Found problems: 85335
A random rectangle (not necessarily a square) with positive integer dimensions is selected from the $2\times4$ grid below. The probability that the selected rectangle contains only white squares can be written as $\frac{a}{b}$ where $a$ and $b$ are relatively prime positive integers. Find $a+b$.
[asy]
fill((2,0)--(3,0)--(3,1)--(2,1)--cycle,blue);
fill((2,1)--(3,1)--(3,2)--(2,2)--cycle,blue);
draw((0,0)--(4,0),black);
draw((0,0)--(0,2),black);
draw((4,0)--(4,2),black);
draw((4,2)--(0,2),black);
draw((0,1)--(4,1),black);
draw((1,0)--(1,2),black);
draw((2,0)--(2,2),black);
draw((3,0)--(3,2),black);
[/asy]
Let $n$ be a positive integer, and $a,b$ be two complex numbers such that $a \neq 1$ and $b^k \neq 1$, for any $k \in \{1,2,\dots ,n\}$. The matrices $A,B \in \mathcal{M}_n(\mathbb{C})$ satisfy the relation $BA=a I_n + bAB$. Prove that $A$ and $B$ are invertible.
Calculate the following indefinite integrals.
[1] $\int (\sin x+\cos x)^4 dx$
[2] $\int \frac{e^{2x}}{e^x+1}dx$
[3] $\int \sin ^ 4 xdx$
[4] $\int \sin 6x\cos 2xdx$
[5] $\int \frac{x^2}{\sqrt{(x+1)^3}}dx$
Determine the smallest positive constant $k$ such that no matter what $3$ lattice points we choose the following inequality holds:
$$ L_{\max} - L_{\min} \ge \frac{1}{\sqrt{k} \cdot L_{max}} $$
where $L_{\max}$, $L_{\min}$ is the maximal and minimal distance between chosen points.
$20$ points with no three collinear are given. How many obtuse triangles can be formed by these points?
$ \textbf{(A)}\ 6 \qquad \textbf{(B)}\ 20 \qquad \textbf{(C)}\ 2{{10}\choose{3}} \qquad \textbf{(D)}\ 3{{10}\choose{3}} \qquad \textbf{(E)}\ {{20}\choose{3}}$
We say that an integer $a$ is a quadratic, cubic, or quintic residue modulo $n$ if there exists an integer $x$ such that $x^2\equiv a \pmod n$, $x^3 \equiv a \pmod n$, or $x^5 \equiv a \pmod n$, respectively. Further, an integer $a$ is a primitive residue modulo $n$ if it is exactly one of these three types of residues modulo $n$.
How many integers $1 \le a \le 2015$ are primitive residues modulo $2015$?
[i] Proposed by Michael Ren [/i]
Determine $w^2+x^2+y^2+z^2$ if
\[ \begin{array}{l} \displaystyle \frac{x^2}{2^2-1}+\frac{y^2}{2^2-3^2}+\frac{z^2}{2^2-5^2}+\frac{w^2}{2^2-7^2}=1 \\ \displaystyle \frac{x^2}{4^2-1}+\frac{y^2}{4^2-3^2}+\frac{z^2}{4^2-5^2}+\frac{w^2}{4^2-7^2}=1 \\ \displaystyle \frac{x^2}{6^2-1}+\frac{y^2}{6^2-3^2}+\frac{z^2}{6^2-5^2}+\frac{w^2}{6^2-7^2}=1 \\ \displaystyle \frac{x^2}{8^2-1}+\frac{y^2}{8^2-3^2}+\frac{z^2}{8^2-5^2}+\frac{w^2}{8^2-7^2}=1 \\ \end{array} \]
Several pounamu stones weigh altogether $10$ tons and none of them weigh more than $1$ tonne. A truck can carry a load which weight is at most $3$ tons. What is the smallest number of trucks such that bringing all stones from the quarry will be guaranteed?
For $a,n$ positive integers we show number of different integer 10-tuples $ (x_1,x_2,...,x_{10})$ on $ (mod n)$ satistfying $x_1x_2...x_{10}=a (mod n)$ with $f(a,n)$. Let $a,b$ given positive integers ,
a) Prove that there exist a positive integer $c$ such that for all $n\in \mathbb{Z^+}$ $$\frac {f(a,cn)}{f(b,cn)}$$is constant
b) Find all $(a,b)$ pairs such that minumum possible value of $c$ is 27 where $c$ satisfying condition in $(a)$
Let $A{}$ be a set with $n{}$ $(n\geq3)$ elements. Iterations $f^2,f^2,\ldots$ of the function $f:A\rightarrow A$ are defined as $f^2(x)=f(f(x)), f^{i+1}=f(f^i(x)), \forall i\geq2$. Find the number of functions $f:A\rightarrow A$ with the property: the function $f^{n-2}$ is constant, but $f^{n-3}$ is not.
Two circles with radius $2$, $\omega_1$ and $\omega_2$, are centered at $O_1$ and $O_2$ respectively. The circles $\omega_1$ and $\omega_2$ are externally tangent to each other and internally tangent to a larger circle $\omega$ centered at $O$ at points $A$ and $B$, respectively. Let $M$ be the midpoint of minor arc $AB$. Let $P$ be the intersection of $\omega_1$ and $O_1M$, and let $Q$ be the intersection of $\omega_2$ and $O_2M$. Given that there is a point $R$ on $\omega$ such that $\triangle PQR$ is equilateral, the radius of $\omega$ can be written as $\frac{a+\sqrt{b}}{c}$ where $a$, $b$, and $c$ are positive integers and $a$ and $c$ are relatively prime. Find $a+b+c$.
Two triangles $ABC,ABD$ (with the common side $c=AB$) are given in space. Triangle $ABC$ is right with hypotenuse $AB$, $ABD$ is equilateral. Denote $\varphi$ the dihedral angle between planes $ABC,ABD$.
1) Determine the length of $CD$ in terms of $a=BC,b=CA,c$ and $\varphi$.
2) Determine all values of $\varphi$ such that the tetrahedron $ABCD$ has four sides of the same length.
$m,n$ are non-zero-real numbers, $z\in\mathbb{C}$. Then, the figure of equations $|z+n\text{i}|+|z-m\text{i}|=n$ and $|z+n\text{i}|-|z-m\text{i}|=-m$ in complex plane is ($F_1,F_2$ are focal points)
[img]https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvMS84L2RkYWZjM2JmNTc0N2RmYjJlMGUwMGFmMWRkY2RkZTA4NTljZTUwLnBuZw==&rn=MTI0NTI0NTQucG5n[/img]
In the plane, there are $n$ points ($n\ge 4$) where no 3 of them are collinear. Let $A(n)$ be the number of parallelograms whose vertices are those points with area $1$. Prove the following inequality:
$A(n)\leq \frac{n^2-3n}{4}$ for all $n\ge 4$
At the end of $ 1994$, Walter was half as old as his grandmother. The sum of the years in which they were born was $ 3838$. How old will Walter be at the end of $ 1999$?
$ \textbf{(A)}\ 48 \qquad \textbf{(B)}\ 49\qquad \textbf{(C)}\ 53\qquad \textbf{(D)}\ 55\qquad \textbf{(E)}\ 101$
Show that there exists a sequence of positive integers $x_1, x_2,…x_n,…$ that satisfies the following two conditions:
(i) Every positive integer appears exactly once,
(ii) For every $n=1,2,…$ the partial sum $x_1+x_2+…+x_n$ is divisible by $n^n$.
Show that $e=\sum^{\infty}_{n=0} \frac{1}{n!}$ is irrational.
Let $a_1,a_2,a_3,\ldots,a_n$ be a sequence of non-negative integers, where $n$ is a positive integer. Let
\[ A_n={a_1+a_2+\cdots+a_n\over n}\ . \]
Prove that
\[ a_1!a_2!\ldots a_n!\ge\left(\lfloor A_n\rfloor !\right)^n \]
where $\lfloor A_n\rfloor$ is the greatest integer less than or equal to $A_n$, and $a!=1\times 2\times\cdots\times a$ for $a\ge 1$(and $0!=1$). When does equality hold?
Determine all $x$, $y$, $k$ and $n$ positive integers such that:
$10^x$ + $10^y$ + $n!$ = $2024^k$
Seven spheres are situated in space such that no three centers are collinear, no four centers are coplanar, and every pair of spheres intersect each other at more than one point. For every pair of spheres, the plane on which the intersection of the two spheres lies in is drawn. What is the least possible number of sets of four planes that intersect in at least one point?
Find the value of $|xy|$ given that $x$ and $y$ are integers and \[6x^2y^2+5x^2-18y^2=17253.\]
Prove that there exist infinitely many positive integers $ n$ such that $ p \equal{} nr,$ where $ p$ and $ r$ are respectively the semiperimeter and the inradius of a triangle with integer side lengths.
Determine all integers $n\ge1$ for which there exists $n$ real numbers $x_1,\ldots,x_n$ in the closed interval $[-4,2]$ such that the following three conditions are fulfilled:
- the sum of these real numbers is at least $n$.
- the sum of their squares is at most $4n$.
- the sum of their fourth powers is at least $34n$.
[i](Proposed by Gerhard Woeginger, Austria)[/i]
A $40$ feet high screen is put on a vertical wall $10$ feet above your eye-level. How far should you stand to maximize the angle subtended by the screen (from top to bottom) at your eye?
Given a triangle \(ABC\), an arbitrary point \(D\) is chosen on the side \(AC\). In triangles \(ABD\) and \(CBD\), the angle bisectors \(BK\) and \(BL\) are drawn, respectively. The point \(O\) is the circumcenter of \(\triangle KBL\). Prove that the second intersection point of the circumcircles of triangles \(ABL\) and \(CBK\) lies on the line \(OD\).
[i]Proposed by Anton Trygub[/i]