+j+k$.
[i]Proposed by Eugene Chen
[/i]
Let $m$ be the largest real solution to the equation \[\frac{3}{x-3}+\frac{5}{x-5}+\frac{17}{x-17}+\frac{19}{x-19}= x^2-11x-4.\] There are positive integers $a,b,c$ such that $m = a + \sqrt{b+\sqrt{c}}$. Find $a+b+c$.
$n$ is a natural number. We shall call a permutation $a_1,\dots,a_n$ of $1,\dots,n$ a quadratic(cubic) permutation if $\forall 1\leq i \leq n-1$ we have $a_ia_{i+1}+1$ is a perfect square(cube).
$(a)$ Prove that for infinitely many natural numbers $n$ there exists a quadratic permutation.
$(b)$ Prove that for no natural number $n$ exists a cubic permutation.
The equations $x^2 +2ax+b^2 = 0$ and $x^2 +2bx+c^2 = 0$ both have two diff erent real roots.
Determine the number of real roots of the equation $x^2 + 2cx + a^2 = 0.$
Let $ABCD$ be a cyclic quadrilateral, and $E$ be the intersection of its diagonals. If $m(\widehat{ADB}) = 22.5^\circ$, $|BD|=6$, and $|AD|\cdot|CE|=|DC|\cdot|AE|$, find the area of the quadrilateral $ABCD$.
Find the number of second-degree polynomials $ f(x)$ with integer coefficients and integer zeros for which $ f(0)\equal{}2010$.
Given the equation $3x^2 - 4x + k = 0$ with real roots. The value of $k$ for which the product of the roots of the equation is a maximum is:
$\textbf{(A)}\ \dfrac{16}{9} \qquad
\textbf{(B)}\ \dfrac{16}{3}\qquad
\textbf{(C)}\ \dfrac{4}{9} \qquad
\textbf{(D)}\ \dfrac{4}{3} \qquad
\textbf{(E)}\ -\dfrac{4}{3}$
Let $ x,y$ be positive integers with $ y>3$ and $ x^2\plus{}y^4\equal{}2((x\minus{}6)^2\plus{}(y\plus{}1)^2).$ Prove that: $ x^2\plus{}y^4\equal{}1994.$
Solve in the real numbers the equation:
$$ \cos^2 \frac{(x-2)\pi }{4} +\cos\frac{(x-2)\pi }{3} =\log_3 (x^2-4x+6) $$
[i]Gheorghe Mihai[/i]
Do there exist quadratic polynomials $P(x)$ and $Q(x)$ with real coeffcients such that the polynomial $P(Q(x))$ has precisely the zeros $x = 2, x = 3, x =5$ and $x = 7$?
For any sequence of real numbers $A=(a_1,a_2,a_3,\ldots)$, define $\Delta A$ to be the sequence $(a_2-a_1,a_3-a_2,a_4-a_3,\ldots)$, whose $n^\text{th}$ term is $a_{n+1}-a_n$. Suppose that all of the terms of the sequence $\Delta(\Delta A)$ are $1$, and that $a_{19}=a_{92}=0$. Find $a_1$.
Find positive integer $n$ so that $\tfrac{80-6\sqrt{n}}{n}$ is the reciprocal of $\tfrac{80+6\sqrt{n}}{n}.$
Let $x$ and $y$ be real numbers such that $\frac{\sin{x}}{\sin{y}} = 3$ and $\frac{\cos{x}}{\cos{y}} = \frac{1}{2}$. The value of $\frac{\sin{2x}}{\sin{2y}} + \frac{\cos{2x}}{\cos{2y}}$ can be expressed in the form $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p + q$.
The graphs of $ y\equal{}\frac{x^2\minus{}4}{x\minus{}2}$ and $ y\equal{}2x$ intersect in:
$ \textbf{(A)}\ \text{1 point whose abscissa is 2} \qquad
\textbf{(B)}\ \text{1 point whose abscissa is 0}\\
\textbf{(C)}\ \text{no points} \qquad
\textbf{(D)}\ \text{two distinct points} \qquad
\textbf{(E)}\ \text{two identical points}$
Solve in the integers the diophantine equation
$$x^4-6x^2+1 = 7 \cdot 2^y.$$
Let $S$ be a finite set of integers. Suppose that for every two different elements of $S$, $p$ and $q$, there exist not necessarily distinct integers $a \neq 0$, $b$, $c$ belonging to $S$, such that $p$ and $q$ are the roots of the polynomial $ax^{2}+bx+c$. Determine the maximum number of elements that $S$ can have.
If $\log_{b^2}x+\log_{x^2}b=1, b>0, b \neq 1, x \neq 1$, then $x$ equals:
$ \textbf{(A)}\ 1/b^2 \qquad\textbf{(B)}\ 1/b \qquad\textbf{(C)}\ b^2 \qquad\textbf{(D)}\ b \qquad\textbf{(E)}\ \sqrt{b} $
Determine $ m > 0$ so that $ x^4 \minus{} (3m\plus{}2)x^2 \plus{} m^2 \equal{} 0$ has four real solutions forming an arithmetic series: i.e., that the solutions may be written $ a, a\plus{}b, a\plus{}2b,$ and $ a\plus{}3b$ for suitable $ a$ and $ b$.
A. 1
B. 3
C. 7
D. 12
E. None of these
Both roots of the quadratic equation $ x^2 \minus{} 63x \plus{} k \equal{} 0$ are prime numbers. The number of possible values of $ k$ is
$ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 3 \qquad \textbf{(E)}\ \textbf{more than four}$