Found problems: 85335
Given a positive integer $k\geq2$, set $a_1=1$ and, for every integer $n\geq 2$, let $a_n$ be the smallest solution of equation
\[x=1+\sum_{i=1}^{n-1}\left\lfloor\sqrt[k]{\frac{x}{a_i}}\right\rfloor\]
that exceeds $a_{n-1}$. Prove that all primes are among the terms of the sequence $a_1,a_2,\ldots$
An empty $2020 \times 2020 \times 2020$ cube is given, and a $2020 \times 2020$ grid of square unit cells is drawn on each of its six faces. A [i]beam[/i] is a $1 \times 1 \times 2020$ rectangular prism. Several beams are placed inside the cube subject to the following conditions:
[list=]
[*]The two $1 \times 1$ faces of each beam coincide with unit cells lying on opposite faces of the cube. (Hence, there are $3 \cdot {2020}^2$ possible positions for a beam.)
[*]No two beams have intersecting interiors.
[*]The interiors of each of the four $1 \times 2020$ faces of each beam touch either a face of the cube or the interior of the face of another beam.
[/list]
What is the smallest positive number of beams that can be placed to satisfy these conditions?
[i]Proposed by Alex Zhai[/i]
Let $S$ be a set of $N \ge 3$ points in the plane. Assume that no $3$ points in $S$ are collinear. The segments with both endpoints in $S$ are colored in two colors.
Prove that there is a set of $N - 1$ segments of the same color which don't intersect except in their endpoints such that no subset of them forms a polygon with positive area.
Draw a regular hexagon. Then make a square from each edge of the hexagon. Then form equilateral triangles by drawing an edge between every pair of neighboring squares. If this figure is continued symmetrically off to infi nity, what is the ratio between the number of triangles and the number of squares?
[b]p1.[/b] On SeaBay, green herring costs $\$2.50$ per pound, blue herring costs $\$4.00$ per pound, and red herring costs $\$5,85$ per pound. What must Farmer James pay for $12$ pounds of green herring and $7$ pounds of blue herring, in dollars?
[b]p2.[/b] A triangle has side lengths $3$, $4$, and $6$. A second triangle, similar to the first one, has one side of length $12$. Find the sum of all possible lengths of the second triangle's longest side.
[b]p3.[/b] Hen Hao runs two laps around a track. Her overall average speed for the two laps was $20\%$ slower than her average speed for just the first lap. What is the ratio of Hen Hao's average speed in the first lap to her average speed in the second lap?
[b]p4.[/b] Square $ABCD$ has side length $2$. Circle $\omega$ is centered at $A$ with radius $2$, and intersects line $AD$ at distinct points $D$ and $E$. Let $X$ be the intersection of segments $EC$ and $AB$, and let $Y$ be the intersection of the minor arc $DB$ with segment $EC$. Compute the length of $XY$ .
[b]p5.[/b] Hen Hao rolls $4$ tetrahedral dice with faces labeled $1$, $2$, $3$, and $4$, and adds up the numbers on the faces facing down. Find the probability that she ends up with a sum that is a perfect square.
[b]p6.[/b] Let $N \ge 11$ be a positive integer. In the Eggs-Eater Lottery, Farmer James needs to choose an (unordered) group of six different integers from $1$ to $N$, inclusive. Later, during the live drawing, another group of six numbers from $1$ to $N$ will be randomly chosen as winning numbers. Farmer James notices that the probability he will choose exactly zero winning numbers is the same as the probability that he will choose exactly one winning number. What must be the value of $N$?
[b]p7.[/b] An egg plant is a hollow cylinder of negligible thickness with radius $2$ and height $h$. Inside the egg plant, there is enough space for four solid spherical eggs of radius $1$. What is the minimum possible value for $h$?
[b]p8.[/b] Let $a_1, a_2, a_3, ...$ be a geometric sequence of positive reals such that $a_1 < 1$ and $(a_{20})^{20} = (a_{18})^{18}$. What is the smallest positive integer n such that the product $a_1a_2a_3...a_n$ is greater than $1$?
[b]p9.[/b] In parallelogram $ABCD$, the angle bisector of $\angle DAB$ meets segment $BC$ at $E$, and $AE$ and $BD$ intersect at $P$. Given that $AB = 9$, $AE = 16$, and $EP = EC$, find $BC$.
[b]p10.[/b] Farmer James places the numbers $1, 2,..., 9$ in a $3\times 3$ grid such that each number appears exactly once in the grid. Let $x_i$ be the product of the numbers in row $i$, and $y_i$ be the product of the numbers in column $i$. Given that the unordered sets $\{x_1, x_2, x_3\}$ and $\{y_1, y_2, y_3\}$ are the same, how many possible arrangements could Farmer James have made?
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
Given the function $f(x) =|4 - 4|x||- 2$. How many solutions does the equation $f(f(x)) = x$ have?
Triangle $ABC$ lies entirely in the first quadrant of the Cartesian plane, and its sides have slopes $63$, $73$, $97$. Suppose the curve $\mathcal V$ with equation $y=(x+3)(x^2+3)$ passes through the vertices of $ABC$. Find the sum of the slopes of the three tangents to $\mathcal V$ at each of $A$, $B$, $C$.
[i]Proposed by Akshaj[/i]
Let $A,B\in M_n(C)$ be two square matrices satisfying $A^2+B^2 = 2AB$.
1.Prove that $\det(AB-BA)=0$.
2.If $rank(A-B)=1$, then prove that $AB=BA$.
Let $a,b,c \in \mathbb{R}^+$ and $abc = 1$ such that $a^{2011} + b^{2011} + c^{2011} < \dfrac{1}{a^{2011}} + \dfrac{1}{b^{2011}} + \dfrac{1}{c^{2011}}$. Prove that $a + b + c < \dfrac{1}{a} + \dfrac{1}{b} + \dfrac{1}{c}$.
For all real numbers $x,y$ define $x\star y = \frac{ x+y}{ 1+xy}$. Evaluate the expression \[ ( \cdots (((2 \star 3) \star 4) \star 5) \star \cdots ) \star 1995. \]
[i]Macedonia[/i]
Through vertex $C$ of $\Delta ABC$ are constructed lines $l_1$ and $l_2$ which are symmetrical about the angle bisector $CL_c$. Prove that the projections of $A$ and $B$ on lines $l_1$ and $l_2$ lie on one circle.
Evaluate $ \int_1^e \frac{1\minus{}x(e^x\minus{}1)}{x(1\plus{}xe^x\ln x)}\ dx$.
Let $ABCD$ be a convex quadrilateral and $A',B'C',D'$ be the midpoints of $AB,BC,CD,DA$, respectively. Let $a,b,c,d$ denote the areas of quadrilaterals into which lines $A'C'$ and $B'D'$ divide the quadrilateral $ABCD$ (where a corresponds to vertex $A$ etc.). Prove that $a+c = b+d$.
Let $a$ and $b$ be given positive coprime integers. Then for every integer $n$ there exist integers $x,y$ such that
$$n=ax+by.$$Prove that $n=ab$ is the greatest integer for which $xy\le0$ in all such representations of $n$.
The operation $\otimes$ is defined for all nonzero numbers by $a\otimes b = \dfrac{a^2}{b}$. Determine $[(1\otimes 2)\otimes 3] - [1\otimes (2\otimes 3)]$.
$\text{(A)}\ -\dfrac{2}{3} \qquad \text{(B)}\ -\dfrac{1}{4} \qquad \text{(C)}\ 0 \qquad \text{(D)}\ \dfrac{1}{4} \qquad \text{(E)}\ \dfrac{2}{3}$
There are four spheres each of radius $1$ whose centers form a triangular pyramid where each side has length $2$. There is a 5th sphere which touches all four other spheres and has radius less than $1$. What is its radius?
Among the positive integers that can be expressed as the sum of 2005 consecutive integers, which occupies the 2005th position when arranged in order?
[i]Roland Hablutzel, Venezuela[/i]
[hide="Remark"]The original question was: Among the positive integers that can be expressed as the sum of 2004 consecutive integers, and also as the sum of 2005 consecutive integers, which occupies the 2005th position when arranged in order?[/hide]
Positive integers $a, b, c$ satisfying the equality $a^2 + b^2 = c^2$.
Show that the number $\frac12(c - a) (c - b)$ is square of an integer.
Pedro and Tiago are playing a game with a deck of n cards, numbered from $1$ to $n$. Starting with Pedro, they choose cards alternately, and receive the number of points indicated by the cards. However, whenever the player chooses the card with the highest number among those remaining in the deck, he is forced to pass his next turn, not choosing any card. When the deck runs out, the player with the most points wins. Knowing that Tiago can at least draw, regardless of Pedro's moves, how many cards are in the deck? Indicates all possibilities,
Let $ f(x)\equal{}a_{2n}x^{2n}\plus{}a_{2n\minus{}1}x^{2n\minus{}1}\plus{}\cdots\plus{}a_1x\plus{}a_0$, with $ a_i\equal{}a_{2n\minus{}1}$ for all $ i\equal{}1,2,\ldots,n$ and $ a_{2n}\ne0$. Prove that there exists a polynomial $ g(x)$ of degree $ n$ such that $ g\left(x\plus{}\frac1x\right)x^n\equal{}f(x)$.
A uniform round tabletop of diameter $\text{4.0 m}$ and mass $\text{50.0 kg}$ rests on massless, evenly spaced legs of length $\text{1.0 m}$ and spacing $\text{3.0 m}$. A carpenter sits on the edge of the table. What is the maximum mass of the carpenter such that the table remains upright? Assume that the force exerted by the carpenter on the table is vertical and at the edge of the table.
[asy]
size(10cm);
import graph;
xaxis(-3.6,3.6);
//The legs
draw((-1.4,1.4)--(-1.4,0));
draw((-1.6,1.4)--(-1.6,0));
draw((1.4,1.4)--(1.4,0));
draw((1.6,1.4)--(1.6,0));
//The tabletop
draw((-2.5,1.6)--(-2.5,1.4)--(2.5,1.4)--(2.5,1.6)--cycle);
path a = ((-3.2,0.1)--(-3.2,1.5));
draw(a,Arrows);
label("1.0m",a,E);
draw((-2.6,1.6)--(-3.6,1.6),dashed+linewidth(0.4));
path c = (-1.4,-0.5)--(1.4,-0.5);
draw(c,Arrows);
label("3.0m",c,2*S);
draw((-1.5,-0.1)--(-1.5,-0.8),dashed+linewidth(0.4));
draw((1.5,-0.1)--(1.5,-0.8),dashed+linewidth(0.4));
draw((-2.5,-0.1)--(-2.5,-1.8),dashed+linewidth(0.4));
draw((2.5,-0.1)--(2.5,-1.8),dashed+linewidth(0.4));
path d = (-2.4,-1.5)--(2.4,-1.5);
draw(d,Arrows);
label("4.0m",d,2*S);
//The Man
pair A = (2.4,1.7);
draw(circle((2.5,3.7),0.5),linewidth(1.6));
draw((2.4,1.7)--(2.5,3.2),linewidth(1.6));
draw((2.15,2.8)--(2.85,2.8),linewidth(1.6));
draw(A--A+(0.6,0)--A+(0.4,-0.8)--A+(0.6,-1.2),linewidth(1.6));
draw(A--(A+(0.64,-0.12)--A+(0.73,-0.9)--A+(1,-0.9)),linewidth(1.6));
draw((2.15,2.8)--(2.15,2.4)--(2.9,2),linewidth(1.6));
draw((2.85,2.8)--(3,2.3)--(3.5,2.1),linewidth(1.6));
[/asy]
(a) $\text{67 kg}$
(b) $\text{75 kg}$
(c) $\text{81 kg}$
(d) $\text{150 kg}$
(e) $\text{350 kg}$
Suppose we have a necklace of $n$ beads. Each bead is labelled with an integer and the sum of all these labels is $n-1$. Prove that we can cut the necklace to form a string whose consecutive labels $x_1, x_2,\cdots , x_n$ satisfy
\[ \sum_{i=1}^{k}x_i\le k-1\quad \forall \;\;1\le k\le n \]
1. Let $f(n) = n^2+6n+11$ be a function defined on positive integers. Find the sum of the first three prime values $f(n)$ takes on.
[i]Proposed by winnertakeover[/i]
The numbers from $1$ to $2015$ are written on sheets so that if if $n-m$ is a prime, then $n$ and $m$ are on different sheets. What is the minimum number of sheets required?
Say that a positive integer $n$ is [i]smooth[/i] if $\frac{1}{n}$ has a terminating decimal expansion. (Note that 1 is smooth.) Compute the value of the infinite series
\[
\sum_n \frac{1}{n^3} \, ,
\]
where $n$ ranges over all smooth positive integers.