Found problems: 85335
Let $ABCD$ be a Rhombus and let $w$ be it's incircle. Let $M$ be the midpoint of $AB$ the point $K$ is on $w$ and inside $ABCD$ such that $MK$ is tangent to $w$. Prove that $CDKM$ is cyclic.
What is the measure of the acute angle formed by the hands of the clock at $4:20$ PM?
$\textbf{(A)}\ 0 \qquad
\textbf{(B)}\ 5 \qquad
\textbf{(C)}\ 8 \qquad
\textbf{(D)}\ 10 \qquad
\textbf{(E)}\ 12$
Let $n>1$ be an integer and $d_1<d_2<\dots<d_m$ the list of its positive divisors, including $1$ and $n$. The $m$ instruments of a mathematical orchestra will play a musical piece for $m$ seconds, where the instrument $i$ will play a note of tone $d_i$ during $s_i$ seconds (not necessarily consecutive), where $d_i$ and $s_i$ are positive integers. This piece has "sonority" $S=s_1+s_2+\dots s_n$.
A pair of tones $a$ and $b$ are harmonic if $\frac ab$ or $\frac ba$ is an integer. If every instrument plays for at least one second and every pair of notes that sound at the same time are harmonic, show that the maximum sonority achievable is a composite number.
Let be a finite set $ A $ of real numbers, and define the sets $ S_{\pm }=\{ x\pm y| x,y\in A \} . $
Show that $ \left| A \right|\cdot\left| S_{-} \right| \le \left| S_{+} \right|^2 . $
Two arbitrary distinct lattice points are selected on the coordinate plane within the square marked by the points $(0,0)$, $(3,0)$, $(0,3)$, and $(3,3)$ (the lattice points may lie on a side or a corner of the square). What is the probability that the distance between the two points is at most $\sqrt{2}$?
Let $G$ be a group with $m$ elements and let $H$ be a proper subgroup of $G$ with $n$ elements. For each $x\in G$ we denote $H^x = \{ xhx^{-1} \mid h \in H \}$ and we suppose that $H^x \cap H = \{e\}$, for all $x\in G - H$ (where by $e$ we denoted the neutral element of the group $G$).
a) Prove that $H^x=H^y$ if and only if $x^{-1}y \in H$;
b) Find the number of elements of the set $\bigcup_{x\in G} H^x$ as a function of $m$ and $n$.
[i]Calin Popescu[/i]
The vertices of two acute-angled triangles lie on the same circle. The Euler circle (nine-point circle) of one of the triangles passes through the midpoints of two sides of the other triangle. Prove that the triangles have the same Euler circle.
EDIT by pohoatza (in concordance with Luis' PS): [hide=Alternate/initial version ]Let $ABC$ be a triangle with circumcenter $\Gamma$ and nine-point center $\gamma$. Let $X$ be a point on $\Gamma$ and let $Y$, $Z$ be on $\Gamma$ so that the midpoints of segments $XY$ and $XZ$ are on $\gamma$. Prove that the midpoint of $YZ$ is on $\gamma$.[/hide]
For the real numbers $x$ and $y$, $[\sqrt{x}] = 10$ and $[\sqrt{y}] =14$.
How large is $\left[\sqrt{[ \sqrt{x+y} ]}\right]$ ?
(Note: the square roots are the positive values and $[x]$ is the largest integer less than or equal to x.)
Let $p$ be an odd prime. Prove that
\[1^{p-2}+2^{p-2}+\cdots+\left(\frac{p-1}{2}\right)^{p-2}\equiv\frac{2-2^p}{p}\pmod p.\]
Points $A$, $B$, $C$, $D$ and $E$ are placed on the circle. In how many ways can the resulting five arcs be designated by the letters $a$, $b$, $c$, $d$ and $e$, if it is forbidden to designate an arc with the same letter as one of its ends? (For example, an arc with ends $A$ and $B$ cannot be designated by the letter $a$ or $b$.)
Find all pairs of integers $(a,b)$ such that $a^2+ab+1\mid b^2+ab+a+b-1$.
What is the area of the polygon whose vertices are the points of intersection of the curves $x^2+y^2=25$ and $(x-4)^2+9y^2=81$?
${{ \textbf{(A)}\ 24\qquad\textbf{(B)}\ 27\qquad\textbf{(C)}\ 36\qquad\textbf{(D)}\ 37.5}\qquad\textbf{(E)}\ 42} $
Given seven points in the plane, some of them are connected by segments such that:
[b](i)[/b] among any three of the given points, two are connected by a segment;
[b](ii)[/b] the number of segments is minimal.
How many segments does a figure satisfying [b](i)[/b] and [b](ii)[/b] have? Give an example of such a figure.
Let $ABC$ be a triangle with $AB = 9$ and $AC=12$. Point $B'$ is chosen on line $AC$ such that the midpoint of $B$ and $B'$ is equidistant from $A$ and $C$. Point $C'$ is chosen similarly. Given that the circumcircle of $AB'C'$ is tangent to $BC$, compute $BC^2$.
[i]2021 CCA Math Bonanza Team Round #8[/i]
Find all natural numbers $n$ greater than $2$ such that there exist $n$ natural numbers $a_{1},a_{2},\ldots,a_{n}$ such that they are not all equal, and the sequence $a_{1}a_{2},a_{2}a_{3},\ldots,a_{n}a_{1}$ forms an arithmetic progression with nonzero common difference.
Let $Q$ be a quadratic polynomial with a unique zero. Suppose $Q(12)=Q(16)$ and $Q(20)=24$. Compute $Q(28)$.
Let $n, k \geq 3$ be integers, and let $S$ be a circle. Let $n$ blue points and $k$ red points be
chosen uniformly and independently at random on the circle $S$. Denote by $F$ the intersection of the
convex hull of the red points and the convex hull of the blue points. Let $m$ be the number of vertices
of the convex polygon $F$ (in particular, $m=0$ when $F$ is empty). Find the expected value of $m$.
Let $D$ be the midpoint of side $BC$ of triangle $ABC$ and $E$ the midpoint of median $AD$. Line $BE$ intersects side $CA$ at $F$. Prove that the area of quadrilateral $CDEF$ is $\frac{5}{12}$ the area of triangle $ABC$.
In a school there are $1200$ students. Each student is part of exactly $k$ clubs. For any $23$ students, they are part of a common club. Finally, there is no club to which all students belong. Find the smallest possible value of $k$.
We are given twelve coins, one of which is a fake with a different mass from the other eleven. Determine that coin with three weighings and whether it is heavier or lighter than the others.
The teacher gave each of her $37$ students $36$ pencils in different colors. It turned out that each pair of students received exactly one pencil of the same color. Determine the smallest possible number of different colors of pencils distributed.
Is there a power of $2$ such that it is possible to rearrange the digits giving another power of $2$?
Find all posible pairs of positive integers $x,y$ such that $$\text{lcm}(x,y+3001)=\text{lcm}(y,x+3001)$$
Given a rectangle $\mathcal{R}$ of area $100000 $, Pancho must completely cover the rectangle $\mathcal{R}$ with a finite number of rectangles with sides parallel to the sides of $\mathcal{R}$ . Next, Martín colors some rectangles of Pancho's cover red so that no two red rectangles have interior points in common. If the red area is greater than $0.00001$, Martin wins. Otherwise, Pancho wins. Prove that Pancho can cover to ensure victory,
Let $x_{0}$, $x_{1}$, $x_{2}$, $\cdots$ be a sequence of numbers, where each $x_{k}$ is either $0$ or $1$. For each positive integer $n$, define
\[S_{n} = \displaystyle\sum^{n-1}_{k=0}{x_{k}2^{k}}\]
Suppose $7S_{n} \equiv 1\pmod {2^{n}}$ for all $n\geq 1$. What is the value of the sum
\[x_{2019}+2x_{2020}+4x_{2021}+8x_{2022}?\]
$ \textbf{(A)}\ 6 \qquad
\textbf{(B)}\ 7 \qquad
\textbf{(C)}\ 12 \qquad
\textbf{(D)}\ 14 \qquad
\textbf{(E)}\ 15$