Found problems: 85335
When a bucket is two-thirds full of water, the bucket and water weigh $ a$ kilograms. When the bucket is one-half full of water the total weight is $ b$ kilograms. In terms of $ a$ and $ b$, what is the total weight in kilograms when the bucket is full of water?
$ \textbf{(A)}\ \frac23a\plus{}\frac13b\qquad
\textbf{(B)}\ \frac32a\minus{}\frac12b\qquad
\textbf{(C)}\ \frac32a\plus{}b$
$ \textbf{(D)}\ \frac32a\plus{}2b\qquad
\textbf{(E)}\ 3a\minus{}2b$
In the country of East Westmore, statisticians estimate there is a baby born every 8 hours and a death every day. To the nearest hundred, how many people are added to the population of East Westmore each year?
$\textbf{(A)}\hspace{.05in}600 \qquad \textbf{(B)}\hspace{.05in}700 \qquad \textbf{(C)}\hspace{.05in}800 \qquad \textbf{(D)}\hspace{.05in}900 \qquad \textbf{(E)}\hspace{.05in}1000 $
Let $a_1,a_2,...,a_n$ be integers . Show that there exists integers $k$ and $r$ such that the sum
$a_k+a_{k+1}+...+a_{k+r}$
is divisible by $n$ .
Let $g$ and $h$ be two distinct elements of a group $G$, and let $n$ be a positive integer. Consider a sequence $w=(w_1,w_2,\dots)$ which is not eventually periodic and where each $w_i$ is either $g$ or $h$. Denote by $H$ the subgroup of $G$ generated by all elements of the form $w_kw_{k+1}\dotsc w_{k+n-1}$ with $k \ge 1$. Prove that $H$ does not depend on the choice of the sequence $w$ (but may depend on $n$).
Prove that there is no polynomial $P(x)$ with real coefficients that satisfies
$$P'(x)P''(x)>P(x)P'''(x)\qquad\text{for all }x\in\mathbb R.$$Is this statement true for all of the thrice differentiable real functions?
Suppose $f(x)=x^2+a$. Define $f^1(x)=f(x)$, $f^n(x)=f(f^{n-1}(x))$, $n=2, 3, \cdots$, and let $M=\{a\in\mathbb{R}| |f^n(0)|\le 2, \text{for any } n\in\mathbb{N}\} $. Prove that $M=[-2, \frac{1}{4}]$.
The gure below shows a large square divided into $9$ congruent smaller squares. A shaded square bounded by some of the diagonals of those smaller squares has area $14$. Find the area of the large square.
[img]https://cdn.artofproblemsolving.com/attachments/5/e/bad21be1b3993586c3860efa82ab27d340dbcb.png[/img]
Let $ABCD$ be a tetrahedron and suppose that $M$ is a point inside it such that $\angle MAD=\angle MBC$ and $\angle MDB=\angle MCA$. Prove that $$MA\cdot MB+MC\cdot MD<\max(AD\cdot BC,AC\cdot BD).$$
Prove that if the product of positive numbers $a,b$ and $c$ equals one, then $\frac{1}{a(a+1)}+\frac{1}{b(b+1)}+\frac{1}{c(c+1)}\ge \frac{3}{2}$
Let $a=1+10^{-4}$. Consider some $2023\times 2023$ matrix with each entry a real in $[1,a]$. Let $x_i$ be the sum of the elements of the $i$-th row and $y_i$ be the sum of the elements of the $i$-th column for each integer $i\in [1,n]$. Find the maximum possible value of $\dfrac{y_1y_2\cdots y_{2023}}{x_1x_2\cdots x_{2023}}$ (the answer may be expressed in terms of $a$).
Let $ABCDE$ be a convex pentagon such that angles $CAB$, $BCA$, $ECD$, $DEC$ and $AEC$ are equal. Prove that $CE$ bisects $BD$.
Natural number $k$ has $n$ digits in its decimal notation. It was rounded up to tens, then the obtained number was rounded up to hundreds, and so on $(n-1)$ times. Prove that the obtained number $m$ satisfies inequality $m < \frac{18k}{13}$.
(Examples of rounding: $191\to190\to 200, 135\to140\to 100$.)
There are arbitrary 7 points in the plane. Circles are drawn through every 4 possible concyclic points. Find the maximum number of circles that can be drawn.
In a quadrilateral $ABCD$ of area $1$, the parallel sides $BC$ and $AD$ are in the ratio $1 :2$ . $K$ is the midpoint of the diagonal $AC$ and $L$ is the point of intersection of the line $DK$ and the side $AB$. Determine the area of the quadrilateral $BCKL$ .
(M G Sonkin)
Consider triangles $ABC$ in space.
(a) What condition must the angles $\angle A, \angle B , \angle C$ of $\triangle ABC$ fulfill in order that there is a point $P$ in space such that $\angle APB, \angle BPC, \angle CPA$ are right angles?
(b) Let $d$ be the longest of the edges $PA,PB,PC$ and let $h$ be the longest altitude of $\triangle ABC$. Show that $\frac{1}{3}\sqrt6 h \le d \le h$.
Find all positive integers $n$ for which there are distinct integer numbers $a_1, a_2, ... , a_n$ such that
$$\frac{1}{a_1}+\frac{2}{a_2}+...+\frac{n}{a_n}=\frac{a_1 + a_2 + ... + a_n}{2}$$
A positive integer $n$ is lucky if $2n+1$, $3n+1$, and $4n+1$ are all composite numbers. Compute the smallest lucky number.
[i]Proposed by Michael Tang[/i]
Let $a,b,c$ be given pairwise coprime positive integers where $a>bc$. Let $m<n$ be positive integers. We call $m$ to be a grandson of $n$ if and only if, for all possible piles of stones whose total mass adds up to $n$ and consist of stones with masses $a,b,c$, it's possible to take some of the stones out from this pile in a way that in the end, we can obtain a new pile of stones with total mass of $m$. Find the greatest possible number that doesn't have any grandsons.
Alireza is currently standing at the point $(0, 0)$ in the $x-y$ plane. At any given time, Alireza can move from the point $(x, y)$ to the point $(x + 1, y)$ or the point $(x, y + 1)$. However, he cannot move to any point of the form $(x, y)$ where $y \equiv 2x\,\, (\mod \,\,5)$. Let $p_k$ be the number of paths Alireza can take starting from the point $(0, 0)$ to the point $(k + 1, 2k + 1)$. Evaluate the sum $$\sum^{\infty}_{k=1} \frac{p_k}{5^k}.$$.
In a blackboard there are $K$ circles in a row such that one of the numbers $1,...,K$ is assigned to each circle from the left to the right.
Change of situation of a circle is to write in it or erase the number which is assigned to it.At the beginning no number is written in its own circle.
For every positive divisor $d$ of $K$ ,$1\leq d\leq K$ we change the situation of the circles in which their assigned numbers are divisible by $d$,performing for each divisor $d$ $K$ changes of situation.
Determine the value of $K$ for which the following holds;when this procedure is applied once for all positive divisors of $K$ ,then all numbers $1,2,3,...,K$ are written in the circles they were assigned in.
[b]p1.[/b] The length of the first side of a triangle is $1$, the length of the second side is $11$, and the length of the third side is an integer. Find that integer.
[b]p2.[/b] Suppose $a, b$, and $c$ are positive numbers such that $a + b + c = 1$. Prove that $ab + ac + bc \le \frac13$.
[b]p3.[/b] Prove that $1 +\frac12+\frac13+\frac14+ ... +\frac{1}{100}$ is not an integer.
[b]p4.[/b] Find all of the four-digit numbers n such that the last four digits of $n^2$ coincide with the digits of $n$.
[b]p5.[/b] (Bonus) Several ants are crawling along a circle with equal constant velocities (not necessarily in the same direction). If two ants collide, both immediately reverse direction and crawl with the same velocity. Prove that, no matter how many ants and what their initial positions are, they will, at some time, all simultaneously return to the initial positions.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
Prove that the set $\{2,2^2,\ldots,2^n,\ldots\}$ satisfies the following properties:
[b](1)[/b] For every $a\in A, b\in\mathbb{N}$, if $b<2a-1$, then $b(b+1)$ isn't a multiple of $2a$;
[b](2)[/b] For every positive integer $a\notin A,a\ne 1$, there exists a positive integer $b$, such that $b<2a-1$ and $b(b+1)$ is a multiple of $2a$.
Among the $100$ constants $a_1,a_2,a_3,...,a_{100}$, there are $39$ equal to $-1$ and $61$ equal to $+1$. Find the sum of all the products $a_ia_j$, where $1\le i < j \le 100$.
Two concentric circles have radii $1$ and $2$. Two points on the outer circle are chosen independently and uniformly at random. What is the probability that the chord joining the two points intersects the inner circle?
$\textbf{(A) }\frac{1}{6}\qquad\textbf{(B) }\frac{1}{4}\qquad\textbf{(C) }\frac{2-\sqrt{2}}{2}\qquad\textbf{(D) }\frac{1}{3}\qquad\textbf{(E) }\frac{1}{2}\qquad$
Determine all triples of positive integers $(a,b,c)$ such that $a \leq b \leq c$ and $a +b + c + ab+ bc +ca = abc +1$.