Found problems: 85335
3.Let ABCD be a parallelogram with side AB longer than AD and acute angle
$\angle DAB$. The bisector of ∠DAB meets side CD at L and line BC at K. If O is
the circumcenter of triangle LCK, prove that the points B,C,O,D lie on a circle.
Two squares are arranged as shown in the picture. Prove that the areas of shaded quadrilaterals are equal.
[img]https://3.bp.blogspot.com/-W50DOuizFvY/XT6wh3-L6sI/AAAAAAAAKaw/pIW2RKmttrwPAbrKK3bpahJz7hfIZwM8QCK4BGAYYCw/s400/Oral%2BSharygin%2B2016%2B10.11%2Bp3.png[/img]
The number $a$ is such that both fractions
$$(1-3a)/(2a + 3) \,\,\, and \,\,\, (17 + 4a)/(7 + a)$$
are positive. Which one is closer to $\sqrt5$?
What is the largest area of a right triangle, the vertices of which are located at distances $a$, $b$ and $c$ from a certain point (where $a$ is the distance to the vertex of the right angle)?
A [i]normal magic square[/i] of order $n$ is an arrangement of the integers from $1$ to $n^2$ in a square such that the $n$ numbers in each row, each column, and each of the two diagonals sum to a constant, called the [i]magic sum[/i] of the magic square. Compute the magic sum of a normal magic square of order $8$.
Prove the identity: $$\cos \frac{\pi}{7} \cdot \cos \frac{4\pi}{7} \cdot \cos \frac{5\pi}{7} = \frac{1}{8}$$
Let $\triangle ABC$ a scalene triangle with incenter $I$, circumcenter $O$ and circumcircle $\Gamma$. The incircle of $\triangle ABC$ is tangent to $BC, CA$ and $AB$ at points $D, E$ and $F$, respectively. The line $AI$ meet $EF$ and $\Gamma$ at $N$ and $M\neq A$, respectively. $MD$ meet $\Gamma$ at $L\neq M$ and $IL$ meet $EF$ at $K$. The circumference of diameter $MN$ meet $\Gamma$ at $P\neq M$. Prove that $AK, PN$ and $OI$ are concurrent.
Let $n \geq 3$ be a positive integer. Find the maximum number of diagonals in a regular $n$-gon one can select, so that any two of them do not intersect in the interior or they are perpendicular to each other.
Let $a,b,c,d$ be positive real numbers. What is the minimum value of $$ \frac{(a^2+b^2+2c^2+3d^2)(2a^2+3b^2+6c^2+6d^2)}{(a+b)^2(c+d)^2}$$
Trapezoid $ABCD$ is an isosceles trapezoid with $AD=BC$. Point $P$ is the intersection of the diagonals $AC$ and $BD$. If the area of $\triangle ABP$ is $50$ and the area of $\triangle CDP$ is $72$, what is the area of the entire trapezoid?
[i]Proposed by David Altizio
A unit square is divided into polygons, so that all sides of a polygon are parallel to sides of the given square. If the total length of the segments inside the square (without the square) is $2n$ (where $n$ is a positive real number), prove that there exists a polygon whose area is greater than $\frac{1}{(n+1)^2}$.
Compute
$$\int\frac{(\sin x+\cos x)(4-2\sin2x-\sin^22x)e^x}{\sin^32x}dx$$
where $x\in\left(0,\frac\pi2\right)$.
[i]Proposed by Mihály Bencze[/i]
A set $S$ of $100$ points, no four in a plane, is given in space. Prove that there are no more than $4 .101^2$ tetrahedra with the vertices in $S$, such that any two of them have at most two vertices in common.
Let $B$ and $C$ be two fixed points in the plane. For each point $A$ of the plane, outside of the line $BC$, let $G$ be the barycenter of the triangle $ABC$. Determine the locus of points $A$ such that $\angle BAC + \angle BGC = 180^{\circ}$.
Note: The locus is the set of all points of the plane that satisfies the property.
Determine all possible values of integer $k$ for which there exist positive integers $a$ and $b$ such that $\dfrac{b+1}{a} + \dfrac{a+1}{b} = k$.
Let $a, b$ and $c$ be positive integers such that $ab + 1, bc + 1$ and $ca + 1$ are all integer squares.
a) Give an example of such numbers $a, b$ and $c$.
b) Prove that at least one of the numbers $a, b$ and $c$ is divisible by $4$
Consider quadrilateral $ABCD$. It is given that $\angle DAC=70^\circ$, $\angle BAC=40^\circ$, $\angle BDC=20^\circ$, $\angle CBD=35^\circ$. Let $P$ be the intersection of $AC$ and $BD$. Find $\angle BPC$.
Let $a,b,c$ and $m$ ($0 \le m \le 26$) be integers such that $a + b + c = (a - b)(b- c)(c - a) = m$ (mod $27$) then $m$ is
(A): $0$, (B): $1$, (C): $25$, (D): $26$ (E): None of the above.
For each positive integer $n$, let $\tau\left(n\right)$ be the number of positive divisors of $n$. It is well-known that if $a$ and $b$ are relatively prime positive integers then $\tau\left(ab\right)=\tau\left(a\right)\tau\left(b\right)$. Does the converse hold? That is, if $a$ and $b$ are positive integers such that $\tau\left(ab\right)=\tau\left(a\right)\tau\left(b\right)$, then is it necessarily true that $a$ and $b$ are relatively prime? Either give a proof, or find a counter-example.
For which digits $a$ do exist integers $n \geq 4$ such that each digit of $\frac{n(n+1)}{2}$ equals $a \ ?$
The points of this lattice $4\times 4 = 16$ points can be vertices of squares.
[asy]
unitsize(1 cm);
int i, j;
for (i = 0; i <= 3; ++i) {
draw((i,0)--(i,3));
draw((0,i)--(3,i));
}
draw((1,1)--(2,2)--(1,3)--(0,2)--cycle);
for (i = 0; i <= 3; ++i) {
for (j = 0; j <= 3; ++j) {
dot((i,j));
}}
[/asy]
Calculate the number of different squares that can be formed in a lattice of $100\times 100$ points.
There are $ n$ websites $ 1,2,\ldots,n$ ($ n \geq 2$). If there is a link from website $ i$ to $ j$, we can use this link so we can move website $ i$ to $ j$.
For all $ i \in \left\{1,2,\ldots,n - 1 \right\}$, there is a link from website $ i$ to $ i+1$.
Prove that we can add less or equal than $ 3(n - 1)\log_{2}(\log_{2} n)$ links so that for all integers $ 1 \leq i < j \leq n$, starting with website $ i$, and using at most three links to website $ j$. (If we use a link, website's number should increase. For example, No.7 to 4 is impossible).
Sorry for my bad English.
We call the figure shown in the picture consisting of five unit squares a $\emph{plus}$, and each rectangle consisting of two such squares a $\emph{minus}$. Does there exist an odd integer $n$ with the property that a square with side length $n$ can be dissected into pluses and minuses? Justify your answer.
[img]
https://wiki-images.artofproblemsolving.com//6/6a/18-1-6.png
[/img]
We say that a set $S$ of integers is [i]rootiful[/i] if, for any positive integer $n$ and any $a_0, a_1, \cdots, a_n \in S$, all integer roots of the polynomial $a_0+a_1x+\cdots+a_nx^n$ are also in $S$. Find all rootiful sets of integers that contain all numbers of the form $2^a - 2^b$ for positive integers $a$ and $b$.
Let $\mathbb N$ denote the set of positive integers. Find all functions $f: \mathbb{N} \to \mathbb{N}$ such that:
(i) The greatest common divisor of the sequence $f(1), f(2), \dots$ is $1$.
(ii) For all sufficiently large integers $n$, we have $f(n) \neq 1$ and \[ f(a)^n \mid f(a+b)^{a^{n-1}} - f(b)^{a^{n-1}} \] for all positive integers $a$ and $b$.
[i]Proposed by Yang Liu[/i]