Found problems: 85335
Let $a,b,c$ be positive real numbers . Prove that$$ \frac{1}{ab(b+1)(c+1)}+\frac{1}{bc(c+1)(a+1)}+\frac{1}{ca(a+1)(b+1)}\geq\frac{3}{(1+abc)^2}.$$
Given that: $\tfrac{6}{11}-\tfrac{10}{19}=\tfrac{9}{19}-\tfrac{n}{11}$, find $n$.
Determine all positive integers $ n$, for which $ 2^{n\minus{}1}n\plus{}1$ is a perfect square.
Five girls and five boys randomly sit in ten seats that are equally spaced around a circle. The probability that there is at least one diameter of the circle with two girls sitting on opposite ends of the diameter is $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
Let $(a_1,a_2,a_3,a_4,a_5)$ be a random permutation of the integers from $1$ to $5$ inclusive. Find the expected value of $$\sum^5_{i=1} |a_i -i | = |a_1 -1|+|a_2 -2|+|a_3 -3|+|a_4 -4|+|a_5 -5|.$$
[i]Proposed by Muztaba Syed[/i]
Let $p$ be a prime number. Prove that there exist infinitely many positive integers $n$ such that $p$ divides $1^n + 2^n +... + (p + 1)^n.$
Let $a=-\sqrt{3}+\sqrt{5}+\sqrt{7}, b=\sqrt{3}-\sqrt{5}+\sqrt{7}, c=\sqrt{3}+\sqrt{5}-\sqrt{7}$. Evaluate \[\frac{a^4}{(a-b)(a-c)}+\frac{b^4}{(b-c)(b-a)}+\frac{c^4}{(c-a)(c-b)}.\]
Find the least natural number $ n$ such that, if the set $ \{1,2, \ldots, n\}$ is arbitrarily divided into two non-intersecting subsets, then one of the subsets contains 3 distinct numbers such that the product of two of them equals the third.
$WHY$ is a triangle with angle $W \ge 90$ degrees. On the side $HY$, two distinct points $M$ and $E$ are chosen such that angle $HWM$ is equivalent to angle $ MWE$ and $HM * YE = HY * ME$. Find the angle $MWY$.
$11$ students take a test. For any two question in a test, there are at least $6$ students who solved exactly one of those two questions. Prove that there are no more than $12$ questions in this test. Showing the equality case is not needed.
Find the smallest positive interger number $n$ such that there exists $n$ real numbers $a_1,a_2,\ldots,a_n$ satisfied three conditions as follow:
a. $a_1+a_2+\cdots+a_n>0$;
b. $a_1^3+a_2^3+\cdots+a_n^3<0$;
c. $a_1^5+a_2^5+\cdots+a_n^5>0$.
An $n \times n$ complex matrix $A$ is called \emph{t-normal} if
$AA^t = A^t A$ where $A^t$ is the transpose of $A$. For each $n$,
determine the maximum dimension of a linear space of complex $n
\times n$ matrices consisting of t-normal matrices.
Proposed by Shachar Carmeli, Weizmann Institute of Science
A $12$-sided polygon is inscribed in a circle with radius $r$. The polygon has six sides of length $6\sqrt3$ that alternate with six sides of length $2$. Find $r^2$.
A cube $ABCDA'B'C'D'$ is given with an edge of length $2$ and vertices marked as in the figure. The point $K$ is center of the edge $AB$. The plane containing the points $B',D', K$ intersects the edge $AD$ at point $L$. Calculate the volume of the pyramid with apex $A$ and base the quadrilateral $D'B'KL$.
[img]https://cdn.artofproblemsolving.com/attachments/7/9/721989193ffd830fd7ad43bdde7e177c942c76.png[/img]
In the quadrilateral $ABCD$ it is known that $\angle ABD= \angle DBC$ and $AD= CD$. Let $DH$ be the altitude of $\vartriangle ABD$. Prove that $| BC - BH | = HA$.
(Hilko Danilo)
Let $\mathcal{P}$ be the set of all primes, and let $M$ be a subset of $\mathcal{P}$, having at least three elements, and such that for any proper subset $A$ of $M$ all of the prime factors of the number $ -1+\prod_{p\in A}p$ are found in $M$. Prove that $M= \mathcal{P}$.
[i]Valentin Vornicu[/i]
Let $n$ be a positive integer. Find all polynomials $Q(x)$ with integer coefficients so that the degree of $Q(x)$ is less than $n$ and there exists an integer $m\geq 1$ for which
\[x^n-1\mid Q(x)^m-1\]
Let $ABC$ be an acute triangle and $D, E, F$ are the midpoints of $BC, CA, AB$, respectively. The circle with diameter $AD$ intersects the lines $AB$ and $AC$ at points $P$ and $Q$ , respectively. The lines through $P$ and $Q$ parallel to $BC$ intersect $DE$ at point $R$ and $DF$ at point $S$, respectively. The circumcircle of $DPR$ intersects $AB$ at $X$, the circumcircle of $DQS$ intersects $AC$ in $Y$, and these two circles intersect again point $Z$. Prove that $Z$ is the midpoint of $XY$.
$ n $ lines meet at a point. Each one of the $ 2n $ disjoint angles formed around this point by these lines has either $ 7^{\circ} $ or $ 17^{\circ} . $
[b]a)[/b] Find $ n. $
[b]b)[/b] Prove that among these lines there are at least two perpendicular ones.
Prove that the equation $x^{19} + x^{17} = x^{16 }+ x^7 + a$ for any $a \in R$ has at least two imaginary roots
Suppose that ${x_1, x_2, \dots , x_n}$ are positive integers for which $x_1 + x_2 + \cdots+ x_n = 2(n + 1)$. Show that there exists an integer $r$ with $0 \leq r \leq n - 1$ for which the following $n - 1$ inequalities hold:
\[x_{r+1} + \cdots + x_{r+i} \leq 2i+ 1, \qquad \qquad \forall i, 1 \leq i \leq n - r; \]
\[x_{r+1} + \cdots + x_n + x_1 + \cdots+ x_i \leq 2(n - r + i) + 1, \qquad \qquad \forall i, 1 \leq i \leq r - 1.\]
Prove that if all the inequalities are strict, then $r$ is unique and that otherwise there are exactly two such $r.$
Given a prime number $p{}$ and integers $x{}$ and $y$, find the remainder of the sum $x^0y^{p-1}+x^1y^{p-2}+\ldots+x^{p-2}y^1+x^{p-1}y^0$ upon division by $p{}$.
Given is an acute triangle $ABC$. Its heights $BB_1$ and $CC_1$ are extended past points $B_1$ and $C_1$. On these extensions, points $P$ and $Q$ are chosen, such that angle $PAQ$ is right. Let $AF$ be a height of triangle $APQ$. Prove that angle $BFC$ is a right angle.
Players $1,2,\ldots n$ are seated around a table, and each has a single penny. Player $1$ passes a penny to Player $2$, who then passes two pennies to Player $3$, who then passes one penny to player $4$, who then passes two pennies to Player $5$ and so on, players alternately pass one or two pennies to the next player who still has some pennies. The player who runs out of pennies drops out of the game and leaves the table. Find an infinite set of numbers $n$ for which some player ends up with all the $n$ pennies.
Determine all integer values of n for which the number $\frac{14n+25}{2n+1}$ 'is a perfect square.