Found problems: 329
2017 District Olympiad, 4
Let be a natural number $ n\ge 2, $ and a matrix $ A\in\mathcal{M}_n\left( \mathbb{C} \right) $ whose determinant vanishes. Show that
$$ \left( A^* \right)^2 =A^*\cdot\text{tr} A^*, $$
where $ A^* $ is the adjugate of $ A. $
2005 All-Russian Olympiad, 4
Integers $x>2,\,y>1,\,z>0$ satisfy an equation $x^y+1=z^2$. Let $p$ be a number of different prime divisors of $x$, $q$ be a number of different prime divisors of $y$. Prove that $p\geq q+2$.
2005 USAMTS Problems, 4
Find, with proof, all triples of real numbers $(a, b, c)$ such that all four roots of the polynomial $f(x) = x^4 +ax^3 +bx^2 +cx+b$ are positive integers. (The four roots need not be distinct.)
2011 Iran MO (3rd Round), 4
For positive real numbers $a,b$ and $c$ we have $a+b+c=3$. Prove
$\frac{a}{1+(b+c)^2}+\frac{b}{1+(a+c)^2}+\frac{c}{1+(a+b)^2}\le \frac{3(a^2+b^2+c^2)}{a^2+b^2+c^2+12abc}$.
[i]proposed by Mohammad Ahmadi[/i]
2008 AIME Problems, 4
There exist unique positive integers $ x$ and $ y$ that satisfy the equation $ x^2 \plus{} 84x \plus{} 2008 \equal{} y^2$. Find $ x \plus{} y$.
2023 CCA Math Bonanza, L5.3
Estimate the number of characters, excluding spaces, in the \LaTeX~source file for this Lightning Round, which includes the answer sheets and exactly one Asymptote diagram. Your score is determined by the function $max\{0, 20 - \lfloor \frac{|A - E|}{20}\rfloor\}$where $A$ is the actual answer, and $E$ is your estimate?
[i]Lightning 5.3[/i]
2007 Germany Team Selection Test, 2
Find all quadruple $ (m,n,p,q) \in \mathbb{Z}^4$ such that \[ p^m q^n \equal{} (p\plus{}q)^2 \plus{} 1.\]
1989 APMO, 3
Let $A_1$, $A_2$, $A_3$ be three points in the plane, and for convenience, let $A_4= A_1$, $A_5 = A_2$. For $n = 1$, $2$, and $3$, suppose that $B_n$ is the midpoint of $A_n A_{n+1}$, and suppose that $C_n$ is the midpoint of $A_n B_n$. Suppose that $A_n C_{n+1}$ and $B_n A_{n+2}$ meet at $D_n$, and that $A_n B_{n+1}$ and $C_n A_{n+2}$ meet at $E_n$.
Calculate the ratio of the area of triangle $D_1 D_2 D_3$ to the area of triangle $E_1 E_2 E_3$.
PEN A Problems, 19
Let $f(x)=x^3 +17$. Prove that for each natural number $n \ge 2$, there is a natural number $x$ for which $f(x)$ is divisible by $3^n$ but not $3^{n+1}$.
2006 AIME Problems, 11
A sequence is defined as follows $a_1=a_2=a_3=1$, and, for all positive integers $n$, $a_{n+3}=a_{n+2}+a_{n+1}+a_n$. Given that $a_{28}=6090307$, $a_{29}=11201821$, and $a_{30}=20603361$, find the remainder when $\displaystyle \sum^{28}_{k=1} a_k$ is divided by 1000.
2016 USAMTS Problems, 1:
Another round, another diagram...
[asy]
unitsize(1cm);
defaultpen(linewidth(0.45));
real[][] arr = {
{0,0,0,0},
{0,0,0,0},
{0,0,0,0},
{0,0,0,0}};
for (int i=0; i<4; ++i){
for (int j=0; j<4; ++j){
if(arr[3-j][i] != 0){
label((string) arr[3-j][i], (i+0.5, j+0.5));
}
}
}
label("$+$", (-0.5, 4.5), dir(-45));
label("$-$", (4.5, -0.5), dir(135));
label("\Large 13", (-0.5, 3.5));
label("\Large 28", (-0.5, 2.5));
label("\Large 23", (-0.5, 0.5));
label("\Large 7", (4.5, 3.5));
label("\Large 8", (4.5, 1.5));
label("\Large 8", (4.5, 0.5));
label("\Large 12", (3.5, -0.5));
label("\Large 12", (2.5,-0.5));
label("\Large 7", (0.5, -0.5));
label("\Large 23", (3.5, 4.5 ));
label("\Large 25", (2.5,4.5));
label("\Large 28", (1.5,4.5));
label("\Large 13", (0.5,4.5));
for(int i = 1; i <= 3; ++i){
draw((i, 0)--(i, 4));
draw((0, i)--(4, i));
}
draw((0,0)--(0,4)--(4,4)--(4,0)--cycle, linewidth(1.5));
draw((-0.8,-0.8)--(0,0), linewidth(1.5));
draw((4,4 )--(4.8,4.8), linewidth(1.5));
[/asy]
Use [code]\begin{asy}
\end{asy}[/code]environment to render the diagram correctly in a latex document. Remember to write [code]\usepackage{asymptote}[/code] in the preamble.
And of course, replace the 0's in the array at the beginning of the code with the numbers you wish to fill it in with.
2013 Online Math Open Problems, 45
Let $N$ denote the number of ordered 2011-tuples of positive integers $(a_1,a_2,\ldots,a_{2011})$ with $1\le a_1,a_2,\ldots,a_{2011} \le 2011^2$ such that there exists a polynomial $f$ of degree $4019$ satisfying the following three properties:
[list] [*] $f(n)$ is an integer for every integer $n$; [*] $2011^2 \mid f(i) - a_i$ for $i=1,2,\ldots,2011$; [*] $2011^2 \mid f(n+2011) - f(n)$ for every integer $n$. [/list]
Find the remainder when $N$ is divided by $1000$.
[i]Victor Wang[/i]
2014 Iran Team Selection Test, 3
we named a $n*n$ table $selfish$ if we number the row and column with $0,1,2,3,...,n-1$.(from left to right an from up to down)
for every {$ i,j\in{0,1,2,...,n-1}$} the number of cell $(i,j)$ is equal to the number of number $i$ in the row $j$.
for example we have such table for $n=5$
1 0 3 3 4
1 3 2 1 1
0 1 0 1 0
2 1 0 0 0
1 0 0 0 0
prove that for $n>5$ there is no $selfish$ table
2002 AMC 12/AHSME, 19
The graph of the function $ f$ is shown below. How many solutions does the equation $ f(f(x)) \equal{} 6$ have?
[asy]size(220);
defaultpen(fontsize(10pt)+linewidth(.8pt));
dotfactor=4;
pair P1=(-7,-4), P2=(-2,6), P3=(0,0), P4=(1,6), P5=(5,-6);
real[] xticks={-7,-6,-5,-4,-3,-2,-1,1,2,3,4,5,6};
real[] yticks={-6,-5,-4,-3,-2,-1,1,2,3,4,5,6};
draw(P1--P2--P3--P4--P5);
dot("(-7, -4)",P1);
dot("(-2, 6)",P2,LeftSide);
dot("(1, 6)",P4);
dot("(5, -6)",P5);
xaxis("$x$",-7.5,7,Ticks(xticks),EndArrow(6));
yaxis("$y$",-6.5,7,Ticks(yticks),EndArrow(6));[/asy]$ \textbf{(A)}\ 2 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 6 \qquad \textbf{(E)}\ 7$
2007 AMC 12/AHSME, 14
Point $ P$ is inside equilateral $ \triangle ABC$. Points $ Q, R$ and $ S$ are the feet of the perpendiculars from $ P$ to $ \overline{AB}, \overline{BC}$, and $ \overline{CA}$, respectively. Given that $ PQ \equal{} 1, PR \equal{} 2$, and $ PS \equal{} 3$, what is $ AB$?
$ \textbf{(A)}\ 4 \qquad \textbf{(B)}\ 3 \sqrt {3}\qquad \textbf{(C)}\ 6 \qquad \textbf{(D)}\ 4 \sqrt {3}\qquad \textbf{(E)}\ 9$
2005 IMC, 3
3) $f$ cont diff, $R\rightarrow ]0,+\infty[$, prove $|\int_{0}^{1}f^{3}-{f(0)}^{2}\int_{0}^{1}f| \leq \max_{[0,1]} |f'|(\int_{0}^{1}f)^{2}$
2008 IMS, 1
Let $ A_1,A_2,\dots,A_n$ be idempotent matrices with real entries. Prove that:
\[ \mbox{N}(A_1)\plus{}\mbox{N}(A_2)\plus{}\dots\plus{}\mbox{N}(A_n)\geq \mbox{rank}(I\minus{}A_1A_2\dots A_n)\]
$ \mbox{N}(A)$ is $ \mbox{dim}(\mbox{ker(A)})$
2013 F = Ma, 21
A simple pendulum experiment is constructed from a point mass $m$ attached to a pivot by a massless rod of length $L$ in a constant gravitational field. The rod is released from an angle $\theta_0 < \frac{\pi}{2}$ at rest and the period of motion is found to be $T_0$. Ignore air resistance and friction.
The experiment is repeated with a new pendulum with a rod of length $4L$, using the same angle $\theta_0$, and the period of motion is found to be $T$. Which of the following statements is correct?
$\textbf{(A) } T = 2T_0 \text{ regardless of the value of } \theta_0\\
\textbf{(B) } T > 2T_0 \text{ with } T \approx 2T_0 \text{ if } \theta_0 \ll 1\\
\textbf{(C) } T < 2T_0 \text{ with } T \approx 2T_0 \text{ if } \theta_0 \ll 1\\
\textbf{(D) } T < 2T_0 \text{ with some values of } \theta_0 \text{ and } T > 2T_0 \text{ for other values of } \theta_0\\
\textbf{(E) } T \text{ and } T_0 \text{ are not defined because the motion is not periodic unless } \theta_0 \ll 1$
2005 Polish MO Finals, 1
Find all triplets $(x,y,n)$ of positive integers which satisfy:
\[ (x-y)^n=xy \]
2006 Peru IMO TST, 1
[color=blue][size=150]PERU TST IMO - 2006[/size]
Saturday, may 20.[/color]
[b]Question 01[/b]
Find all $(x,y,z)$ positive integers, such that:
$\sqrt{\frac{2006}{x+y}} + \sqrt{\frac{2006}{y+z}} + \sqrt{\frac{2006}{z+x}},$
is an integer.
---
[url=http://www.mathlinks.ro/Forum/viewtopic.php?t=88509]Spanish version[/url]
$\text{\LaTeX}{}$ed by carlosbr
1998 Putnam, 5
Let $N$ be the positive integer with 1998 decimal digits, all of them 1; that is,
\[N=1111\cdots 11.\]
Find the thousandth digit after the decimal point of $\sqrt N$.
2013 Tuymaada Olympiad, 6
Solve the equation $p^2-pq-q^3=1$ in prime numbers.
[i]A. Golovanov[/i]
1997 Romania National Olympiad, 2
Prove that:
$\int_{-1}^1f^2(x)dx\ge \frac 1 2 (\int_{-1}^1f(x)dx)^2 +\frac 3 2(\int_{-1}^1xf(x)dx)^2$
Please give a proof without using even and odd functions. (the oficial proof uses those and seems to be un-natural) :D
1998 China Team Selection Test, 3
For any $h = 2^{r}$ ($r$ is a non-negative integer), find all $k \in \mathbb{N}$ which satisfy the following condition: There exists an odd natural number $m > 1$ and $n \in \mathbb{N}$, such that $k \mid m^{h} - 1, m \mid n^{\frac{m^{h}-1}{k}} + 1$.
2007 AMC 10, 17
Point $ P$ is inside equilateral $ \triangle ABC$. Points $ Q, R$ and $ S$ are the feet of the perpendiculars from $ P$ to $ \overline{AB}, \overline{BC}$, and $ \overline{CA}$, respectively. Given that $ PQ \equal{} 1, PR \equal{} 2$, and $ PS \equal{} 3$, what is $ AB$?
$ \textbf{(A)}\ 4 \qquad \textbf{(B)}\ 3 \sqrt {3}\qquad \textbf{(C)}\ 6 \qquad \textbf{(D)}\ 4 \sqrt {3}\qquad \textbf{(E)}\ 9$