Found problems: 1132
2016 Indonesia TST, 2
Let $a,b$ be two positive integers, such that $ab\neq 1$. Find all the integer values that $f(a,b)$ can take, where \[ f(a,b) = \frac { a^2+ab+b^2} { ab- 1} . \]
2014 Tuymaada Olympiad, 5
For two quadratic trinomials $P(x)$ and $Q(x)$ there is a linear function $\ell(x)$ such that $P(x)=Q(\ell(x))$ for all real $x$. How many such linear functions $\ell(x)$ can exist?
[i](A. Golovanov)[/i]
2009 Indonesia TST, 1
Find the smallest odd integer $ k$ such that: for every $ 3\minus{}$degree polynomials $ f$ with integer coefficients, if there exist $ k$ integer $ n$ such that $ |f(n)|$ is a prime number, then $ f$ is irreducible in $ \mathbb{Z}[n]$.
2011 APMO, 1
Let $a,b,c$ be positive integers. Prove that it is impossible to have all of the three numbers $a^2+b+c,b^2+c+a,c^2+a+b$ to be perfect squares.
1998 National Olympiad First Round, 8
$ a_{1} \equal{}1$, $ a_{n\plus{}1} \equal{}\frac{a_{n} }{\sqrt{1\plus{}4a_{n}^{2} } }$ for $ n\ge 1$. What is the least $ k$ such that $ a_{k} <10^{\minus{}2}$ ?
$\textbf{(A)}\ 2501 \qquad\textbf{(B)}\ 251 \qquad\textbf{(C)}\ 2499 \qquad\textbf{(D)}\ 249 \qquad\textbf{(E)}\ \text{None}$
2001 JBMO ShortLists, 4
The discriminant of the equation $x^2-ax+b=0$ is the square of a rational number and $a$ and $b$ are integers. Prove that the roots of the equation are integers.
2004 Finnish National High School Mathematics Competition, 1
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 $x^2 + 2cx + a^2 = 0.$
2020 MMATHS, I10
Let $f(x)$ be a quadratic polynomial such that $f(f(1)) = f(-f(-1)) = 0$ and $f(1) \neq -f(-1)$. Suppose furthermore that the quadratic $2f(x)$ has coefficients that are nonzero integers. Find $f(0)$.
[i]Proposed by Andrew Wu[/i]
2011 All-Russian Olympiad, 2
Nine quadratics, $x^2+a_1x+b_1, x^2+a_2x+b_2,...,x^2+a_9x+b_9$ are written on the board. The sequences $a_1, a_2,...,a_9$ and $b_1, b_2,...,b_9$ are arithmetic. The sum of all nine quadratics has at least one real root. What is the the greatest possible number of original quadratics that can have no real roots?
1989 AIME Problems, 6
Two skaters, Allie and Billie, are at points $A$ and $B$, respectively, on a flat, frozen lake. The distance between $A$ and $B$ is $100$ meters. Allie leaves $A$ and skates at a speed of $8$ meters per second on a straight line that makes a $60^\circ$ angle with $AB$. At the same time Allie leaves $A$, Billie leaves $B$ at a speed of $7$ meters per second and follows the straight path that produces the earliest possible meeting of the two skaters, given their speeds. How many meters does Allie skate before meeting Billie?
[asy]
defaultpen(linewidth(0.8));
draw((100,0)--origin--60*dir(60), EndArrow(5));
label("$A$", origin, SW);
label("$B$", (100,0), SE);
label("$100$", (50,0), S);
label("$60^\circ$", (15,0), N);[/asy]
2010 Today's Calculation Of Integral, 612
For $f(x)=\frac{1}{x}\ (x>0)$, prove the following inequality.
\[f\left(t+\frac 12 \right)\leq \int_t^{t+1} f(x)\ dx\leq \frac 16\left\{f(t)+4f\left(t+\frac 12\right)+f(t+1)\right\}\]
1951 AMC 12/AHSME, 22
The values of $ a$ in the equation: $ \log_{10}(a^2 \minus{} 15a) \equal{} 2$ are:
$ \textbf{(A)}\ \frac {15\pm\sqrt {233}}{2} \qquad\textbf{(B)}\ 20, \minus{} 5 \qquad\textbf{(C)}\ \frac {15 \pm \sqrt {305}}{2}$
$ \textbf{(D)}\ \pm20 \qquad\textbf{(E)}\ \text{none of these}$
2008 Harvard-MIT Mathematics Tournament, 21
Let $ ABC$ be a triangle with $ AB \equal{} 5$, $ BC \equal{} 4$ and $ AC \equal{} 3$. Let $ \mathcal P$ and $ \mathcal Q$ be squares inside $ ABC$ with disjoint interiors such that they both have one side lying on $ AB$. Also, the two squares each have an edge lying on a common line perpendicular to $ AB$, and $ \mathcal P$ has one vertex on $ AC$ and $ \mathcal Q$ has one vertex on $ BC$. Determine the minimum value of the sum of the areas of the two squares.
[asy]import olympiad;
import math;
import graph;
unitsize(1.5cm);
pair A, B, C;
A = origin;
B = A + 5 * right;
C = (9/5, 12/5);
pair X = .7 * A + .3 * B;
pair Xa = X + dir(135);
pair Xb = X + dir(45);
pair Ya = extension(X, Xa, A, C);
pair Yb = extension(X, Xb, B, C);
pair Oa = (X + Ya)/2;
pair Ob = (X + Yb)/2;
pair Ya1 = (X.x, Ya.y);
pair Ya2 = (Ya.x, X.y);
pair Yb1 = (Yb.x, X.y);
pair Yb2 = (X.x, Yb.y);
draw(A--B--C--cycle);
draw(Ya--Ya1--X--Ya2--cycle);
draw(Yb--Yb1--X--Yb2--cycle);
label("$A$", A, W);
label("$B$", B, E);
label("$C$", C, N);
label("$\mathcal P$", Oa, origin);
label("$\mathcal Q$", Ob, origin);[/asy]
2014 Cezar Ivănescu, 2
While there do not exist pairwise distinct real numbers $a,b,c$ satisfying $a^2+b^2+c^2 = ab+bc+ca$, there do exist complex numbers with that property. Let $a,b,c$ be complex numbers such that $a^2+b^2+c^2 = ab+bc+ca$ and $|a+b+c| = 21$. Given that $|a-b| = 2\sqrt{3}$, $|a| = 3\sqrt{3}$, compute $|b|^2+|c|^2$.
[hide="Clarifications"]
[list]
[*] The problem should read $|a+b+c| = 21$. An earlier version of the test read $|a+b+c| = 7$; that value is incorrect.
[*] $|b|^2+|c|^2$ should be a positive integer, not a fraction; an earlier version of the test read ``... for relatively prime positive integers $m$ and $n$. Find $m+n$.''[/list][/hide]
[i]Ray Li[/i]
PEN A Problems, 1
Show that if $x, y, z$ are positive integers, then $(xy+1)(yz+1)(zx+1)$ is a perfect square if and only if $xy+1$, $yz+1$, $zx+1$ are all perfect squares.
2002 Iran Team Selection Test, 12
We call a permutation $ \left(a_1, a_2, ..., a_n\right)$ of $ \left(1, 2, ..., n\right)$ [i]quadratic[/i] if there exists at least a perfect square among the numbers $ a_1$, $ a_1 \plus{} a_2$, $ ...$, $ a_1 \plus{} a_2 \plus{} ... \plus{} a_n$. Find all natural numbers $ n$ such that all permutations in $ S_n$ are quadratic.
[i]Remark.[/i] $ S_{n}$ denotes the $ n$-th symmetric group, the group of permutations on $ n$ elements.
1989 India National Olympiad, 1
Prove that the Polynomial $ f(x) \equal{} x^{4} \plus{} 26x^{3} \plus{} 56x^{2} \plus{} 78x \plus{} 1989$ can't be expressed as a product $ f(x) \equal{} p(x)q(x)$ , where $ p(x)$ and $ q(x)$ are both polynomial with integral coefficients and with degree at least $ 1$.
2011 ELMO Problems, 5
Let $p>13$ be a prime of the form $2q+1$, where $q$ is prime. Find the number of ordered pairs of integers $(m,n)$ such that $0\le m<n<p-1$ and
\[3^m+(-12)^m\equiv 3^n+(-12)^n\pmod{p}.\]
[i]Alex Zhu.[/i]
[hide="Note"]The original version asked for the number of solutions to $2^m+3^m\equiv 2^n+3^n\pmod{p}$ (still $0\le m<n<p-1$), where $p$ is a Fermat prime.[/hide]
2009 Sharygin Geometry Olympiad, 22
Construct a quadrilateral which is inscribed and circumscribed, given the radii of the respective circles and the angle between the diagonals of quadrilateral.
2011 AMC 10, 19
What is the product of all the roots of the equation \[\sqrt{5|x|+8} = \sqrt{x^2-16}. \]
$ \textbf{(A)}\ -64 \qquad
\textbf{(B)}\ -24 \qquad
\textbf{(C)}\ -9 \qquad
\textbf{(D)}\ 24 \qquad
\textbf{(E)}\ 576 $
2010 Romania National Olympiad, 1
Let $(a_n)_{n\ge0}$ be a sequence of positive real numbers such that
\[\sum_{k=0}^nC_n^ka_ka_{n-k}=a_n^2,\ \text{for any }n\ge 0.\]
Prove that $(a_n)_{n\ge0}$ is a geometric sequence.
[i]Lucian Dragomir[/i]
1978 AMC 12/AHSME, 1
If $1-\frac{4}{x}+\frac{4}{x^2}=0$, then $\frac{2}{x}$ equals
$\textbf{(A) }-1\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }-1\text{ or }2\qquad \textbf{(E) }-1\text{ or }-2$
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.
2013 China Team Selection Test, 2
Prove that: there exists a positive constant $K$, and an integer series $\{a_n\}$, satisfying:
$(1)$ $0<a_1<a_2<\cdots <a_n<\cdots $;
$(2)$ For any positive integer $n$, $a_n<1.01^n K$;
$(3)$ For any finite number of distinct terms in $\{a_n\}$, their sum is not a perfect square.
2006 Turkey Team Selection Test, 1
For all integers $n\geq 1$ we define $x_{n+1}=x_1^2+x_2^2+\cdots +x_n^2$, where $x_1$ is a positive integer. Find the least $x_1$ such that 2006 divides $x_{2006}$.