Found problems: 85335
Bridget bakes $48$ loaves of bread for her bakery. She sells half of them in the morning for $\$2.50$ each. In the afternoon she sells two thirds of what she has left, and because they are not fresh, she charges only half price. In the late afternoon she sells the remaining loaves at a dollar each. Each loaf costs $\$0.75$ for her to make. In dollars, what is her profit for the day?
${ \textbf{(A)}\ 24\qquad\textbf{(B)}\ 36\qquad\textbf{(C)}\ 44\qquad\textbf{(D)}}\ 48\qquad\textbf{(E)}\ 52$
Let $n \ge 2$ be a natural number and suppose that positive numbers $a_0,a_1,...,a_n$ satisfy the equality
$(a_{k-1}+a_{k})(a_{k}+a_{k+1})=a_{k-1}-a_{k+1}$ for each $k =1,2,...,n -1$. Prove that $a_n< \frac{1}{n-1}$
$2023$ balls are divided into several buckets such that no bucket contains more than $99$ balls. We can remove balls from any bucket or remove an entire bucket, as many times as we want. Prove that we can remove them in such a way that each of the remaining buckets will have an equal number of balls and the total number of remaining balls will be at least $100$.
$3^3+3^3+3^3 = $
$\text{(A)}\ 3^4 \qquad \text{(B)}\ 9^3 \qquad \text{(C)}\ 3^9 \qquad \text{(D)}\ 27^3 \qquad \text{(E)}\ 3^{27}$
Let $n$ be an odd positive integer. There is a graph $G$ with $2n$ vertices such that if you partition the vertices into two groups $A$ and $B$ with $n$ vertices each, then the subgraph consisting of only vertices and edges within $A$ is connected and has a closed path containing all of its edges, starting and ending with the same vertex. The same condition is true for $B$ as well. Is $G$ necessarily a clique?
[i](Proposed by Ho Janson)[/i]
Prove that for all positive real numbers $a, b$, and $c$ , $\sqrt[3]{abc}+\frac{1}{a}+\frac{1}{b}+\frac{1}{c} \ge 2\sqrt3$.
When does the equality occur?
Planet Ypsilon has a calendar similar to ours: A year consists of $365$ days, and every month has $28$, $30$ or $31$ days.
Prove that on Planet Ypsilon, a year must have $12$ months.
On a $ 50 \times 50$ board, the centers of several unit squares are colored black. Find the maximum number of centers that can be colored black in such a way that no three black points form a right-angled triangle.
[b]polyhedral[/b]
we call a $12$-gon in plane good whenever:
first, it should be regular, second, it's inner plane must be filled!!, third, it's center must be the origin of the coordinates, forth, it's vertices must have points $(0,1)$,$(1,0)$,$(-1,0)$ and $(0,-1)$.
find the faces of the [u]massivest[/u] polyhedral that it's image on every three plane $xy$,$yz$ and $zx$ is a good $12$-gon.
(it's obvios that centers of these three $12$-gons are the origin of coordinates for three dimensions.)
time allowed for this question is 1 hour.
Show that for any real numbers $ a,b, $ there exists $ c\in [-2,1] $ such that $ \big| c^3+ac+b\big| \ge 1. $
Find all prime numbers $p$ such that $y^2 = x^3+4x$ has exactly $p$ solutions in integers modulo $p$.
In other words, determine all prime numbers $p$ with the following property:
there exist exactly $p$ ordered pairs of integers $(x,y)$ such that $x,y \in \{0,1,\dots,p-1\}$
and \[ p \text{ divides } y^2 - x^3 - 4x. \]
A circle inscribed in triangle $ABC$ has center $O$ and touches side $AC$ at point $K$. A second circle also has center $O$, intersects all sides of triangle $ABC$. Let $E$ and $F$ be the corresponding points of intersection with sides $AB$ and $BC$, closest to vertex $B$; $B_1$ and $B_2$ are the points of its intersection with side $AC$, and $B_1$ is closer to $A$. Prove that points $B$, $K$ and point $P$, the intersections of the segments $B_2E$ and $B_1F$ lie on the same straight line.
Determine the smallest possible value of the sum $S (a, b, c) = \frac{ab}{c}+\frac{bc}{a}+\frac{ca}{b}$ where $a, b, c$ are three positive real numbers with $a^2 + b^2 + c^2 = 1$
Prove that any triangle can be divided into $22$ triangles, each of which has an angle of $22^o$, and another $23$ triangles, each of which has an angle of $23^o$.
There are two equal circles of radius $1$ placed inside the triangle $ABC$ with side $BC = 6$. The circles are tangent to each other, one is inscribed in angle $B$, the other one is inscribed in angle $C$.
(a) Prove that the centroid $M$ of the triangle $ABC$ does not lie inside any of the given circles.
(b) Prove that if $M$ lies on one of the circles, then the triangle $ABC$ is isosceles.
Let $a,b,c,d$ be positive integers such that $ab\equal{}cd$. Prove that $a\plus{}b\plus{}c\plus{}d$ is a composite number.
Harold tosses a nickel four times. The probability that he gets at least as many heads as tails is
$\text{(A)}\ \frac{5}{16} \qquad \text{(B)}\ \frac{3}{8} \qquad \text{(C)}\ \frac{1}{2} \qquad \text{(D)}\ \frac{5}{8} \qquad \text{(E)}\ \frac{11}{16}$
Equilateral triangle $ABC$ has been creased and folded so that vertex $A$ now rests at $A'$ on $\overline{BC}$ as shown. If $BA' = 1$ and $A'C = 2$ then the length of crease $\overline{PQ}$ is
[asy]
size(170);
defaultpen(linewidth(0.7)+fontsize(10));
pair B=origin, A=(1.5,3*sqrt(3)/2), C=(3,0), D=(1,0), P=B+1.6*dir(B--A), Q=C+1.2*dir(C--A);
draw(B--P--D--B^^P--Q--D--C--Q);
draw(Q--A--P, linetype("4 4"));
label("$A$", A, N);
label("$B$", B, W);
label("$C$", C, E);
label("$A'$", D, S);
label("$P$", P, W);
label("$Q$", Q, E);
[/asy]
$ \textbf{(A)}\ \frac{8}{5}\qquad\textbf{(B)}\ \frac{7}{20}\sqrt{21}\qquad\textbf{(C)}\ \frac{1+\sqrt{5}}{2}\qquad\textbf{(D)}\ \frac{13}{8}\qquad\textbf{(E)}\ \sqrt{3} $
Let $M$ be the midpoint of $BC$ of triangle $ABC$. The circle with diameter $BC$, $\omega$, meets $AB,AC$ at $D,E$ respectively. $P$ lies inside $\triangle ABC$ such that $\angle PBA=\angle PAC, \angle PCA=\angle PAB$, and $2PM\cdot DE=BC^2$. Point $X$ lies outside $\omega$ such that $XM\parallel AP$, and $\frac{XB}{XC}=\frac{AB}{AC}$. Prove that $\angle BXC +\angle BAC=90^{\circ}$.
An integer $a$ is called friendly if the equation $(m^2+n)(n^2+m)=a(m-n)^3$ has a solution over the positive integers.
[b]a)[/b] Prove that there are at least $500$ friendly integers in the set $\{ 1,2,\ldots ,2012\}$.
[b]b)[/b] Decide whether $a=2$ is friendly.
Every card in a deck has a picture of one shape - circle, square, or triangle, which is painted in one of the three colors - red, blue, or green. Furthermore, each color is applied in one of three shades - light, medium, or dark. The deck has 27 cards, with every shape-color-shade combination represented. A set of three cards from the deck is called complementary if all of the following statements are true:
i. Either each of the three cards has a different shape or all three of the card have the same shape.
ii. Either each of the three cards has a different color or all three of the cards have the same color.
iii. Either each of the three cards has a different shade or all three of the cards have the same shade.
How many different complementary three-card sets are there?
For all real number $x$ consider the family $F(x)$ of all sequences $(a_{n})_{n\geq 0}$ satisfying the equation \[a_{n+1}=x-\frac{1}{a_{n}}\quad (n\geq 0).\] A positive integer $p$ is called a [i]minimal period[/i] of the family $F(x)$ if
(a) each sequence $\left(a_{n}\right)\in F(x)$ is periodic with the period $p$,
(b) for each $0<q<p$ there exists $\left(a_{n}\right)\in F(x)$ such that $q$ is not a period of $\left(a_{n}\right)$.
Prove or disprove that for each positive integer $P$ there exists a real number $x=x(P)$ such that the family $F(x)$ has the minimal period $p>P$.
Isabella must take four 100-point tests in her math class. Her goal is to achieve an average grade of 95 on the tests. Her first two test scores were 97 and 91. After seeing her score on the third test, she realized she can still reach her goal. What is the lowest possible score she could have made on the third test?
$\textbf{(A)}\hspace{.05in}90 \qquad \textbf{(B)}\hspace{.05in}92 \qquad \textbf{(C)}\hspace{.05in}95 \qquad \textbf{(D)}\hspace{.05in}96 \qquad \textbf{(E)}\hspace{.05in}97 $
Given a positive integer $k$, find the least integer $n_k$ for which there exist five sets $S_1, S_2, S_3, S_4, S_5$ with the following properties:
\[|S_j|=k \text{ for } j=1, \cdots , 5 , \quad |\bigcup_{j=1}^{5} S_j | = n_k ;\]
\[|S_i \cap S_{i+1}| = 0 = |S_5 \cap S_1|, \quad \text{for } i=1,\cdots ,4 \]
[b]p1.[/b] The serpent of fire and the serpent of ice play a game. Since the serpent of ice loves the lucky number $6$, he will roll a fair $6$-sided die with faces numbered $1$ through $6$. The serpent of fire will pay him $\log_{10} x$, where $x$ is the number he rolls. The serpent of ice rolls the die $6$ times. His expected total amount of winnings across the $6$ rounds is $k$. Find $10^k$.
[b]p2.[/b] Let $a = \log_3 5$, $b = \log_3 4$, $c = - \log_3 20$, evaluate $\frac{a^2+b^2}{a^2+b^2+ab} +\frac{b^2+c^2}{b^2+c^2+bc} +\frac{c^2+a^2}{c^2+a^2+ca}$.
[b]p3.[/b] Let $\vartriangle ABC$ be an isosceles obtuse triangle with $AB = AC$ and circumcenter $O$. The circle with diameter $AO$ meets $BC$ at points $X, Y$ , where X is closer to $B$. Suppose $XB = Y C = 4$, $XY = 6$, and the area of $\vartriangle ABC$ is $m\sqrt{n}$ for positive integers $m$ and $n$, where $n$ does not contain any square factors. Find $m + n$.
[b]p4.[/b] Alice is not sure what to have for dinner, so she uses a fair $6$-sided die to decide. She keeps rolling, and if she gets all the even numbers (i.e. getting all of $2, 4, 6$) before getting any odd number, she will reward herself with McDonald’s. Find the probability that Alice could have McDonald’s for dinner.
[b]p5.[/b] How many distinct ways are there to split $50$ apples, $50$ oranges, $50$ bananas into two boxes, such that the products of the number of apples, oranges, and bananas in each box are nonzero and equal?
[b]p6.[/b] Sujay and Rishabh are taking turns marking lattice points within a square board in the Cartesian plane with opposite vertices $(1, 1)$,$(n, n)$ for some constant $n$. Sujay loses when the two-point pattern $P$ below shows up:[img]https://cdn.artofproblemsolving.com/attachments/1/9/d1fe285294d4146afc0c7a2180b15586b04643.png[/img]
That is, Sujay loses when there exists a pair of points $(x, y)$ and $(x + 2, y + 1)$. He and Rishabh stop marking points when the pattern $P$ appears on the board. If Rishabh goes first, let $S$ be the set of all integers $3 \le n \le 100$ such that Rishabh has a strategy to always trick Sujay into being the one who creates $P$. Find the sum of all elements of $S$.
[b]p7.[/b] Let $a$ be the shortest distance between the origin $(0, 0)$ and the graph of $y^3 = x(6y -x^2)-8$. Find $\lfloor a^2 \rfloor $. ($\lfloor x\rfloor $ is the largest integer not exceeding $x$)
[b]p8.[/b] Find all real solutions to the following equation:
$$2\sqrt2x^2 + x -\sqrt{1 - x^2 } -\sqrt2 = 0.$$
[b]p9.[/b] Given the expression $S = (x^4 - x)(x^2 - x^3)$ for $x = \cos \frac{2\pi}{5 }+ i\sin \frac{2\pi}{5 }$, find the value of $S^2$
.
[b]p10.[/b] In a $32$ team single-elimination rock-paper-scissors tournament, the teams are numbered from $1$ to $32$. Each team is guaranteed (through incredible rock-paper-scissors skill) to win any match against a team with a higher number than it, and therefore will lose to any team with a lower number. Each round, teams who have not lost yet are randomly paired with other teams, and the losers of each match are eliminated. After the $5$ rounds of the tournament, the team that won all $5$ rounds is ranked $1$st, the team that lost the 5th round is ranked $2$nd, and the two teams that lost the $4$th round play each other for $3$rd and $4$th place. What is the probability that the teams numbered $1, 2, 3$, and $4$ are ranked $1$st, 2nd, 3rd, and 4th respectively? If the probability is $\frac{m}{n}$ for relatively prime integers $m$ and $n$, find $m$.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].