Found problems: 85335
How many different integers can be expressed as the sum of three distinct members of the set $ \{1, 4, 7, 10, 13, 16, 19\}$?
$ \textbf{(A)}\ 13 \qquad
\textbf{(B)}\ 16 \qquad
\textbf{(C)}\ 24 \qquad
\textbf{(D)}\ 30 \qquad
\textbf{(E)}\ 35$
If $f\left(\frac{x}{x - 1}\right) = \frac{1}{x}$ for all $x \ne 0,1$ and $0 < \theta < \frac{\pi}{2}$, then $f(\sec^{2}\theta) =$
$ \textbf{(A) }\sin^{2}\theta\qquad\textbf{(B) }\cos^{2}\theta\qquad\textbf{(C) }\tan^{2}\theta\qquad\textbf{(D) }\cot^{2}\theta\qquad\textbf{(E) }\csc^{2}\theta $
$N$ unfair coins (with heads and tails on the sides) are thrown, with the $k^{th}$ coin has got a chance of $\frac{1}{2k + 1}$ to land on tails.How high is the probability that an odd number of coins will show tails?
Consider the following figure where $AC$ is tangent to the circle of center $O$, $\angle BCD = 35^o$, $\angle BAD = 40^o$ and the measure of the minor arc $DE$ is $70^o$. Prove that points $B, O, E$ are collinear.
[img]https://cdn.artofproblemsolving.com/attachments/4/0/fd5f8d3534d9d0676deebd696d174999c2ad75.png[/img]
A fox stands in the centre of the field which has the form of an equilateral triangle, and a rabbit stands at one of its vertices. The fox can move through the whole field, while the rabbit can move only along the border of the field. The maximal speeds of the fox and rabbit are equal to $u$ and $v$, respectively. Prove that:
(a) If $2u>v$, the fox can catch the rabbit, no matter how the rabbit moves.
(b) If $2u\le v$, the rabbit can always run away from the fox.
Figures $I$, $II$, and $III$ are squares. The perimeter of $I$ is $12$ and the perimeter of $II$ is $24$. The perimeter of $III$ is
[asy]
draw((0,0)--(15,0)--(15,6)--(12,6)--(12,9)--(0,9)--cycle);
draw((9,0)--(9,9));
draw((9,6)--(12,6));
label("$III$",(4.5,4),N);
label("$II$",(12,2.5),N);
label("$I$",(10.5,6.75),N);
[/asy]
$\text{(A)}\ 9 \qquad \text{(B)}\ 18 \qquad \text{(C)}\ 36 \qquad \text{(D)}\ 72 \qquad \text{(D)}\ 81$
Petya and Vasya only know positive integers not exceeding $10^9-4000$. Petya considers numbers as good which are representable in the form $abc+ab+ac+bc$, where $a,b$ and $c$ are natural numbers not less than $100$. Vasya considers numbers as good which are representable in the form $xyz-x-y-z$, where $x,y$ and $z$ are natural numbers strictly bigger than $100$. For which of them are there more good numbers?
[i]Proposed by I. Bogdanov[/i]
Let $P$ be a point inside the square $ABCD$ such that $\angle PAC = \angle PCD = 17^o$ (see Figure 1). Calculate $\angle APB$?
[img]https://cdn.artofproblemsolving.com/attachments/d/0/0b20ebee1fe28e9c5450d04685ac8537acda07.png[/img]
Assume real numbers $a_i,b_i\,(i=0,1,\cdots,2n)$ satisfy the following conditions:
(1) for $i=0,1,\cdots,2n-1$, we have $a_i+a_{i+1}\geq 0$;
(2) for $j=0,1,\cdots,n-1$, we have $a_{2j+1}\leq 0$;
(2) for any integer $p,q$, $0\leq p\leq q\leq n$, we have $\sum_{k=2p}^{2q}b_k>0$.
Prove that $\sum_{i=0}^{2n}(-1)^i a_i b_i\geq 0$, and determine when the equality holds.
Let $p=4k+1$ be a prime and let $|x| \leq \frac{p-1}{2}$ such that $\binom{2k}{k}\equiv x \pmod p$. Show that $|x| \leq 2\sqrt{p}$.
Let $\mathbb{Z} _{>0}$ be the set of positive integers. Find all functions $f: \mathbb{Z} _{>0}\rightarrow \mathbb{Z} _{>0}$ such that
\[ m^2 + f(n) \mid mf(m) +n \]
for all positive integers $m$ and $n$.
Let $(G, \cdot)$ a finite group with order $n \in \mathbb{N}^{*},$ where $n \geq 2.$ We will say that group $(G, \cdot)$ is arrangeable if there is an ordering of its elements, such that
\[
G = \{ a_1, a_2, \ldots, a_k, \ldots , a_n \} = \{ a_1 \cdot a_2, a_2 \cdot a_3, \ldots, a_k \cdot a_{k + 1}, \ldots , a_{n} \cdot a_1 \}.
\]
a) Determine all positive integers $n$ for which the group $(Z_n, +)$ is arrangeable.
b) Give an example of a group of even order that is arrangeable.
The distance between the $5^\text{th}$ and $26^\text{th}$ exits on an interstate highway is $118$ miles. If any two exits are at least $5$ miles apart, then what is the largest number of miles there can be between two consecutive exits that are between the $5^\text{th}$ and $26^\text{th}$ exits?
$\text{(A)}\ 8 \qquad \text{(B)}\ 13 \qquad \text{(C)}\ 18 \qquad \text{(D)}\ 47 \qquad \text{(E)}\ 98$
Is it possible to partition a triangle, with line segments, into exactly five isosceles triangles? All the triangles in concern are assumed to be nondegenerated triangles.
What number should be removed from the list
\[1,2,3,4,5,6,7,8,9,10,11\]
so that the average of the remaining numbers is $6.1$?
$\text{(A)}\ 4 \qquad \text{(B)}\ 5 \qquad \text{(C)}\ 6 \qquad \text{(D)}\ 7 \qquad \text{(E)}\ 8$
Points $A' ,B' ,C'$ on the respective sides $BC,CA,AB$ of triangle $ABC$ satisfy $\frac{AC' }{AB} = \frac{BA' }{BC} = \frac{CB' }{CA} = k$. The lines $AA' ,BB' ,CC' $ form a triangle $A_1B_1C_1$ (possibly degenerate). Given $k$ and the area $S$ of $\triangle ABC$, compute the area of $\triangle A_1B_1C_1$.
Suppose $a _ { n } > 0$ and $\sum _ { n = 1 } ^ { \infty } a _ { n }$ diverges. Determine whether $\sum _ { n = 1 } ^ { \infty } a _ { n } / S _ { n } ^ { 2 }$ converges, where $S _ { n } = a _ { 1 } + a _ { 2 } + \dots + a _ { n } .$
Show that there do not exist any distinct natural numbers $ a$, $ b$, $ c$, $ d$ such that $ a^3\plus{}b^3\equal{}c^3\plus{}d^3$ and $ a\plus{}b\equal{}c\plus{}d$.
Milo rolls five fair dice which have 4, 6, 8, 12, and 20 sides respectively (and each one is labeled $1$-$n$ for appropriate $n$. How many distinct ways can they roll a full house (three of one number and two of another)? The same numbers appearing on different dice are considered distinct full houses, so $(1,1,1,2,2)$ and $(2,2,1,1,1)$ would both be counted.
[i]Proposed by Robert Trosten[/i]
A circle cuts each side of a rhombus twice thus dividing each side into three segments. Let us go around the perimeter of the rhombus clockwise beginning at a vertex and paint these segments successively in red, white and blue. Prove that the sum of lengths of the blue segments equals that of the red ones.
(V Proizvolov)
On the coast of a circular island there are eight different cities. Initially there are no routes between the cities. We have to construct five straight two-way routes, which do not intersect, so that from each city there are one or two routes. In how many ways can this happen?
Prove that there are $100$ distinct positive integers $n_1,n_2,\dots,n_{99},n_{100}$ such that $\frac{n_1^3+n_2 ^3+\dots +n_{100}^3}{100}$ is a perfect cube.
If $ f(x) \equal{} 5x^2 \minus{} 2x \minus{} 1$, then $ f(x \plus{} h) \minus{} f(x)$ equals:
$ \textbf{(A)}\ 5h^2 \minus{} 2h \qquad \textbf{(B)}\ 10xh \minus{} 4x \plus{} 2 \qquad \textbf{(C)}\ 10xh \minus{} 2x \minus{} 2 \\
\textbf{(D)}\ h(10x \plus{} 5h \minus{} 2) \qquad \textbf{(E)}\ 3h$
Three athletes ran at different constant speeds along a track of length $1$. They started moving at the same time at one end of the track. Having reached one of the ends of the track, the athlete immediately turned around and continued running in the opposite direction. After a while, all three athletes met at the start and finished training. At what maximum $S$ can we knowingly say that at some point the sum of the pairwise distances between athletes was at least $S$?
[i]Proposed by A. Golovanov, I. Rubanov[/i]
Triangle $ABC$ has circumcenter $O$. $M$ is the midpoint of arc $BC$ not containing $A$. $S$ is a point on $(ABC)$ such that $AS$ and $BC$ intersect on the line passing through $O$ and perpendicular to $AM$. $D$ is a point such that $ABDC$ is a parallelogram. Prove that $D$ lies on the line $SM$.