Found problems: 85335
The chord $[CD]$ of the circle with diameter $[AB]$ is perpendicular to $[AB]$. Let $M$ and $N$ be the midpoints of $[BC]$ and $[AD]$, respectively. If $|BC|=6$ and $|AD|=2\sqrt{3}$, then $|MN|=?$
$ \textbf{(A)}\ 4 \qquad \textbf{(B)}\ 3 \sqrt 2 \qquad \textbf{(C)}\ \sqrt{21} \qquad \textbf{(D)}\ 5 \qquad \textbf{(E)}\ \text{None}$
Let $a_0, a_1, a_2, \ldots$ be an arbitrary infinite sequence of positive numbers. Show that the inequality $1 + a_n > a_{n-1} \sqrt[n]{2}$ holds for infinitely many positive integers $n$.
In a video game, there is a board divided into squares, with $27$ rows and $27$ columns.
The squares are painted alternately in black, gray and white as follows:
$\bullet$ in the first row, the first square is black, the next is gray, the next is white, the next is black, and so on;
$\bullet$ in the second row, the first is white, the next is black, the next is gray, the next is white, and so on;
$\bullet$ in the third row, the order would be gray-white-black-gray and so on;
$\bullet$ the fourth row is painted the same as the first, the fifth the same as the second,
$\bullet$ the sixth the same as the third, and so on.
In the box in row $i$ and column $j$, there are $ij$ coins.
For example, in the box in row $15$ and column $20$ there are $(15) (20) = 300$ coins.
Verify that in total there are, in the black squares, $9^2 (13^2 + 14^2 + 15^2)$ coins.
Let $a, b, c$ be the side lengths of an non-degenerate triangle with $a \le b \le c$. With $t (a, b, c)$ denote the minimum of the quotients $\frac{b}{a}$ and $\frac{c}{b}$ . Find all values that $t (a, b, c)$ can take.
In baseball, a player's [i]batting average[/i] is the number of hits divided by the number of at bats, rounded to three decimal places. Danielle's batting average is $.399$. What is the fewest number of at bats that Danielle could have?
Let $a,b,c,d,e,f$ be positive real numbers. Given that $def+de+ef+fd=4$, show that \[ ((a+b)de+(b+c)ef+(c+a)fd)^2 \geq\ 12(abde+bcef+cafd). \][i]Proposed by Allen Liu[/i]
Prove that there are no $x\in\mathbb{R}^+$ such that $$x^{\lfloor x \rfloor }=\frac92.$$
Let $ ABCD$ be a convex cuadrilateral inscribed in a circumference centered at $ O$ such that $ AC$ is a diameter. Pararellograms $ DAOE$ and $ BCOF$ are constructed. Show that if $ E$ and $ F$ lie on the circumference then $ ABCD$ is a rectangle.
Prove that the equation $a^2 +b^2 =c^2 +3$ has infinitely many integer solutions $(a, b, c)$.
Find the number of ordered pairs of positive integers $(m,n)$, where $m,n\leq 10$, such that $m!+n!$ is a multiple of $10$.
Does there exist a piecewise linear continuous function $f:\mathbb{R}\to \mathbb{R}$ such that for any two-way infinite sequence $a_n\in[0,1]$, $n\in\mathbb{Z}$, there exists an $x\in\mathbb{R}$ with
\[
\limsup_{K\to \infty} \frac{\#\{k\le K\,:\, k\in\mathbb{N},f^k(x)\in[n,n+1)\}}{K}=a_n
\]
for all $n\in\mathbb{Z}$, where $f^k=f\circ f\circ \dots\circ f$ stands for the $k$-fold iterate of $f$?
Let $x$ satisfy $x^4+x^3+x^2+x+1=0$. Compute the value of $(5x+x^2)(5x^2+x^4)(5x^3+x^6)(5x^4+x^8)$.
[i]Proposed by Andrew Zhao[/i]
A planar country has an odd number of cities separated by pairwise distinct distances. Some of these cities are connected by direct two-way flights. Each city is directly connected to exactly two ther cities, and the latter are located farthest from it. Prove that, using these flights, one may go from any city to any other city
Find all continuous function $f:\mathbb{R}^{\geq 0}\rightarrow \mathbb{R}^{\geq 0}$ such that :
\[f(xf(y))+f(f(y)) = f(x)f(y)+2 \: \: \forall x,y\in \mathbb{R}^{\geq 0}\]
[i]Proposed by Mohammad Ahmadi[/i]
The leader of an IMO team chooses positive integers $n$ and $k$ with $n > k$, and announces them to the deputy leader and a contestant. The leader then secretly tells the deputy leader an $n$-digit binary string, and the deputy leader writes down all $n$-digit binary strings which differ from the leader’s in exactly $k$ positions. (For example, if $n = 3$ and $k = 1$, and if the leader chooses $101$, the deputy leader would write down $001, 111$ and $100$.) The contestant is allowed to look at the strings written by the deputy leader and guess the leader’s string. What is the minimum number of guesses (in terms of $n$ and $k$) needed to guarantee the correct answer?
Consider the sytem of equations
\[ a_{11}x_1+a_{12}x_2+a_{13}x_3 = 0 \]\[a_{21}x_1+a_{22}x_2+a_{23}x_3 =0\]\[a_{31}x_1+a_{32}x_2+a_{33}x_3 = 0 \] with unknowns $x_1, x_2, x_3$. The coefficients satisfy the conditions:
a) $a_{11}, a_{22}, a_{33}$ are positive numbers;
b) the remaining coefficients are negative numbers;
c) in each equation, the sum ofthe coefficients is positive.
Prove that the given system has only the solution $x_1=x_2=x_3=0$.
Find the smallest positive integer $N$ of two or more digits that has the following property: If we insert any non-null digit $d$ between any two adjacent digits of $N$ we obtain a number that is a multiple of $d$.
Find all integers $n$ and $m$, $n > m > 2$, and such that a regular $n$-sided polygon can be inscribed in a regular $m$-sided polygon so that all the vertices of the $n$-gon lie on the sides of the $m$-gon.
Ivan bought $50$ cats consisting of five different breeds. He records the number of cats of each breed and after multiplying these five numbers he obtains the number $100000$. How many cats of each breed does he have?
Show an infinite sequence $a_1, a_2, \ldots$ of integers with both of the following properties:
• $a_i \neq 0$ for every positive integer $i$, that is, no term in the sequence is equal to zero;
• for all positive integer $n$, $a_n + a_{2n} + \ldots + a_{2023n} = 0$.
Over all real numbers $x$ and $y$ such that $$x^3=3x+y \qquad \text{and} \qquad y^3=3y+x,$$ compute the sum of all possible values of $x^2+y^2.$
The figure below shows a right triangle $ \triangle ABC$.
[asy]unitsize(15);
pair A = (0, 4);
pair B = (0, 0);
pair C = (4, 0);
draw(A -- B -- C -- cycle);
pair D = (2, 0);
real p = 7 - 3sqrt(3);
real q = 4sqrt(3) - 6;
pair E = p + (4 - p)*I;
pair F = q*I;
draw(D -- E -- F -- cycle);
label("$A$", A, N);
label("$B$", B, S);
label("$C$", C, S);
label("$D$", D, S);
label("$E$", E, NE);
label("$F$", F, W);[/asy]
The legs $ \overline{AB}$ and $ \overline{BC}$ each have length $ 4$. An equilateral triangle $ \triangle DEF$ is inscribed in $ \triangle ABC$ as shown. Point $ D$ is the midpoint of $ \overline{BC}$. What is the area of $ \triangle DEF$?
Find all one-to-one mappings $f:\mathbb{N}\to\mathbb{N}$ such that for all positive integers $n$ the following relation holds:
\[ f(f(n)) \leq \frac {n+f(n)} 2 . \]
Denote by $\Omega$ the circumcircle of the acute triangle $ABC$. Point $D$ is the midpoint of the arc $BC$ of $\Omega$ not containing $A$. Circle $\omega$ centered at $D$ is tangent to the segment $BC$ at point $E$. Tangents to the circle $\omega$ passing through point $A$ intersect line $BC$ at points $K$ and $L$ such that points $B, K, L, C$ lie on the line $BC$ in that order. Circle $\gamma_1$ is tangent to the segments $AL$ and $BL$ and to the circle $\Omega$ at point $M$. Circle $\gamma_2$ is tangent to the segments $AK$ and $CK$ and to the circle $\Omega$ at point $N$. Lines $KN$ and $LM$ intersect at point $P$. Prove that $\sphericalangle KAP = \sphericalangle EAL$.
Let $a,b,c$ be positive real numbers . Prove that$$ \frac{1}{ab(b+1)(c+1)}+\frac{1}{bc(c+1)(a+1)}+\frac{1}{ca(a+1)(b+1)}\geq\frac{3}{(1+abc)^2}.$$