Found problems: 1148
1999 Harvard-MIT Mathematics Tournament, 1
If $a@b=\dfrac{a^3-b^3}{a-b}$, for how many real values of $a$ does $a@1=0$?
1955 AMC 12/AHSME, 49
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}$
1976 AMC 12/AHSME, 20
Let $a,~b,$ and $x$ be positive real numbers distinct from one. Then \[4(\log_ax)^2+3(\log_bx)^2=8(\log_ax)(\log_bx)\]
$\textbf{(A) }\text{for all values of }a,~b,\text{ and }x\qquad$
$\textbf{(B) }\text{if and only if }a=b^2\qquad$
$\textbf{(C) }\text{if and only if }b=a^2\qquad$
$\textbf{(D) }\text{if and only if }x=ab\qquad$
$ \textbf{(E) }\text{for none of these}$
2005 Bulgaria Team Selection Test, 4
Let $a_{i}$ and $b_{i}$, where $i \in \{1,2, \dots, 2005 \}$, be real numbers such that the inequality $(a_{i}x-b_{i})^{2} \ge \sum_{j=1, j \not= i}^{2005} (a_{j}x-b_{j})$ holds for all $x \in \mathbb{R}$ and all $i \in \{1,2, \dots, 2005 \}$. Find the maximum possible number of positive numbers amongst $a_{i}$ and $b_{i}$, $i \in \{1,2, \dots, 2005 \}$.
2018 Romania National Olympiad, 3
Let $f,g : \mathbb{R} \to \mathbb{R}$ be two quadratics such that, for any real number $r,$ if $f(r)$ is an integer, then $g(r)$ is also an integer.
Prove that there are two integers $m$ and $n$ such that $$g(x)=mf(x)+n, \: \forall x \in \mathbb{R}$$
2006 Stanford Mathematics Tournament, 5
There exist two positive numbers $ x$ such that $ \sin(\arccos(\tan(\arcsin x)))\equal{}x$. Find the product of the two possible $ x$.
1963 Vietnam National Olympiad, 2
For what values of $ m$ does the equation $ x^2 \plus{} (2m \plus{} 6)x \plus{} 4m \plus{} 12 \equal{} 0$ has two real roots, both of them greater than $ \minus{}1$.
1998 Tournament Of Towns, 6
In a function $f (x) = (x^2 + ax + b )/ (x^2 + cx + d)$ , the quadratics $x^2 + ax + b$ and $x^2 + cx + d$ have no common roots. Prove that the next two statements are equivalent:
(i) there is a numerical interval without any values of $f(x)$ ,
(ii) $f(x)$ can be represented in the form $f (x) = f_1 (f_2( ... f_{n-1} (f_n (x))... ))$ where each of the functions $f_j$ is o f one of the three forms $k_j x + b_j, 1/x, x^2$ .
(A Kanel)
2001 USAMO, 3
Let $a, b, c \geq 0$ and satisfy \[ a^2+b^2+c^2 +abc = 4 . \] Show that \[ 0 \le ab + bc + ca - abc \leq 2. \]
2003 China Team Selection Test, 2
Positive integer $n$ cannot be divided by $2$ and $3$, there are no nonnegative integers $a$ and $b$ such that $|2^a-3^b|=n$. Find the minimum value of $n$.
1992 Vietnam Team Selection Test, 2
Find all pair of positive integers $(x, y)$ satisfying the equation
\[x^2 + y^2 - 5 \cdot x \cdot y + 5 = 0.\]
2006 Taiwan TST Round 1, 2
Let $p,q$ be two distinct odd primes. Calculate
$\displaystyle \sum_{j=1}^{\frac{p-1}{2}}\left \lfloor \frac{qj}{p}\right \rfloor +\sum_{j=1}^{\frac{q-1}{2}}\left \lfloor \frac{pj}{q}\right\rfloor$.
2008 AIME Problems, 13
Let
\[ p(x,y) \equal{} a_0 \plus{} a_1x \plus{} a_2y \plus{} a_3x^2 \plus{} a_4xy \plus{} a_5y^2 \plus{} a_6x^3 \plus{} a_7x^2y \plus{} a_8xy^2 \plus{} a_9y^3.
\]Suppose that
\begin{align*}p(0,0) &\equal{} p(1,0) \equal{} p( \minus{} 1,0) \equal{} p(0,1) \equal{} p(0, \minus{} 1) \\&\equal{} p(1,1) \equal{} p(1, \minus{} 1) \equal{} p(2,2) \equal{} 0.\end{align*}
There is a point $ \left(\tfrac {a}{c},\tfrac {b}{c}\right)$ for which $ p\left(\tfrac {a}{c},\tfrac {b}{c}\right) \equal{} 0$ for all such polynomials, where $ a$, $ b$, and $ c$ are positive integers, $ a$ and $ c$ are relatively prime, and $ c > 1$. Find $ a \plus{} b \plus{} c$.
2013 Finnish National High School Mathematics Competition, 1
The coefficients $a,b,c$ of a polynomial $f:\mathbb{R}\to\mathbb{R}, f(x)=x^3+ax^2+bx+c$ are mutually distinct integers and different from zero. Furthermore, $f(a)=a^3$ and $f(b)=b^3.$ Determine $a,b$ and $c$.
1991 IberoAmerican, 5
Let $P(x,\, y)=2x^{2}-6xy+5y^{2}$. Let us say an integer number $a$ is a value of $P$ if there exist integer numbers $b$, $c$ such that $P(b,\, c)=a$.
a) Find all values of $P$ lying between 1 and 100.
b) Show that if $r$ and $s$ are values of $P$, then so is $rs$.
2014 Iran Team Selection Test, 4
$n$ 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.
2011 Austria Beginners' Competition, 2
Let $p$ and $q$ be real numbers. The quadratic equation $$x^2 + px + q = 0$$
has the real solutions $x_1$ and $x_2$. In addition, the following two conditions apply:
(i) The numbers $x_1$ and $x_2$ differ from each other by exactly $ 1$.
(ii) The numbers $p$ and $q$ differ from each other by exactly $ 1$.
Show that then $p$, $q$, $x_1$ and $x_2$ are integers.
(G. Kirchner, University of Innsbruck)
2008 Czech-Polish-Slovak Match, 1
Prove that there exists a positive integer $n$, such that the number $k^2+k+n$ does not have a prime divisor less than $2008$ for any integer $k$.
2005 AMC 10, 10
There are two values of $ a$ for which the equation $ 4x^2 \plus{} ax \plus{} 8x \plus{} 9 \equal{} 0$ has only one solution for $ x$. What is the sum of these values of $ a$?
$ \textbf{(A)}\ \minus{}16\qquad
\textbf{(B)}\ \minus{}8\qquad
\textbf{(C)}\ 0\qquad
\textbf{(D)}\ 8\qquad
\textbf{(E)}\ 20$
2007 IMO Shortlist, 6
Let $ k$ be a positive integer. Prove that the number $ (4 \cdot k^2 \minus{} 1)^2$ has a positive divisor of the form $ 8kn \minus{} 1$ if and only if $ k$ is even.
[url=http://www.mathlinks.ro/viewtopic.php?p=894656#894656]Actual IMO 2007 Problem, posed as question 5 in the contest, which was used as a lemma in the official solutions for problem N6 as shown above.[/url]
[i]Author: Kevin Buzzard and Edward Crane, United Kingdom [/i]
2005 Taiwan TST Round 2, 1
Prove that for any quadratic polynomial $f(x)=x^2+px+q$ with integer coefficients, it is possible to find another polynomial $q(x)=2x^2+rx+s$ with integer coefficients so that \[\{f(x)|x \in \mathbb{Z} \} \cap \{g(x)|x \in \mathbb{Z} \} = \emptyset .\]
1964 Czech and Slovak Olympiad III A, 3
Determine all values of parameter $\alpha\in [0,2\pi]$ such that the equation $$(2\cos\alpha-1)x^2+4x+4\cos\alpha+2=0$$ has
1) a positive root $x_1$,
2) if a second root $x_2$ exists and if $x_2\neq x_1$, the $x_2\leq 0$.
2011 Moldova Team Selection Test, 2
Find all pairs of real number $x$ and $y$ which simultaneously satisfy the following 2 relations:
$x+y+4=\frac{12x+11y}{x^2+y^2}$
$y-x+3=\frac{11x-12y}{x^2+y^2}$
1995 Niels Henrik Abels Math Contest (Norwegian Math Olympiad) Round 2, 2
On the figure, the quadrilateral $ ABCD$ is a rectangle, $ P$ lies on $ AD$ and $ Q$ on $ AB.$ The triangles $ PAQ, QBC,$ and $ PCD$ all have the same areas, and $ BQ \equal{} 2.$ How long is $ AQ$?
[img]http://i250.photobucket.com/albums/gg265/geometry101/NielsHenrikAbel1995Number2.jpg[/img]
A. 7/2
B. $ \sqrt{7}$
C. $ 2 \sqrt{3}$
D. $ 1 \plus{} \sqrt{5}$
E. Not uniquely determined
2010 N.N. Mihăileanu Individual, 1
Let be two real reducible quadratic polynomials $ P,Q $ in one variable. Prove that if $ P-Q $ is irreducible, then $ P+Q $ is reducible.