Found problems: 85335
A nondegenerate rhombus has side length $l$, and its area is twice that of its inscribed circle. Find the radius of the inscribed circle.
Suppose all of the 200 integers lying in between (and including) 1 and 200 are written on a blackboard. Suppose we choose exactly 100 of these numbers and circle each one of them. By the [i]score[/i] of such a choice, we mean the square of the difference between the sum of the circled numbers and the sum of the non-circled numbers. What is the average scores over all possible choices for 100 numbers?
Among $18$ parts placed in a row, some three in a row weigh $99 $ g each, and all the rest weigh $100$ g each. On a scale with an arrow, identify all $99$-gram parts.
In convex quadrilateral $ABCD, \angle BAC=\angle CAD.$ $E$ lies on segment $CD$, and $BE$ and $AC$ intersect at $F,$ $DF$ and $BC$ intersect at $G.$ Prove that $\angle GAC=\angle EAC.$
During the softball season, Judy had $35$ hits. Among her hits were $1$ home run, $1$ triple and $5$ doubles. The rest of her hits were single. What percent of her hits were single?
$\text{(A)}\ 28\% \qquad \text{(B)}\ 35\% \qquad \text{(C)}\ 70\% \qquad \text{(D)}\ 75\% \qquad \text{(E)}\ 80\% $
Find all real numbers $a>0$ for which the series
$$\sum_{n=1}^\infty\frac{a^{f(n)}}{n^2}$$is convergent; $f(n)$ denotes the number of $0$'s in the decimal expansion of $f$.
Let $s_1, s_2, s_3$ be the respective sums of $n$, $2n$, $3n$ terms of the same arithmetic progression with $a$ as the first term and $d$ as the common difference. Let $R=s_3-s_2-s_1$. Then $R$ is dependent on:
$ \textbf{(A)}\ a \text{ } \text{and} \text{ } d\qquad\textbf{(B)}\ d \text{ } \text{and} \text{ } n\qquad\textbf{(C)}\ a \text{ } \text{and} \text{ } n\qquad\textbf{(D)}\ a, d, \text{ } \text{and} \text{ } n\qquad$
$\textbf{(E)}\ \text{neither} \text{ } a \text{ } \text{nor} \text{ } d \text{ } \text{nor} \text{ } n $
Alex starts at the origin $O$ of a hexagonal lattice. Every second, he moves to one of the six vertices adjacent to the vertex he is currently at. If he ends up at $X$ after $2018$ moves, then let $p$ be the probability that the shortest walk from $O$ to $X$ (where a valid move is from a vertex to an adjacent vertex) has length $2018$. Then $p$ can be expressed as $\tfrac{a^m-b}{c^n}$, where $a$, $b$, and $c$ are positive integers less than $10$; $a$ and $c$ are not perfect squares; and $m$ and $n$ are positive integers less than $10000$. Find $a+b+c+m+n$.
Regular hexagon $ABCDEF$ has side length $1$. Given that $P$ is a point inside $ABCDEF$, compute the minimum of $AP \sqrt3 + CP + DP + EP\sqrt3$.
Find all ordered pairs $(a,b)$ of positive integers for which the numbers $\dfrac{a^3b-1}{a+1}$ and $\dfrac{b^3a+1}{b-1}$ are both positive integers.
Since $24 = 3+5+7+9$, the number $24$ can be written as the sum of at least two consecutive odd positive integers.
(a) Can $2005$ be written as the sum of at least two consecutive odd positive integers? If yes, give an example of how it can be done. If no, provide a proof why not.
(b) Can $2006$ be written as the sum of at least two consecutive odd positive integers? If yes, give an example of how it can be done. If no, provide a proof why not.
Let $H$ be the orthocenter of an acute-angled triangle $ABC$. Point $X_A$ lying on the tangent at $H$ to the circumcircle of triangle $BHC$ is such that $AH=AX_A$ and $X_A \not= H$. Points $X_B,X_C$ are defined similarly. Prove that the triangle $X_AX_BX_C$ and the orthotriangle of $ABC$ are similar.
Let $d_i$ be the first decimal digit of $2^i$ for every non-negative integer $i$. Prove that for each positive integer $n$ there exists a decimal digit other than $0$ which can be found in the sequence $d_0, d_1, \dots, d_{n-1}$ strictly less than $\frac{n}{17}$ times.
Suppose that $ T$ is a tree with $ k$ edges. Prove that the $ k$-dimensional cube can be partitioned to graphs isomorphic to $ T$.
On each day, more than half of the inhabitants of Évora eats [i]sericaia[/i] as dessert. Show that there is a group of 10 inhabitants of Évora such that, on each of the last 2010 days, at least one of the inhabitants ate [i]sericaia[/i] as dessert.
Determine the minimum possible amount of distinct prime divisors of $19^{4n}+4$, for a positive integer $n$.
Find all functions $f, g: \mathbb{R} \rightarrow \mathbb{R}$ satisfying $f (0)=2022$ and $f (x+g(y)) =xf(y)+(2023-y)f(x)+g(x)$ for all $x, y \in \mathbb{R}$.
Two identical jars are filled with alcohol solutions, the ratio of the volume of alcohol to the volume of water being $p : 1$ in one jar and $q : 1$ in the other jar. If the entire contents of the two jars are mixed together, the ratio of the volume of alcohol to the volume of water in the mixture is
$\textbf{(A) }\frac{p+q}{2}\qquad\textbf{(B) }\frac{p^2+q^2}{p+q}\qquad\textbf{(C) }\frac{2pq}{p+q}\qquad\textbf{(D) }\frac{2(p^2+pq+q^2)}{3(p+q)}\qquad\textbf{(E) }\frac{p+q+2pq}{p+q+2}$
Andrew Mellon found a piece of melon that is shaped like a octagonal prism where the bases are regular. Upon slicing it in half once, he found that he created a cross-section that is an equilateral hexagon. What is the minimum possible ratio of the height of the melon piece to the side length of the base?
[i]Proposed by Lohith Tummala[/i]
Consolidated Megacorp is planning to send a salesperson to Elbonia who needs to visit every town there. It is possible to travel between any two towns of Elbonia directly either by barge or by mule cart (the same type of travel is available in either direction, and these are the only types of travel available). Show that it is possible to choose a starting town so that the salesperson can complete a round trip visiting each town exactly once and returning to her starting point, while changing the type of transportation used at most one time (this is desirable, since it’s hard to arrange for the merchandise to be transferred from barge to cart or vice versa).
There are $n$ MOPpers $p_1,...,p_n$ designing a carpool system to attend their morning class. Each $p_i$'s car fits $\chi (p_i)$ people ($\chi : \{p_1,...,p_n\} \to \{1,2,...,n\}$). A $c$-fair carpool system is an assignment of one or more drivers on each of several days, such that each MOPper drives $c$ times, and all cars are full on each day. (More precisely, it is a sequence of sets $(S_1, ...,S_m)$ such that $|\{k: p_i\in S_k\}|=c$ and $\sum_{x\in S_j} \chi(x) = n$ for all $i,j$. )
Suppose it turns out that a $2$-fair carpool system is possible but not a $1$-fair carpool system. Must $n$ be even?
[i]Proposed by Nathan Ramesh and Palmer Mebane
Let $(m,n,N)$ be a triple of positive integers. Bruce and Duncan play a game on an m\times n array, where the entries are all initially zeroes. The game has the following rules.
$\bullet$ The players alternate turns, with Bruce going first.
$\bullet$ On Bruce's turn, he picks a row and either adds $1$ to all of the entries in the row or subtracts $1$ from all the entries in the row.
$\bullet$ On Duncan's turn, he picks a column and either adds $1$ to all of the entries in the column or subtracts $1$ from all of the entries in the column.
$\bullet$ Bruce wins if at some point there is an entry $x$ with $|x|\ge N$.
Find all triples $(m, n,N)$ such that no matter how Duncan plays, Bruce has a winning strategy.
Let $f(x)=2x^3-2x$. For what positive values of $a$ do there exist distinct $b,c,d$ such that $(a,f(a)),(b,f(b)),(c,f(c)),(d,f(d))$ is a rectangle?
An electric circuit has a battery of EMF $E$ and it is connected to a resistor system as shown in the image below.
The resistor system alone is put in an adiabatic box (with circuit still connected) filled with ideal gas and containing a thermally conducting plate (plate shewn below) with coefficient of areal thermal expansion $\beta$ and heat capacity $c$. $R_2,R_5,R_3$ are fixed and $R_1,R_4$ are variable. Assume temperature change doesn't alter any of the macroscopically noticeable attributes of the wire and resistors.
[list]
[*]Find the condition on the resistors (all of them non-zero) for which the rate thermal expansion attains maximum.[/*]
[*] Find the equivalent resistance in such a condition described above.[/*]
[*] Draw how the plate will look like after a time $t$ and describe its size qualitatively.(Just the shape matters in drawing).[/*]
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Let $A_1 A_2…A_{66}$ be a convex 66-gon. What’s the greatest number of pentagons $A_i A_{i+1} A_{i+2} A_{i+3} A_{i+4},1\leq i\leq 66,$ which have an inscribed circle? ($A_{66+i}\equiv A_i$).