Found problems: 85335
Let $n$ be an integer such that $n > 3$. Suppose that we choose three numbers from the set $\{1, 2, \ldots, n\}$. Using each of these three numbers only once and using addition, multiplication, and parenthesis, let us form all
possible combinations.
(a) Show that if we choose all three numbers greater than $\frac{n}{2}$, then the values of these combinations are all distinct.
(b) Let $p$ be a prime number such that $p \leq \sqrt{n}$. Show that the number of ways of choosing three numbers so that the smallest one is $p$ and the values of the combinations are not all distinct is precisely the number of positive divisors of $p - 1$.
If the sum of two numbers is 1 and their product is 1, then the sum of their cubes is:
$\text{(A)} \ 2 \qquad \text{(B)} \ -2-\frac{3i\sqrt{3}}{4} \qquad \text{(C)} \ 0 \qquad \text{(D)} \ -\frac{3i\sqrt{3}}{4} \qquad \text{(E)} \ -2$
Let $A=\left(
\begin{array}{ccc}
1 & 1& 0 \\
0 & 1& 0 \\
0 &0 & 2
\end{array}
\right),\ B=\left(
\begin{array}{ccc}
a & 1& 0 \\
b & 2& c \\
0 &0 & a+1
\end{array}
\right)\ (a,\ b,\ c\in{\mathbb{C}}).$
(1) Find the condition for $a,\ b,\ c$ such that ${\text{rank} (AB-BA})\leq 1.$
(2) Under the condition of (1), find the condition for $a,\ b,\ c$ such that $B$ is diagonalizable.
A pair of standard 6-sided fair dice is rolled once. The sum of the numbers rolled determines the diameter of a circle. What is the probability that the numerical value of the area of the circle is less than the numerical value of the circle's circumference?
$\textbf{(A)}\,\frac{1}{36} \qquad\textbf{(B)}\,\frac{1}{12} \qquad\textbf{(C)}\,\frac{1}{6} \qquad\textbf{(D)}\,\frac{1}{4} \qquad\textbf{(E)}\,\frac{5}{18}$
In triangle $ABC$, side $AB$ is $1$. It is known that one of the angle bisectors of triangle $ABC$ is perpendicular to one of its medians, and some other angle bisector is perpendicular to the other median. What can be the perimeter of triangle $ABC$?
Source: 1976 Euclid Part B Problem 3
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$I$ is the centre of the inscribed circle of $\triangle{ABC}$. $AI$ meets the circumcircle of $\triangle{ABC}$ at $D$. Prove that $D$ is equidistant from $I$, $B$, and $C$.
Find all positive integers $k$ for which there exist $a$, $b$, and $c$ positive integers such that \[\lvert (a-b)^3+(b-c)^3+(c-a)^3\rvert=3\cdot2^k.\]
The radius of the circumscribed circle of an acute-angled triangle is $23$ and the radius of its Inscribed circle is $9$. Common external tangents to its ex-circles, other than straight lines containing the sides of the original triangle, form a triangle. Find the radius of its inscribed circle.
Find all positive integers that are equal to $700$ times the sum of its digits.
We say that a real $a\geq-1$ is philosophical if there exists a sequence $\epsilon_1,\epsilon_2,\dots$, with $\epsilon_i \in\{-1,1\}$ for all $i\geq1$, such that the sequence $a_1,a_2,a_3,\dots$, with $a_1=a$, satisfies
$$a_{n+1}=\epsilon_{n}\sqrt{a_{n}+1},\forall n\geq1$$
and is periodic. Find all philosophical numbers.
Determine the parity of the positive integer $N$, where $$N = \lfloor \frac{2002!}{2001 \cdot2003} \rfloor.$$
Find the sum of all positive integers $n$ where the mean and median of $\{20, 42, 69, n\}$ are both integers.
[i]Proposed by bissue[/i]
Let two circles $C_1$ and $C_2$ intersect in points $A$ and $B$. The tangent to $C_1$ at $A$ intersects $C_2$ in $P$ and the line $PB$ intersects $C_1$ for the second time in $Q$ (suppose that $Q$ is outside $C_2$). The tangent to $C_2$ from $Q$ intersects $C_1$ and $C_2$ in $C$ and $D$, respectively. (The points $A$ and $D$ lie on different sides of the line $PQ$.) Show that $AD$ is the bisector of $\angle CAP$.
[i]Proposed by Iman Maghsoudi[/i]
Find the largest positive integer $n$ ($n \ge 3$), so that there is a convex $n$-gon, the tangent of each interior angle is an integer.
Find all integers $n$ for which there exists an equiangular $n$-gon whose side lengths are distinct rational numbers.
Let $A=a_{ij}$ is simetrical real matrix. Prove that :
$\sum_i e^{a_{ii}} \leq tr (e^A)$
Professor Gamble buys a lottery ticket, which requires that he pick six different integers from $ 1$ through $ 46$, inclusive. He chooses his numbers so that the sum of the base-ten logarithms of his six numbers is an integer. It so happens that the integers on the winning ticket have the same property--- the sum of the base-ten logarithms is an integer. What is the probability that Professor Gamble holds the winning ticket?
$ \textbf{(A)}\ 1/5 \qquad \textbf{(B)}\ 1/4 \qquad \textbf{(C)}\ 1/3 \qquad \textbf{(D)}\ 1/2 \qquad \textbf{(E)}\ 1$
Given positive numbers ${a_1},{a_2},...,{a_n}$ with ${a_1} + {a_2} + ... + {a_n} = 1$. Prove that $\left( {\frac{1}{{a_1^2}} - 1} \right)\left( {\frac{1}{{a_2^2}} - 1} \right)...\left( {\frac{1}{{a_n^2}} - 1} \right) \geqslant {({n^2} - 1)^n}$.
Consider a regular $2n$-gon and the $n$ diagonals of it that pass through its center. Let $P$ be a point of the inscribed circle and let $a_1, a_2, \ldots , a_n$ be the angles in which the diagonals mentioned are visible from the point $P$. Prove that
\[\sum_{i=1}^n \tan^2 a_i = 2n \frac{\cos^2 \frac{\pi}{2n}}{\sin^4 \frac{\pi}{2n}}.\]
Let $ABC$ and $DEF$ be two triangles, such that $AB=DE=20$, $BC=EF=13$, and $\angle A = \angle D$. If $AC-DF=10$, determine the area of $\triangle ABC$.
[i]Proposed by Lewis Chen[/i]
Pete puts 100 stones in a row: black one, white one, black one, white one, ..., black one, white one. In a single move either Pete chooses two black stones with only white stones between them, and repaints all these white stones in black, or Pete chooses two white stones with only black stones between them, and repaints all these black stones in white. Can Pete with a sequence of moves described above obtain a row of 50 black stones followed by 50 white stones?
Egor Bakaev
a) Prove that if $n$ is an odd perfect number then $n$ has the following form \[ n=p^sm^2 \] where $p$ is prime has form $4k+1$, $s$ is positive integers has form $4h+1$, and $m\in\mathbb{Z}^+$, $m$ is not divisible by $p$.
b) Find all $n\in\mathbb{Z}^+$, $n>1$ such that $n-1$ and $\frac{n(n+1)}{2}$ is perfect number
A function f has the following property: If $k > 1, j > 1$, and $\gcd(k, j) = m$, then $f(kj) = f(m) (f\left(\frac km\right) + f\left(\frac jm\right))$. What values can $f(1984)$ and $f(1985)$ take?
In a convex polygon $P$ some diagonals have been drawn, without intersections inside $P$. Show that there exist at least two vertices of $P$, neither one of them being an endpoint of any one of those diagonals.
Solve the equation $tan^2 2x+2 tan2x tan3x = 1$