Found problems: 1132
2007 Bundeswettbewerb Mathematik, 1
For which numbers $ n$ is there a positive integer $ k$ with the following property: The sum of digits for $ k$ is $ n$ and the number $ k^2$ has sum of digits $ n^2.$
2007 China Team Selection Test, 1
$ u,v,w > 0$,such that $ u \plus{} v \plus{} w \plus{} \sqrt {uvw} \equal{} 4$
prove that $ \sqrt {\frac {uv}{w}} \plus{} \sqrt {\frac {vw}{u}} \plus{} \sqrt {\frac {wu}{v}}\geq u \plus{} v \plus{} w$
2001 Saint Petersburg Mathematical Olympiad, 9.2
Define a quadratic trinomial to be "good", if it has two distinct real roots and all of its coefficients are distinct. Do there exist 10 positive integers such that there exist 500 good quadratic trinomials coefficients of which are among these numbers?
[I]Proposed by F. Petrov[/i]
1992 AIME Problems, 13
Triangle $ABC$ has $AB=9$ and $BC: AC=40: 41$. What's the largest area that this triangle can have?
2004 Manhattan Mathematical Olympiad, 2
Assume $a,b,c$ are odd integers. Show that the quadratic equation
\[ ax^2 + bx + c = 0 \]
has no rational solutions. (A number is said to be [i]rational[/i], if it can be written as a fraction: $\frac{\text{integer}}{\text{integer}}$.)
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]$.
2010 Postal Coaching, 2
Find all non-negative integers $m,n,p,q$ such that \[ p^mq^n = (p+q)^2 +1 . \]
2006 Regional Competition For Advanced Students, 1
Let $ 0 < x <y$ be real numbers. Let
$ H\equal{}\frac{2xy}{x\plus{}y}$ , $ G\equal{}\sqrt{xy}$ , $ A\equal{}\frac{x\plus{}y}{2}$ , $ Q\equal{}\sqrt{\frac{x^2\plus{}y^2}{2}}$
be the harmonic, geometric, arithmetic and root mean square (quadratic mean) of $ x$ and $ y$. As generally known $ H<G<A<Q$. Arrange the intervals $ [H,G]$ , $ [G,A]$ and $ [A,Q]$ in ascending order by their length.
1985 Balkan MO, 2
Let $a,b,c,d \in [-\frac{\pi}{2}, \frac{\pi}{2}]$ be real numbers such that
$\sin{a}+\sin{b}+\sin{c}+\sin{d}=1$ and $\cos{2a}+\cos{2b}+\cos{2c}+\cos{2d}\geq \frac{10}{3}$.
Prove that $a,b,c,d \in [0, \frac{\pi}{6}]$
1996 Canadian Open Math Challenge, 1
The roots of the equation $x^2+4x-5 = 0$ are also the roots of the equation $2x^3+9x^2-6x-5 = 0$. What is the third root of the second equation?
2005 AMC 12/AHSME, 9
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$
2018-2019 Fall SDPC, 6
Alice and Bob play a game. Alice writes an equation of the form $ax^2 + bx + c =0$, choosing $a$, $b$, $c$ to be real numbers (possibly zero). Bob can choose to add (or subtract) any real number to each of $a$, $b$, $c$, resulting in a new equation. Bob wins if the resulting equation is quadratic and has two distinct real roots; Alice wins otherwise. For which choices of $a$, $b$, $c$ does Alice win, no matter what Bob does?
2013 Hanoi Open Mathematics Competitions, 12
If $f(x) = ax^2 + bx + c$ satisfies the condition $|f(x)| < 1; \forall x \in [-1, 1]$,
prove that the equation $f(x) = 2x^2 - 1$ has two real roots.
2012 Indonesia TST, 4
Determine all natural numbers $n$ such that for each natural number $a$ relatively prime with $n$ and $a \le 1 + \left\lfloor \sqrt{n} \right\rfloor$ there exists some integer $x$ with $a \equiv x^2 \mod n$.
Remark: "Natural numbers" is the set of positive integers.
2014 AIME Problems, 6
The graphs of $y=3(x-h)^2+j$ and $y=2(x-h)^2+k$ have $y$-intercepts of $2013$ and $2014$, respectively, and each graph has two positive integer $x$-intercepts. Find $h$.
1997 Turkey MO (2nd round), 1
Find all pairs of integers $(x, y)$ such that $5x^{2}-6xy+7y^{2}=383$.
2014 National Olympiad First Round, 18
Which one below cannot be expressed in the form $x^2+y^5$, where $x$ and $y$ are integers?
$
\textbf{(A)}\ 59170
\qquad\textbf{(B)}\ 59149
\qquad\textbf{(C)}\ 59130
\qquad\textbf{(D)}\ 59121
\qquad\textbf{(E)}\ 59012
$
1993 AIME Problems, 9
Two thousand points are given on a circle. Label one of the points 1. From this point, count 2 points in the clockwise direction and label this point 2. From the point labeled 2, count 3 points in the clockwise direction and label this point 3. (See figure.) Continue this process until the labels $1, 2, 3, \dots, 1993$ are all used. Some of the points on the circle will have more than one label and some points will not have a label. What is the smallest integer that labels the same point as 1993?
[asy]
int x=101, y=3*floor(x/4);
draw(Arc(origin, 1, 360*(y-3)/x, 360*(y+4)/x));
int i;
for(i=y-2; i<y+4; i=i+1) {
dot(dir(360*i/x));
}
label("3", dir(360*(y-2)/x), dir(360*(y-2)/x));
label("2", dir(360*(y+1)/x), dir(360*(y+1)/x));
label("1", dir(360*(y+3)/x), dir(360*(y+3)/x));[/asy]
2004 Vietnam National Olympiad, 3
Let $ S(n)$ be the sum of decimal digits of a natural number $ n$. Find the least value of $ S(m)$ if $ m$ is an integral multiple of $ 2003$.
2014 Harvard-MIT Mathematics Tournament, 1
Given that $x$ and $y$ are nonzero real numbers such that $x+\frac{1}{y}=10$ and $y+\frac{1}{x}=\frac{5}{12}$, find all possible values of $x$.
1959 AMC 12/AHSME, 16
The expression $\frac{x^2-3x+2}{x^2-5x+6}\div \frac{x^2-5x+4}{x^2-7x+12}$, when simplified is:
$ \textbf{(A)}\ \frac{(x-1)(x-6)}{(x-3)(x-4)} \qquad\textbf{(B)}\ \frac{x+3}{x-3}\qquad\textbf{(C)}\ \frac{x+1}{x-1}\qquad\textbf{(D)}\ 1\qquad\textbf{(E)}\ 2$
2005 Taiwan TST Round 1, 3
$n$ teams take part in a tournament, in which every two teams compete exactly once, and that no draws are possible. It is known that for any two teams, there exists another team which defeated both of the two teams. Find all $n$ for which this is possible.
2013 Korea National Olympiad, 3
Prove that there exist monic polynomial $f(x) $ with degree of 6 and having integer coefficients such that
(1) For all integer $m$, $f(m) \ne 0$.
(2) For all positive odd integer $n$, there exist positive integer $k$ such that $f(k)$ is divided by $n$.
2010 All-Russian Olympiad, 3
Lines tangent to circle $O$ in points $A$ and $B$, intersect in point $P$. Point $Z$ is the center of $O$. On the minor arc $AB$, point $C$ is chosen not on the midpoint of the arc. Lines $AC$ and $PB$ intersect at point $D$. Lines $BC$ and $AP$ intersect at point $E$. Prove that the circumcentres of triangles $ACE$, $BCD$, and $PCZ$ are collinear.
1968 Vietnam National Olympiad, 2
$L$ and $M$ are two parallel lines a distance $d$ apart. Given $r$ and $x$, construct a triangle $ABC$, with $A$ on $L$, and $B$ and $C$ on $M$, such that the inradius is $r$, and angle $A = x$. Calculate angles $B$ and $C$ in terms of $d$, $r$ and $x$. If the incircle touches the side $BC$ at $D$, find a relation between $BD$ and $DC$