Found problems: 85335
Does there exist a sequence $\{a_n\}$ of positive integers satisfying the following conditions:
$a)$ every natural number occurs in this sequence and exactly once;
$b)$ $a_1 + a_2 +... + a_n$ is divisible by $n^n$ for each $n = 1,2,3, ...$
?
[hide=C stands for Cantor, G stands for Gauss]they had two problem sets under those two names[/hide]
[u] Set 4[/u]
[b]C.16 / G.6[/b] Let $a, b$, and $c$ be real numbers. If $a^3 + b^3 + c^3 = 64$ and $a + b = 0$, what is the value of $c$?
[b]C.17 / G.8[/b] Bender always turns $60$ degrees clockwise. He walks $3$ meters, turns, walks $2$ meters, turns, walks $1$ meter, turns, walks $4$ meters, turns, walks $1$ meter, and turns. How many meters does Bender have to walk to get back to his original position?
[b]C.18 / G.13[/b] Guang has $4$ identical packs of gummies, and each pack has a red, a blue, and a green gummy. He eats all the gummies so that he finishes one pack before going on to the next pack, but he never eats two gummies of the same color in a row. How many different ways can Guang eat the gummies?
[b]C.19[/b] Find the sum of all digits $q$ such that there exists a perfect square that ends in $q$.
[b]C.20 / G.14[/b] The numbers $5$ and $7$ are written on a whiteboard. Every minute Stev replaces the two numbers on the board with their sum and difference. After $2017$ minutes the product of the numbers on the board is $m$. Find the number of factors of $m$.
[u]Set 5[/u]
[b]C.21 / G.10[/b] On the planet Alletas, $\frac{32}{33}$ of the people with silver hair have purple eyes and $\frac{8}{11}$ of the people with purple eyes have silver hair. On Alletas, what is the ratio of the number of people with purple eyes to the number of people with silver hair?
[b]C.22 / G.15[/b] Let $P$ be a point on $y = -1$. Let the clockwise rotation of $P$ by $60^o$ about $(0, 0)$ be $P'$. Find the minimum possible distance between $P'$ and $(0, -1)$.
[b]C.23 / G.18[/b] How many triangles can be made from the vertices and center of a regular hexagon? Two congruent triangles with different orientations are considered distinct.
[b]C.24[/b] Jeremy and Kevin are arguing about how cool a sweater is on a scale of $1-5$. Jeremy says “$3$”, and Kevin says “$4$”. Jeremy angrily responds “$3.5$”, to which Kevin replies “$3.75$”. The two keep going at it, responding with the average of the previous two ratings. What rating will they converge to (and settle on as the coolness of the sweater)?
[b]C.25 / G.20[/b] An even positive integer $n$ has an [i]odd factorization[/i] if the largest odd divisor of $n$ is also the smallest odd divisor of $n$ greater than $1$. Compute the number of even integers $n$ less than $50$ with an odd factorization.
[u]Set 6[/u]
[b]C.26 / G.26[/b] When $2018! = 2018 \times 2017 \times ... \times 1$ is multiplied out and written as an integer, find the number of $4$’s.
If the correct answer is $A$ and your answer is $E$, you will receive $12 \min\, \, (A/E, E/A)^3$points.
[b]C.27 / G.27[/b] A circle of radius $10$ is cut into three pieces of equal area with two parallel cuts. Find the width of the center piece.
[img]https://cdn.artofproblemsolving.com/attachments/e/2/e0ab4a2d51052ee364dd14336677b053a40352.png[/img]
If the correct answer is $A$ and your answer is $E$, you will receive $\max \, \,(0, 12 - 6|A - E|)$points.
[b]C.28 / G.28[/b] An equilateral triangle of side length $1$ is randomly thrown onto an infinite set of lines, spaced $1$ apart. On average, how many times will the boundary of the triangle intersect one of the lines?
[img]https://cdn.artofproblemsolving.com/attachments/0/1/773c3d3e0dfc1df54945824e822feaa9c07eb7.png[/img]
For example, in the above diagram, the boundary of the triangle intersects the lines in $2$ places.
If the correct answer is $A$ and your answer is $E$, you will receive $\max\, \,(0, 12-120|A-E|/A)$ points.
[b]C.29 / G.29[/b] Call an ordered triple of integers $(a, b, c)$ nice if there exists an integer $x$ such that $ax^2 + bx + c = 0$. How many nice triples are there such that $-100 \le a, b, c \le 100$?
If the correct answer is $A$ and your answer is $E$, you will receive $12 \min\, \,(A/E, E/A)$ points.
[b]C.30 / G.30[/b] Let $f(i)$ denote the number of MBMT volunteers to be born in the $i$th state to join the United States. Find the value of $1f(1) + 2f(2) + 3f(3) + ... + 50f(50)$.
Note 1: Maryland was the $7$th state to join the US.
Note 2: Last year’s MBMT competition had $42$ volunteers.
If the correct answer is $A$ and your answer is $E$, you will receive $\max\, \,(0, 12 - 500(|A -E|/A)^2)$ points.
PS. You should use hide for answers. C1-15/ G1-10 have been posted [url=https://artofproblemsolving.com/community/c3h2790674p24540132]here [/url] and G16-25 [url=https://artofproblemsolving.com/community/c3h2790679p24540159]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
The diagram below shows a large square with each of its sides divided into four equal segments. The shaded square whose sides are diagonals drawn to these division points has area $13$. Find the area of the large square.
[img]https://cdn.artofproblemsolving.com/attachments/8/3/bee223ef39dea493d967e7ebd5575816954031.png[/img]
Let there be a figure with $9$ disks and $11$ edges, as shown below.
We will write a real number in each and every disk. Then, for each edge, we will write the square of the difference between the two real numbers written in the two disks that the edge connects. We must write $0$ in disk $A$, and $1$ in disk $I$. Find the minimum sum of all real numbers written in $11$ edges.
A circle $C$ touches three pairwise disjoint circles whose centers are collinear and none of which contains any of the others. Show that its radius must be larger than the radius of the middle of the three circles.
A square region of side $12$ contains a water source that supplies an irrigation system constituted by several straight channels forming polygonal lines. Considers the source as a point and each channel as a line segment.
Knowing that a point is irrigated if it is not more than $1$ distance from any channel and that the system was designed so that the entire region is irrigated, proves that the total length of irrigation channels exceeds $70$.
Ten boxes are arranged in a circle. Each box initially contains a positive number of golf balls. A move consists of taking all of the golf balls from one of the boxes and placing them into the boxes that follow it in a counterclockwise direction, putting one ball into each box. Prove that if the next move always starts with the box where the last ball of the previous move was placed, then after some number of moves, we get back to the initial distribution of golf balls in the boxes.
Given
$$a = bc$$
$$b = ca$$
$$c = a + b$$
$$c > a$$
Evaluate $a+b+c$.
[i]2022 CCA Math Bonanza Lightning Round 1.1[/i]
It is given that there exists a unique triple of positive primes $(p,q,r)$ such that $p<q<r$ and \[\dfrac{p^3+q^3+r^3}{p+q+r} = 249.\] Find $r$.
Let $ABC$ be a triangle with circumcircle $w$. Let $D$ be the midpoint of arc $BC$ that contains $A$. Define $E$ and $F$ similarly. Let the incircle of $ABC$ touches $BC,CA,AB$ at $K,L,M$ respectively. Prove that $DK,EL,FM$ are concurrent.
Consider an odd prime $p$. A comutative ring $(A,+, \cdot)$ has the property that $ab=0$ implies $a^p=0$ or $b^p=0$. Moreover, $\underbrace{1+1+\cdots +1}_{p \textnormal{ times}} =0$. Take $x,y\in A$ such that there exist $m,n\geq 1$, $m\neq n$ with $x+y=x^my=x^ny$, and also $y$ is not invertible. \\ \\
$\textbf{(a)}$ Prove that $(a+b)^p=a^p+b^p$ and $(a+b)^{p^2}=a^{p^2}+b^{p^2}$ for all $a,b\in A$.\\
$\textbf{(b)}$ Prove that $x$ and $y$ are nilpotent.\\
$\textbf{(c)}$ If $y$ is invertible, does the conclusion that $x$ is nilpotent stand true?
\\ \\
[i] (Bogdan Blaga)[/i]
Sequence ${u_n}$ is defined with $u_0=0,u_1=\frac{1}{3}$ and
$$\frac{2}{3}u_n=\frac{1}{2}(u_{n+1}+u_{n-1})$$ $\forall n=1,2,...$
Show that $|u_n|\leq1$ $\forall n\in\mathbb{N}.$
Let $a$, $b$ and $c$ be positive real numbers such that $a+b+c=3$. Prove the inequality
$$\frac{a^3}{a^2+1}+\frac{b^3}{b^2+1}+\frac{c^3}{c^2+1} \geq \frac{3}{2}.$$
[i]Proposed by Anastasija Trajanova[/i]
Show that a sequence $(a_n)$ of $+1$ and $-1$ is periodic with period a power of $2$ if and only if $a_n=(-1)^{P(n)}$, where $P$ is an integer-valued polynomial with rational coefficients.
On side $AB$ of triangle $ABC$, points $M$ and $K$ are taken ($M$ on segment $AK$). It is known that $AM: MK: MB = a: b: c$. Straight lines $CM$ and $CK$ intersect for the second time the circumscribed circle of the triangle $ABC$ at points $E$ and $F$, respectively. In what ratio does the circumscribed circle of the triangle $BMF$ divide the segment $BE$?
Prove that for any positive numbers $x,y$ it holds that $x^xy^y \ge x^yy^x$.
If one can find a student with at least $k$ friends in any class which has $21$ students such that at least two of any three of these students are friends, what is the largest possible value of $k$?
$
\textbf{(A)}\ 8
\qquad\textbf{(B)}\ 9
\qquad\textbf{(C)}\ 10
\qquad\textbf{(D)}\ 11
\qquad\textbf{(E)}\ 12
$
Consider the sequence $u_0, u_1, u_2, ...$ defined by $u_0 = 0, u_1 = 1,$ and $u_n = 6u_{n - 1} + 7u_{n - 2}$ for $n \ge 2$. Show that there are no non-negative integers $a, b, c, n$ such that
$$ab(a + b)(a^2 + ab + b^2) = c^{2022} + 42 = u_n.$$
The road ministry has assigned $80$ informal companies to repair $2400$ roads. These roads connect $100$ cities to each other. Each road is between $2$ cities and there is at most $1$ road between every $2$ cities. We know that each company repairs $30$ roads that it has agencies in each $2$ ends of them. Prove that there exists a city in which $8$ companies have agencies.
In triangle $ ABC$, internal bisector of angle $ A$ intersects with $ BC$ at $ D$. Let $ E$ be a point on $ \left[CB\right.$ such that $ \left|DE\right|\equal{}\left|DB\right|\plus{}\left|BE\right|$. The circle through $ A$, $ D$, $ E$ intersects $ AB$ at $ F$, again. If $ \left|BE\right|\equal{}\left|AC\right|\equal{}7$, $ \left|AD\right|\equal{}2\sqrt{7}$ and $ \left|AB\right|\equal{}5$, then $ \left|BF\right|$ is
$\textbf{(A)}\ \frac {7\sqrt {5} }{5} \qquad\textbf{(B)}\ \sqrt {7} \qquad\textbf{(C)}\ 2\sqrt {2} \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ \sqrt {10}$
Let $n$ be positive integer.A square $A$ of side length $n$ is divided by $n^2$ unit squares. All unit squares are painted in $n$ distinct colors such that each color appears exactly $n$ times. Prove that there exists a positive integer $N$ , such that for any $n>N$ the following is true: There exists a square $B$ of side length $\sqrt{n}$ and side parallel to the sides of $A$ such that $B$ contains completely cells of $4$ distinct colors.
Let $1,\alpha_1,\alpha_2,...,\alpha_{10}$ be the roots of the polynomial $x^{11}-1$. It is a fact that there exists a unique polynomial of the form $f(x) = x^{10}+c_9x^9+ \dots + c_1x$ such that each $c_i$ is an integer, $f(0) = f(1) = 0$, and for any $1 \leq i \leq 10$ we have $(f(\alpha_i))^2 = -11$. Find $\left|c_1+2c_2c_9+3c_3c_8+4c_4c_7+5c_5c_6\right|$.
A graph $G$ with $n$ vertex is called [i]good [/i] if every vertex could be labelled with distinct positive integers which are less than or equal $\lfloor \frac{n^2}{4} \rfloor$ such that there exists a set of nonnegative integers $D$ with the following property: there exists an edge between $2$ vertices if and only if the difference of their labels is in $D$.
Show that there exists a positive integer $N$ such that for every $n \ge N$, there exist a not-good graph with $n$ vertices.
Calculate the area of the intersection of the interior of the ellipse $\frac{x^2}{16}+ \frac{y^2}{4}= 1$ with the circle bounded by the circumference $(x -2)^2 + (y - 1)^2 = 5$.
It is known that among several banknotes of pairwise distinct face values (which are positive integers) there are exactly $N{}$ fakes. In a single test, a detector determines the sum of the face values of all real banknotes in an arbitrary set we have selected. Prove that by using the detector $N{}$ times, all fake banknotes can be identified, if a) $N=2$ and b) $N=3$.
[i]Proposed by S. Tokarev[/i]