Found problems: 1132
2013 F = Ma, 15
A uniform rod is partially in water with one end suspended, as shown in figure. The density of the rod is $5/9$ that of water. At equilibrium, what portion of the rod is above water?
$\textbf{(A) } 0.25\\
\textbf{(B) } 0.33\\
\textbf{(C) } 0.5\\
\textbf{(D) } 0.67\\
\textbf{(E) } 0.75$
1990 All Soviet Union Mathematical Olympiad, 516
Find three non-zero reals such that all quadratics with those numbers as coefficients have two distinct rational roots.
2015 Saint Petersburg Mathematical Olympiad, 1
Is there a quadratic trinomial $f(x)$ with integer coefficients such that $f(f(\sqrt{2}))=0$ ?
[i]A. Khrabrov[/i]
2012 Indonesia TST, 3
The [i]cross[/i] of a convex $n$-gon is the quadratic mean of the lengths between the possible pairs of vertices. For example, the cross of a $3 \times 4$ rectangle is $\sqrt{ \dfrac{3^2 + 3^2 + 4^2 + 4^2 + 5^2 + 5^2}{6} } = \dfrac{5}{3} \sqrt{6}$.
Suppose $S$ is a dodecagon ($12$-gon) inscribed in a unit circle. Find the greatest possible cross of $S$.
2024 All-Russian Olympiad Regional Round, 11.7
Graph $G_1$ of a quadratic trinomial $y = px^2 + qx + r$ with real coefficients intersects the graph $G_2$ of a quadratic trinomial $y = x^2$ in points $A$, $B$. The intersection of tangents to $G_2$ in points $A$, $B$ is point $C$. If $C \in G_1$, find all possible values of $p$.
2011 Polish MO Finals, 1
Determine all pairs of functions $f,g:\mathbb{R}\rightarrow\mathbb{R}$ such that for any $x,y\in \mathbb{R}$,
\[f(x)f(y)=g(x)g(y)+g(x)+g(y).\]
2011 USAJMO, 1
Find, with proof, all positive integers $n$ for which $2^n + 12^n + 2011^n$ is a perfect square.
2006 Putnam, B1
Show that the curve $x^{3}+3xy+y^{3}=1$ contains only one set of three distinct points, $A,B,$ and $C,$ which are the vertices of an equilateral triangle.
2009 Vietnam Team Selection Test, 3
Let a, b be positive integers. a, b and a.b are not perfect squares.
Prove that at most one of following equations
$ ax^2 \minus{} by^2 \equal{} 1$ and $ ax^2 \minus{} by^2 \equal{} \minus{} 1$
has solutions in positive integers.
2009 AMC 12/AHSME, 17
Let $ a\plus{}ar_1\plus{}ar_1^2\plus{}ar_1^3\plus{}\cdots$ and $ a\plus{}ar_2\plus{}ar_2^2\plus{}ar_2^3\plus{}\cdots$ be two different infinite geometric series of positive numbers with the same first term. The sum of the first series is $ r_1$, and the sum of the second series is $ r_2$. What is $ r_1\plus{}r_2$?
$ \textbf{(A)}\ 0\qquad \textbf{(B)}\ \frac{1}{2}\qquad \textbf{(C)}\ 1\qquad \textbf{(D)}\ \frac{1\plus{}\sqrt{5}}{2}\qquad \textbf{(E)}\ 2$
1998 USAMTS Problems, 2
There are infinitely many ordered pairs $(m,n)$ of positive integers for which the sum
\[ m + ( m + 1) + ( m + 2) +... + ( n - 1 )+n\]
is equal to the product $mn$. The four pairs with the smallest values of $m$ are $(1, 1), (3, 6), (15, 35),$ and $(85, 204)$. Find three more $(m, n)$ pairs.
2005 AIME Problems, 4
The director of a marching band wishes to place the members into a formation that includes all of them and has no unfilled positions. If they are arranged in a square formation, there are 5 members left over. The director realizes that if he arranges the group in a formation with 7 more rows than columns, there are no members left over. Find the maximum number of members this band can have.
2010 AIME Problems, 14
In right triangle $ ABC$ with right angle at $ C$, $ \angle BAC < 45$ degrees and $ AB \equal{} 4$. Point $ P$ on $ AB$ is chosen such that $ \angle APC \equal{} 2\angle ACP$ and $ CP \equal{} 1$. The ratio $ \frac{AP}{BP}$ can be represented in the form $ p \plus{} q\sqrt{r}$, where $ p,q,r$ are positive integers and $ r$ is not divisible by the square of any prime. Find $ p\plus{}q\plus{}r$.
1993 Greece National Olympiad, 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]
2007 Hanoi Open Mathematics Competitions, 15
Let $p = \overline{abcd}$ be a $4$-digit prime number. Prove that the equation $ax^3+bx^2+cx+d=0$ has no rational roots.
2006 Taiwan TST Round 1, 2
Let $p,q$ be two distinct odd primes. Calculate
$\displaystyle \sum_{j=1}^{\frac{p-1}{2}}\left \lfloor \frac{qj}{p}\right \rfloor +\sum_{j=1}^{\frac{q-1}{2}}\left \lfloor \frac{pj}{q}\right\rfloor$.
2022 Auckland Mathematical Olympiad, 5
The teacher wrote on the board the quadratic polyomial $x^2+10x+20$. Then in turn, each of the students came to the board and increased or decreased by $1$ either the coefficient at $x$ or the constant term, but not both at once. As a result, the quadratic polyomial $x^2 + 20x +10$ appeared on the board. Is it true that at some point a quadratic polyomial with integer roots appeared on the board?
2016 Saudi Arabia BMO TST, 1
Let $P_i(x) = x^2 + b_i x + c_i , i = 1,2, ..., n$ be pairwise distinct polynomials of degree $2$ with real coefficients so that for any $0 \le i < j \le n , i, j \in N$, the polynomial $Q_{i,j}(x) = P_i(x) + P_j(x)$ has only one real root. Find the greatest possible value of $n$.
2004 Romania Team Selection Test, 18
Let $p$ be a prime number and $f\in \mathbb{Z}[X]$ given by
\[ f(x) = a_{p-1}x^{p-2} + a_{p-2}x^{p-3} + \cdots + a_2x+ a_1 , \]
where $a_i = \left( \tfrac ip\right)$ is the Legendre symbol of $i$ with respect to $p$ (i.e. $a_i=1$ if $ i^{\frac {p-1}2} \equiv 1 \pmod p$ and $a_i=-1$ otherwise, for all $i=1,2,\ldots,p-1$).
a) Prove that $f(x)$ is divisible with $(x-1)$, but not with $(x-1)^2$ iff $p \equiv 3 \pmod 4$;
b) Prove that if $p\equiv 5 \pmod 8$ then $f(x)$ is divisible with $(x-1)^2$ but not with $(x-1)^3$.
[i]Sugested by Calin Popescu.[/i]
2005 AMC 12/AHSME, 24
Let $ P(x) \equal{} (x \minus{} 1)(x \minus{} 2)(x \minus{} 3)$. For how many polynomials $ Q(x)$ does there exist a polynomial $ R(x)$ of degree 3 such that $ P(Q(x)) \equal{} P(x) \cdot R(x)$?
$ \textbf{(A)}\ 19\qquad
\textbf{(B)}\ 22\qquad
\textbf{(C)}\ 24\qquad
\textbf{(D)}\ 27\qquad
\textbf{(E)}\ 32$
2004 Harvard-MIT Mathematics Tournament, 5
There exists a positive real number $x$ such that $ \cos (\arctan (x)) = x $. Find the value of $x^2$.
1968 AMC 12/AHSME, 13
If $m$ and $n$ are the roots of $x^2+mx+n=0$, $m\ne0$, $n\ne0$, then the sum of the roots is:
$\textbf{(A)}\ -\dfrac{1}{2} \qquad
\textbf{(B)}\ -1 \qquad
\textbf{(C)}\ \dfrac{1}{2} \qquad
\textbf{(D)}\ 1 \qquad
\textbf{(E)}\ \text{Undetermined} $
1997 All-Russian Olympiad, 1
Of the quadratic trinomials $x^2 + px + q$ where $p$; $q$ are integers and $1\leqslant p, q \leqslant 1997$, which are there more of: those having integer roots or those not having real roots?
[i]M. Evdokimov[/i]
2002 AMC 10, 10
Suppose that $ a$ and $ b$ are are nonzero real numbers, and that the equation $ x^2\plus{}ax\plus{}b\equal{}0$ has solutions $ a$ and $ b$. Then the pair $ (a,b)$ is
$ \textbf{(A)}\ (\minus{}2,1) \qquad
\textbf{(B)}\ (\minus{}1,2) \qquad
\textbf{(C)}\ (1,\minus{}2) \qquad
\textbf{(D)}\ (2,\minus{}1) \qquad
\textbf{(E)}\ (4,4)$
2009 AMC 12/AHSME, 11
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}$?
[asy]unitsize(3mm);
defaultpen(linewidth(.8pt)+fontsize(10pt));
path d=(1/2,0)--(0,sqrt(3)/2)--(-1/2,0)--(0,-sqrt(3)/2)--cycle;
marker m=marker(scale(5)*d,Fill);
path f1=(0,0);
path f2=(0,0)--(-1,1)--(1,1)--(1,-1)--(-1,-1);
path[] g2=(-1,1)--(-1,-1)--(0,0)^^(1,-1)--(0,0)--(1,1);
path f3=f2--(-2,-2)--(-2,0)--(-2,2)--(0,2)--(2,2)--(2,0)--(2,-2)--(0,-2);
path[] g3=g2^^(-2,-2)--(0,-2)^^(2,-2)--(1,-1)^^(1,1)--(2,2)^^(-1,1)--(-2,2);
path[] f4=f3^^(-3,-3)--(-3,-1)--(-3,1)--(-3,3)--(-1,3)--(1,3)--(3,3)--
(3,1)--(3,-1)--(3,-3)--(1,-3)--(-1,-3);
path[] g4=g3^^(-2,-2)--(-3,-3)--(-1,-3)^^(3,-3)--(2,-2)^^(2,2)--(3,3)^^
(-2,2)--(-3,3);
draw(f1,m);
draw(shift(5,0)*f2,m);
draw(shift(5,0)*g2);
draw(shift(12,0)*f3,m);
draw(shift(12,0)*g3);
draw(shift(21,0)*f4,m);
draw(shift(21,0)*g4);
label("$F_1$",(0,-4));
label("$F_2$",(5,-4));
label("$F_3$",(12,-4));
label("$F_4$",(21,-4));[/asy]$ \textbf{(A)}\ 401 \qquad \textbf{(B)}\ 485 \qquad \textbf{(C)}\ 585 \qquad \textbf{(D)}\ 626 \qquad \textbf{(E)}\ 761$