Found problems: 85335
Let $ABC$ be a triangle with circumcircle $\omega$, incenter $I$, and $A$-excenter $I_A$. Let the incircle and the $A$-excircle hit $BC$ at $D$ and $E$, respectively, and let $M$ be the midpoint of arc $BC$ without $A$. Consider the circle tangent to $BC$ at $D$ and arc $BAC$ at $T$. If $TI$ intersects $\omega$ again at $S$, prove that $SI_A$ and $ME$ meet on $\omega$.
[i]Amol Aggarwal.[/i]
Given a non-degenerate triangle $ABC$, let $A_{1}, B_{1}, C_{1}$ be the point of tangency of the incircle with the sides $BC, AC, AB$. Let $Q$ and $L$ be the intersection of the segment $AA_{1}$ with the incircle and the segment $B_{1}C_{1}$ respectively. Let $M$ be the midpoint of $B_{1}C_{1}$. Let $T$ be the point of intersection of $BC$ and $B_{1}C_{1}$. Let $P$ be the foot of the perpendicular from the point $L$ on the line $AT$. Prove that the points $A_{1}, M, Q, P$ lie on a circle.
Prove that there are infinitely many prime numbers that divide at least one integer of the form $2^{n^3+1}-3^{n^2+1}+5^{n+1}$ where $n$ is a positive integer.
Let $ a_1 , a_2 , \cdots , a_k $ be numbers such that $ a_i \in \{ 0,1,2,3 \} ( i= 1, 2, \cdots ,k) $. Let $ z = ( x_k , x_{k-1} , \cdots , x_1 )_4 $ be a base 4 expansion of $ z \in \{ 0, 1, 2, \cdots , 4^k -1 \} $. Define $ A $ as follows:
\[ A = \{ z | p(z)=z, z=0, 1, \cdots ,4^k-1 \}\]
where
\[ p(z) = \sum_{i=1}^{k} a_i x_i 4^{i-1} . \]
Prove that the number of elements in $ X $ is a power of 2.
Evaluate the following definite integral.
\[\int_{0}^{\pi}\frac{\cos nx}{2-\cos x}dx\ (n=0,\ 1,\ 2,\ \cdots)\]
Let $ A(2,2)$ and $ B(7,7)$ be points in the plane. Define $ R$ as the region in the first quadrant consisting of those points $ C$ such that $ \triangle ABC$ is an acute triangle. What is the closest integer to the area of the region $ R$?
$ \textbf{(A)}\ 25 \qquad \textbf{(B)}\ 39 \qquad \textbf{(C)}\ 51 \qquad \textbf{(D)}\ 60 \qquad \textbf{(E)}\ 80$
Let $ABC$ be a right isosceles triangle with right angle at $A$. Let $E$ and $F$ be points on A$B$ and $AC$ respectively such that $\angle ECB = 30^o$ and $\angle FBC = 15^o$. Lines $CE$ and $BF$ intersect at $P$ and line $AP$ intersects side $BC$ at $D$. Calculate the measure of angle $\angle FDC$.
In a convex pentagon $ ABCDE$ side $ AB$ is perpendicular to $ CD$ and side $ BC$ is perpendicular to $ DE$. Prove that if $ AB \equal{} AE \equal{} ED \equal{} 1$, then
$ BC \plus{} CD < 1$.
A circle $k$ is given with a non-diameter chord $AC$. On the tangent at point $A$ select point $X \ne A$ and mark $D$ the intersection of the circle $k$ with the interior of the line $XC$ (if any). Let $B$ a point in circle $k$ such that quadrilateral $ABCD$ is a trapezoid . Determine the set of intersections of lines $BC$ and $AD$ belonging to all such trapezoids.
Let $A, B$ be integer $n\times n$-matrices, where $det (A) = 1$ and $det (B) \ne 0$. Show that there is a $k \in N$, for which $BA^kB^{-1}$ is a matrix with integer entries.
Determine all the positive integers with more than one digit, all distinct, such that the sum of its digits is equal to the product of its digits.
$(BEL 3)$ Construct the circle that is tangent to three given circles.
Find all triples $(x,y,z)$ of positive integers satisfying the system of equations
\[\begin{cases} x^2=2(y+z)\\ x^6=y^6+z^6+31(y^2+z^2)\end{cases}\]
Let $n \geq 2$ be a fixed integer. Find the least constant $C$ such the inequality
\[\sum_{i<j} x_{i}x_{j} \left(x^{2}_{i}+x^{2}_{j} \right) \leq C
\left(\sum_{i}x_{i} \right)^4\]
holds for any $x_{1}, \ldots ,x_{n} \geq 0$ (the sum on the left consists of $\binom{n}{2}$ summands). For this constant $C$, characterize the instances of equality.
If $a$ is any number, $\lfloor a \rfloor$ is $a$ rounded down to the nearest integer. For example, $\lfloor \pi \rfloor =$ $3$.
Show that the sequence
$\lfloor \frac{2^{1}}{17} \rfloor$, $\lfloor \frac{2^{2}}{17} \rfloor$, $\lfloor \frac{2^{3}}{17} \rfloor$, $\dots$
contains infinitely many odd numbers.
Determine all possible values of $x+\frac{1}{x}$ , where the real number $x$ satisfies the equation $$x^4+5x^3-4x^2+5x+1=0$$ and solve this equation.
Given the four equations:
$\textbf{(1)}\ 3y-2x=12 \qquad
\textbf{(2)}\ -2x-3y=10 \qquad
\textbf{(3)}\ 3y+2x=12 \qquad
\textbf{(4)}\ 2y+3x=10$
The pair representing the perpendicular lines is:
$\textbf{(A)}\ \text{(1) and (4)} \qquad
\textbf{(B)}\ \text{(1) and (3)} \qquad
\textbf{(C)}\ \text{(1) and (2)} \qquad
\textbf{(D)}\ \text{(2) and (4)} \qquad
\textbf{(E)}\ \text{(2) and (3)}$
Jae and Yoon are playing SunCraft. The probability that Jae wins the $n$-th game is $\frac{1}{n+2}.$ What is the probability that Yoon wins the first six games, assuming there are no ties?
A train is going up a hill with vertical velocity given as a function of $t$ by $\frac{1}{1 - t^4}$ , where $t$ is between $[0, 1)$. Determine its height as a function of $t$.
In a triangle $ABC$, let $I$ be its incenter; $Q$ the point at which the incircle touches the line $AC$; $E$ the midpoint of $AC$ and $K$ the orthocenter of triangle $BIC$. Prove that the line $KQ$ is perpendicular to the line $IE$.
Suppose $S= \{1,2,\dots,n\}$ and $n \geq 3$. There is $f:S^k \longmapsto S$ that if $a,b \in S^k$ and $a$ and $b$ differ in all of elements then $f(a) \neq f(b)$. Prove that $f$ is a function of one of its elements.
$SABCD$ is quadrangular pyramid. Lateral faces are acute triangles with orthocenters lying in one plane. $ABCD$ is base of pyramid and $AC$ and $BD$ intersects at $P$, where $SP$ is height of pyramid. Prove that $AC \perp BD$
On a $5 \times 5$ board, $n$ white markers are positioned, each marker in a distinct $1 \times 1$ square. A smart child got an assignment to recolor in black as many markers as possible, in the following manner: a white marker is taken from the board, it is colored in black, and then put back on the board on an empty square such that none of the neighboring squares contains a white marker (two squares are called neighboring if they share a common side).
If it is possible for the child to succeed in coloring all the markers black, we say that the initial positioning of the markers was [i]good[/i].
a) Prove that if $n = 20$, then a [i]good [/i] initial positioning exists.
b) Prove that if $n = 21$, then a [i]good [/i] initial positioning does not exist.
Let $n$ be a positive integer such that $n \geq 3$. Consider a grid with size $n \times n$ where each square can be white or black, in the beginning they are all white. In every step we can change the colors of cells forming a shape like below
[img] https://imgtr.ee/images/2023/04/04/k0i9m.png [/img]
or any of its rotations. Determine all $n$ such that the whole grid can be black after a finite number of steps.
Let $x$ and $y$ be positive numbers with $x +y=1$. Show that
$$\left(1+\frac{1}{x}\right)\left(1+\frac{1}{y}\right) \ge 9.$$