Found problems: 85335
[b]a)[/b] Find all $ 2\times 2 $ complex matrices $ A $ which have the property that there are two complex numbers $ \alpha ,\gamma $ with $ \alpha \neq \text{tr} (A) $ or $ \gamma\neq \det (A) $ such that $ A^2-\alpha A+\gamma I=0. $
[b]b)[/b] Consider $ B\not\in\{ 0,I\} $ as a matrix having the property mentioned at [b]a).[/b]
Solve in the complex numbers the system $ xB-yI-B^2=xB^2-yI-B^4=0. $
[i]Adrian Troie[/i]
Let $\triangle ABC$ be an acute-angled triangle with $AB>AC$, $O$ be its circumcenter and $D$ the midpoint of side $BC$. The circle with diameter $AD$ meets sides $AB,AC$ again at points $E,F$ respectively. The line passing through $D$ parallel to $AO$ meets $EF$ at $M$. Show that $EM=MF$.
When Cirno walks into her perfect math class today, she sees a polynomial $P(x)=1$ (of degree 0) on the blackboard. As her teacher explains, for her pop quiz today, she will have to perform one of the two actions every minute:
(a) Add a monomial to $P(x)$ so that the degree of $P$ increases by 1 and $P$ remains monic;
(b) Replace the current polynomial $P(x)$ by $P(x+1)$. For example, if the current polynomial is $x^2+2x+3$, then she will change it to $(x+1)^2+2(x+1)+3=x^2+4x+6$.
Her score for the pop quiz is the sum of coefficients of the polynomial at the end of 9 minutes. Given that Cirno (miraculously) doesn't make any mistakes in performing the actions, what is the maximum score that she can get?
[i]Proposed by Yannick Yao[/i]
Points $A$ and $B$ are fixed on a circle $c_1$. Circle $c_2$, whose centre lies on $c_1$, touches line $AB$ at $B$. Another line through $A$ intersects $c_2$ at points $D$ and $E$, where $D$ lies between $A$ and $E$. Line $BD$ intersects $c_1$ again at $F$. Prove that line $EB$ is tangent to $c_1$ if and only if $D$ is the midpoint of the segment $BF$.
Let triangle $ABC$ be a right triangle in the xy-plane with a right angle at $C$. Given that the length of the hypotenuse $AB$ is 60, and that the medians through $A$ and $B$ lie along the lines $y=x+3$ and $y=2x+4$ respectively, find the area of triangle $ABC$.
For $n=1,2,\dots$ let
\[S_n=\log\left(\sqrt[n^2]{1^1 \cdot 2^2 \cdot \dotsc \cdot n^n}\right)-\log(\sqrt{n}),\]
where $\log$ denotes the natural logarithm. Find $\lim_{n \to \infty} S_n$.
In an increasing infinite sequence of positive integers, every term starting from the $2002$-th term divides the sum of all preceding terms. Prove that every term starting from some term is equal to the sum of all preceding terms.
Find the least
a. positive real number
b. positive integer
$t$ such that the equation $(x^2+y^2)^2 + 2tx(x^2 + y^2) = t^2y^2$ has a solution where $x,y$ are positive integers.
From among $2^{1/2},$ $3^{1/3},$ $8^{1/8},$ $9^{1/9}$ those which have the greatest and the next to the greatest values, in that order, are
\[ \begin{array}{rlrlrlrl} \hbox {(A)}& 3^{1/3},\ 2^{1/2} \quad & \hbox {(B)}& 3^{1/3},\ 8^{1/8} \quad & \hbox {(C)}& 3^{1/3},\ 9^{1/9} \quad & \hbox {(D)}& 8^{1/8},\ 9^{1/9} \\ \hbox {(E)}& \multicolumn{3}{l}{\hbox{None of these}} \end{array} \]
$a,b$ are real numbers (neither is $0$).Given two conditions:
A: $a>0$.
B: $a>b$ and $a^{-1}>b^{-1}$.
Then, which one of the followings are true?
$(\text{A})$A is sufficient but unnecessary condition of B.
$(\text{B})$A is necessary but insufficient condition of B.
$(\text{C})$A is sufficient and necessary condition of B.
$(\text{D})$A is insufficient and unnecessary condition of B.
A canister contains two and a half cups of flour. Greg and Sally have a brownie recipe which calls for one and one third cups of flour. Greg and Sally want to make one and a half recipes of brownies. To the nearest whole percent, what percent of the flour in the canister would they use?
Betsy is addicted to chocolate. Every day, she eats $2$ chocolates at breakfast, $3$ chocolates at lunch, $1$ chocolate during her afternoon snack time, and $5$ chocolates at dinner. If she begins eating a bag of $100$ chocolates at breakfast one day, during which meal will she eat the last piece in the bag?
$\textbf{(A) }\text{breakfast}\qquad\textbf{(B) }\text{lunch}\qquad\textbf{(C) }\text{snack time}\qquad\textbf{(D) }\text{dinner}\qquad\textbf{(E) }\text{impossible to determine}$
Can you mark the cube's vertices with the three-digit binary numbers in such a way, that the numbers at all the possible couples of neighbouring vertices differ in at least two digits?
We create a sequence by setting $a_1 = 2010$ and requiring that $a_n-a_{n-1}\leq n$ and $a_n$ is also divisible by $n$.
Show that $a_{100},a_{101},a_{102},\dots$ form an arithmetic sequence.
Let $\triangle ABC$ be an acute angled triangle and let $k$ be its circumscribed circle. A point $O$ is given in the interior of the triangle, such that $CE=CF$, where $E$ and $F$ are on $k$ and $E$ lies on $AO$ while $F$ lies on $BO$. Prove that $O$ is on the angle bisector of $\angle ACB$ if and only if $AC=BC$.
Define a sequence $a_i$ as follows: $a_1 = 181$ and for $i \ge 2$, $a_i = a_{i-1}^2-1$ if $a_{i-1}$ is odd and $a_i = a_{i-1}/2$ if $a_{i-1}$ is even. Find the least $i$ such that $a_i = 0$.
For each positive integer $n$, let $s(n)$ be the sum of the digits of $n$. Find the smallest positive integer $k$ such that
\[s(k) = s(2k) = s(3k) = \cdots = s(2013k) = s(2014k).\]
Find all triples of positive integers $(a,b,p)$ with $a,b$ positive integers and $p$ a prime number such that $2^a+p^b=19^a$
Natural $a,b,c$ are $>100$ and $(a,b,c)=1$. $c|a+b,a|b+c$ Find minimal $b$
$(a)$ A plane $\pi$ passes through the vertex $O$ of the regular tetrahedron $OPQR$. We define $p, q, r$ to be the signed distances of $P,Q,R$ from $\pi$ measured along a directed normal to $\pi$. Prove that
\[p^2 + q^2 + r^2 + (q - r)^2 + (r - p)^2 + (p - q)^2 = 2a^2,\]
where $a$ is the length of an edge of a tetrahedron.
$(b)$ Given four parallel planes not all of which are coincident, show that a regular tetrahedron exists with a vertex on each plane.
[u]Note:[/u] Part $(b)$ is [url=http://www.artofproblemsolving.com/Forum/viewtopic.php?f=49&t=60825&start=0]IMO 1972 Problem 6[/url]
Find the smallest positive integer $n$, such that there exist $n$ integers $x_1, x_2, \dots , x_n$ (not necessarily different), with $1\le x_k\le n$, $1\le k\le n$, and such that
\[x_1 + x_2 + \cdots + x_n =\frac{n(n + 1)}{2},\quad\text{ and }x_1x_2 \cdots x_n = n!,\]
but $\{x_1, x_2, \dots , x_n\} \ne \{1, 2, \dots , n\}$.
Find all derivable functions that have real domain and codomain, and are equal to their second functional power.
Let $ \mathcal{H}$ be a family of finite subsets of an infinite set $ X$ such that every finite subset of $ X$ can be represented as the union of two disjoint sets from $ \mathcal{H}$. Prove that for every positive integer $ k$ there is a subset of $ X$ that can be represented in at least $ k$ different ways as the union of two disjoint sets from $ \mathcal{H}$.
[i]P. Erdos[/i]
Let $P$ be a set of $n$ points and $S$ a set of $l$ segments. It is known that:
$(i)$ No four points of $P$ are coplanar.
$(ii)$ Any segment from $S$ has its endpoints at $P$.
$(iii)$ There is a point, say $g$, in $P$ that is the endpoint of a maximal number of segments from $S$ and that is not a vertex of a tetrahedron having all its edges in $S$.
Prove that $l \leq \frac{n^2}{3}$
Find the minimum number of perfect cubes such that their sum is equal to $ 346^{346} . $