Found problems: 85335
What is the hundreds digit of $2011^{2011}$?
$ \textbf{(A)}\ 1 \qquad
\textbf{(B)}\ 4 \qquad
\textbf{(C)}\ 5 \qquad
\textbf{(D)}\ 6 \qquad
\textbf{(E)}\ 9 $
Let $n = 6901$. There are $6732$ positive integers less than or equal to $n$ that are also relatively prime to $n$. Find the sum of the distinct prime factors of $n$.
$A, B$, and $C$ are vertices of a triangle, and $P$ is a point within the triangle. If angles $\angle BAP$, $\angle BCP$, and $\angle ABP$ are all $30^o$ and angle $\angle ACP$ is $45^o$, what is $\sin(\angle CBP)$?
On each square of an $n$x$n$ board sleeps a dragon. Two dragons are called neighbors if their squares have a side in common. Each turn, Minnie wakes up a dragon which has a living neighbor and Max directs it towards one of its living neighbors. The dragon than breathes fire on that neighbor and destroys it, and then goes back to sleep.
Minnie's goal is to minimize the snoring of the dragons and leave as few living dragons as possible. Max is a member of PETD (People for the Ethical Treatment of Dragons), and he wants to save as many dragons as he can.
How many dragons will stay alive at the end if
1. $n=4$?
2. $n=5$?
If a is a natural number, show that the number of positive integral solutions of the indeterminate equation
$$x_1 + 2x_2 + 3x_3 + ... + nx_n = a \ \ (1) $$
is equal to the number of non-negative integral solutions of
$$y_1 + 2y_2 + 3y_3 + ... + ny_n = a - \frac{n(n + 1)}{2} \ \ (2)$$
[By a solution of equation (1), we mean a set of numbers $\{x_1, x_2,..., x_n\}$ which satisfies equation (1)].
Let $a,b,c,d>0$ . Prove that $a^4+b^4+c^4+d^4 \geq 4abcd+4(a-b)^2 \sqrt{abcd}$
Let $\triangle ABC$ be an acute-angled triangle with its incenter $I$. Suppose that $N$ is the midpoint of the arc $\overarc{BAC}$ of the circumcircle of triangle $\triangle ABC$, and $P$ is a point such that $ABPC$ is a parallelogram.Let $Q$ be the reflection of $A$ over $N$ and $R$ the projection of $A$ on $\overline{QI}$. Show that the line $\overline{AI}$ is tangent to the circumcircle of triangle $\triangle PQR$
[i]Proposed by Patrik Bak - Slovakia[/i]
You have a $7\times 7$ board divided into $49$ boxes. Mateo places a coin in a box.
a) Prove that Mateo can place the coin so that it is impossible for Emi to completely cover the $48$ remaining squares, without gaps or overlaps, using $15$ $3\times1$ rectangles and a cubit of three squares, like those in the figure.
[img]https://cdn.artofproblemsolving.com/attachments/6/9/a467439094376cd95c6dfe3e2ad3e16fe9f124.png[/img]
b) Prove that no matter which square Mateo places the coin in, Emi will always be able to cover the 48 remaining squares using $14$ $3\times1$ rectangles and two cubits of three squares.
In a board school, there are 9 subjects, 512 students, and 256 rooms (two people in each room.) For every student there is a set (a subset of the 9 subjects) of subjects the student is interested in. Each student has a different set of subjects, (s)he is interested in, from all other students. (Exactly one student has no subjects (s)he is interested in.)
Prove that the whole school can line up in a circle in such a way that every pair of the roommates has the two people standing next to each other, and those pairs of students standing next to each other that are not roommates, have the following properties. One of the two students is interested in all the subjects that the other student is interested in, and also exactly one more subject.
Suppose that $n$ is a positive integer number. Consider a regular polygon with $2n$ sides such that one of its largest diagonals is parallel to the $x$-axis. Find the smallest integer $d$ such that there is a polynomial $P$ of degree $d$ whose graph intersects all sides of the polygon on points other than vertices.
Proposed by Mohammad Ahmadi
Say that an ordered triple $(a, b, c)$ is [i]pleasing[/i] if
(a) $a$, $b$, and $c$ are in the set $\{ 1, 2, \dots, 17 \}$, and
(b) both $b - a$ and $c - b$ are greater than $3$, and at least one of them is equal to $4$.
How many pleasing triples are there?
Kelvin the Frog has a pair of standard fair $8$-sided dice (each labelled from $1$ to $8$). Alex the sketchy Kat also has a pair of fair $8$-sided dice, but whose faces are labelled differently (the integers on each Alex's dice need not be distinct). To Alex's dismay, when both Kelvin and Alex roll their dice, the probability that they get any given sum is equal!
Suppose that Alex's two dice have $a$ and $b$ total dots on them, respectively. Assuming that $a \neq b$, find all possible values of $\min \{a,b\}$.
Let $H$ be a subgroup with $h$ elements in a group $G.$ Suppose that $G$ has an element $a$ such that for all $x$ in $H,$ $(xa)^3=1,$ the identity. In $G$, let $P$ be the subset of all products $x_1ax_2a\dots x_na,$ with $n$ a positive integer and the $x_i$ in $H.$
(a) Show that $P$ is a finite set.
(b) Show that, in fact, $P$ has no more that $3h^2$ elements.
Are there infinitely many positive integers $n$ such that $19|1+2^n+3^n+4^n$? Justify your claim.
Circles $W_1,W_2$ meet at $D$and $P$. $A$ and $B$ are on $W_1,W_2$ respectively, such that $AB$ is tangent to $W_1$ and $W_2$. Suppose $D$ is closer than $P$ to the line $AB$. $AD$ meet circle $W_2$ for second time at $C$. Let $M$ be the midpoint of $BC$. Prove that $\angle{DPM}=\angle{BDC}$.
Prove that for all polynomials $P \in \mathbb{R}[x]$ and positive integers $n$, $P(x)-x$ divides $P^n(x)-x$ as polynomials.
A parallelogram of paper with sides $25$ and $10$ is given. The distance between the longer sides is $6$. The paper should be cut into exactly two parts in such a way that one can stick both the pieces together and fold it in a suitable manner to form a cube of suitable edge length without any further cuts and overlaps. Show that it is really possible and describe such a fragmentation.
Prove there aren't positive integers $a, b, c, d$ forming an arithmetic progression such that $ ab + 1, ac + 1, ad + 1, bc + 1, bd + 1, cd + 1 $ are all perfect squares.
Points $P_1,P_2,\ldots,P_n,Q$ are given in space $(n\ge4)$, no four of which are in a plane. Prove that if for any three distinct points $P_\alpha,P_\beta,P_\gamma$ there is a point $P_\delta$ such that the tetrahedron $P_\alpha P_\beta P_\gamma P_\delta$ contains the point $Q$, then $n$ is an even number.
Let $ABC$ be an equilateral triangle whose angle bisectors of $B$ and $C$ intersect at $D$. Perpendicular bisectors of $BD$ and $CD$ intersect $BC$ at points $E$ and $Z$ respectively.
a) Prove that $BE=EZ=ZC$.
b) Find the ratio of the areas of the triangles $BDE$ to $ABC$
Consider two real numbers $ a\in [ - 2,\infty)\ ,\ r\in [0,\infty)$ and the natural number $ n\ge 1$. Show that:
\[ r^{2n} + ar^n + 1\ge (1 - r)^{2n}\]
Find all triples $(a,b,c)$ of positive integers such that
(i) $a \leq b \leq c$;
(ii) $\text{gcd}(a,b,c)=1$; and
(iii) $a^3+b^3+c^3$ is divisible by each of the numbers $a^2b, b^2c, c^2a$.
Let rectangle $ABCD$ have side lengths $AB = 8$, $BC = 6$. Let $ABCD$ be inscribed in a circle with center $O$, as shown in the diagram. Let $M$ be the midpoint of side $\overline{AB}$, and let $X$ be the intersection of ray $\overrightarrow{MO}$ with the circle. Compute the length $AX$.
[img]https://cdn.artofproblemsolving.com/attachments/6/0/a13e7ec6798f57d896265f61fa42df4c6cab15.png[/img]
In triangle $ABC$, point $M$ is the midpoint of $BC$, $P$ the point of intersection of the tangents at points $B$ and $C$ of the circumscribed circle of $ABC$, $N$ is the midpoint of the segment $MP$. The segment $AN$ meets the circumcircle $ABC$ at the point $Q$. Prove that $\angle PMQ = \angle MAQ$.
A sequence $P_1, \dots, P_n$ of points in the plane (not necessarily diferent) is [i]carioca[/i] if there exists a permutation $a_1, \dots, a_n$ of the numbers $1, \dots, n$ for which the segments
$$P_{a_1}P_{a_2}, P_{a_2}P_{a_3}, \dots, P_{a_n}P_{a_1}$$
are all of the same length.
Determine the greatest number $k$ such that for any sequence of $k$ points in the plane, $2023-k$ points can be added so that the sequence of $2023$ points is [i]carioca[/i].