Given positive real numbers $p,q$ with $p+q=1$. Prove that for all positive integers $m,n$ the following inequality holds $(1-p^m)^n+(1-q^n)^m \geq 1$.

[b]p1.[/b] Let $p_b(m)$ be the sum of digits of $m$ when $m$ is written in base $b$. (So, for example, $p_2(5) = 2$). Let $f(0) = 2007^{2007}$, and for $n \ge 0$ let $f(n + 1) = p_7(f(n))$. What is $f(10^{10000})$? [b]p2.[/b] Compute: $$\sum^{\infty}_{n=1}\frac{(-1)^{n+1}4n}{n^4 - 8n^2 + 4}.$$ [b]p3.[/b] $ABCDEFGH$ is an octagon whose eight interior angles all have the same measure. The lengths of the eight sides of this octagon are, in some order, $$2, 2\sqrt2, 4, 4\sqrt2, 6, 7, 7, \,\,\, and \,\,\, 8.$$ Find the area of $ABCDEFGH$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

All the numbers $2, 3, 4, 5, 6, 7$ are assigned to the six faces of a cube, one number to each face. For each of the eight vertices of the cube, a product of three numbers is computed, where the three numbers are the numbers assigned to the three faces that include that vertex. What is the greatest possible value of the sum of these eight products? $\textbf{(A)}\ 312 \qquad \textbf{(B)}\ 343 \qquad \textbf{(C)}\ 625 \qquad \textbf{(D)}\ 729 \qquad \textbf{(E)}\ 1680$

One out of every seven mathematicians is a philosopher, and one out of every nine philosophers is a mathematician. Are there more philosophers or mathematicians?

Pat is to select six cookies from a tray containing only chocolate chip, oatmeal, and peanut butter cookies. There are at least six of each of these three kinds of cookies on the tray. How many different assortments of six cookies can be selected? $ \textbf{(A)}\ 22 \qquad \textbf{(B)}\ 25 \qquad \textbf{(C)}\ 27 \qquad \textbf{(D)}\ 28 \qquad \textbf{(E)}\ 29$

Ephram is growing $3$ different variants of radishes in a row of $13$ radishes total, but he forgot where he planted each radish variant and he can't tell what variant a radish is before he picks it. Ephram knows that he planted at least one of each radish variant, and all radishes of one variant will form a consecutive string, with all such possibilities having an equal chance of occurring. He wants to pick three radishes to bring to the farmers market, and wants them to all be of different variants. Given that he uses optimal strategy, the probability that he achieves this can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. [i]Proposed by Jeff Lin[/i]

Given acute angled traingle $ABC$ and altitudes $AA_1$, $BB_1$, $CC_1$. Let $M$ midpoint of $BC$. $P$ point of intersection of circles $(AB_1C_1)$ and $(ABC)$ . $T$ is point of intersection of tangents to $(ABC)$ at $B$ and $C$. $S$ point of intersection of $AT$ and $(ABC)$. Prove that $P,A_1,S$ and midpoint of $MT$ collinear.

Each vertex of a cube is to be labeled with an integer $1$ through $8$, with each integer being used once, in such a way that the sum of the four numbers on the vertices of a face is the same for each face. Arrangements that can be obtained from each other through rotations of the cube are considered to be the same. How many different arrangements are possible? $\textbf{(A) } 1\qquad\textbf{(B) } 3\qquad\textbf{(C) }6 \qquad\textbf{(D) }12 \qquad\textbf{(E) }24$

[b]p1.[/b] Let $A\% B = BA - B - A + 1$. How many digits are in the number $1\%(3\%(3\%7))$ ? [b]p2. [/b]Three circles, of radii $1, 2$, and $3$ are all externally tangent to each other. A fourth circle is drawn which passes through the centers of those three circles. What is the radius of this larger circle? [b]p3.[/b] Express $\frac13$ in base $2$ as a binary number. (Which, similar to how demical numbers have a decimal point, has a “binary point”.) [b]p4. [/b] Isosceles trapezoid $ABCD$ with $AB$ parallel to $CD$ is constructed such that $DB = DC$. If $AD = 20$, $AB = 14$, and $P$ is the point on $AD$ such that $BP + CP$ is minimized, what is $AP/DP$? [b]p5.[/b] Let $f(x) = \frac{5x-6}{x-2}$ . Define an infinite sequence of numbers $a_0, a_1, a_2,....$ such that $a_{i+1} = f(a_i)$ and $a_i$ is always an integer. What are all the possible values for $a_{2014}$ ? [b]p6.[/b] $MATH$ and $TEAM$ are two parallelograms. If the lengths of $MH$ and $AE$ are $13$ and $15$, and distance from $AM$ to $T$ is $12$, find the perimeter of $AMHE$. [b]p7.[/b] How many integers less than $1000$ are there such that $n^n + n$ is divisible by $5$ ? [b]p8.[/b] $10$ coins with probabilities of $1, 1/2, 1/3 ,..., 1/10$ of coming up heads are flipped. What is the probability that an odd number of them come up heads? [b]p9.[/b] An infinite number of coins with probabilities of $1/4, 1/9, 1/16, ...$ of coming up heads are all flipped. What is the probability that exactly $ 1$ of them comes up heads? [b]p10.[/b] Quadrilateral $ABCD$ has side lengths $AB = 10$, $BC = 11$, and $CD = 13$. Circles $O_1$ and $O_2$ are inscribed in triangles $ABD$ and $BDC$. If they are both tangent to $BD$ at the same point $E$, what is the length of $DA$ ? PS. You had better use hide for answers.

If $~$ $a,\, b,\, c,\, d$ $~$ are positive real numbers such that $~$ $abcd=1$ $~$ then prove that the following inequality holds \[ \frac{1}{bc+cd+da-1} + \frac{1}{ab+cd+da-1} + \frac{1}{ab+bc+da-1} + \frac{1}{ab+bc+cd-1}\; \le\; 2\, . \] When does inequality hold?

There are $330$ seats in the first row of the auditorium. Some of these seats are occupied by $25$ viewers. Prove that among the pairwise distances between these viewers, there are two equal.

Let $P$ be a regular $2n$-sided polygon. A [b]rhombus-ulation[/b] of $P$ is dividing $P$ into rhombuses such that no two intersect and no vertex of any rhombus is on the edge of other rhombuses or $P$. (a) Prove that number of rhombuses is a function of $n$. Find the value of this function. Also find the number of vertices and edges of the rhombuses as a function of $n$. (b) Prove or disprove that there always exists an edge $e$ of $P$ such that by erasing all the segments parallel to $e$ the remaining rhombuses are connected. (c) Is it true that each two rhombus-ulations can turn into each other using the following algorithm multiple times? Algorithm: Take a hexagon -not necessarily regular- consisting of 3 rhombuses and re-rhombus-ulate the hexagon. (d) Let $f(n)$ be the number of ways to rhombus-ulate $P$. Prove that:\[\Pi_{k=1}^{n-1} ( \binom{k}{2} +1) \leq f(n) \leq \Pi_{k=1}^{n-1} k^{n-k} \]

Determine all positive integers $n$ with the following property: for each triple $(a, b, c)$ of positive real numbers there is a triple $(k, \ell, m)$ of non-negative integer numbers so that $an^k$, $bn^{\ell}$ and $cn^m$ are the lengths of the sides of a (non-degenerate) triangle shapes.

Let $ \lfloor x \rfloor$ be the greatest integer less than or equal to $ x$. Then the number of real solutions to $ 4x^2 \minus{} 40 \lfloor x \rfloor \plus{} 51 \equal{} 0$ is $ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 3 \qquad \textbf{(E)}\ 4$

In the figure below, circle $O$ has two tangents, $\overline{AC}$ and $\overline{BC}$. $\overline{EF}$ is drawn tangent to circle $O$ such that $E$ is on $\overline{AC}$, $F$ is on $\overline{BC}$, and $\overline{EF} \perp \overline{FC}$. Given that the diameter of circle $O$ has length $10$ and that $CO = 13$, what is the area of triangle $EFC$? [img]https://cdn.artofproblemsolving.com/attachments/b/d/4a1bc818a5e138ae61f1f3d68f6ee5adc1ed6f.png[/img]

Suppose you have a $6$ sided dice with $3$ faces colored red, $2$ faces colored blue, and $1$ face colored green. You roll this dice $20$ times and record the color that shows up on top. What is the expected value of the product of the number of red faces, blue faces, and green faces? [i]Proposed by Daniel Li[/i]

Let $n$ be positive integer and let $a_1, a_2,...,a_n$ be real numbers. Prove that there exist positive integers $m, k$ $<=n$ , $|$ $(a_1+a_2+...+a_m)$ $-$ $(a_{m+1}+a_{m+2}+...+a_n)$ $|$ $<=$ $|$ $a_k$ $|$

In the plane, four points are marked. It is known that these points are the centers of four circles, three of which are pairwise externally tangent, and all these three are internally tangent to the fourth one. It turns out, however, that it is impossible to determine which of the marked points is the center of the fourth (the largest) circle. Prove that these four points are the vertices of a rectangle.

Let $n > 1$ be a positive integer. Prove that every term of the sequence $$ n - 1, n^n - 1, n^{n^2} - 1, n^{n^3} - 1, \dots $$ has a prime divisor that does not divide any of the previous terms.

Let $z = a + bi$ be the complex number with $|z| = 5$ and $b > 0$ such that the distance between $(1 + 2i)z^3$ and $z^5$ is maximized, and let $z^4 = c + di$. Find $c+d$.

On the sides of a convex hexagon $ ABCDEF$ , equilateral triangles are constructd in its exterior. Prove that the third vertices of these six triangles are vertices of a regular hexagon if and only if the initial hexagon is [i]affine regular[/i]. (A hexagon is called affine regular if it is centrally symmetric and any two opposite sides are parallel to the diagonal determine by the remaining two vertices.)

Let $A$ be the set $\{k^{19}-k: 1<k<20, k\in N\}$. Let $G$ be the GCD of all elements of $A$. Then the value of $G$ is?

Find all functions $f : R \to R$ such that $$f(f(x) + f(y)^2) = f(x)^2 +y^2f(y)^3.$$ Here $R$ denotes the usual real numbers.

To be able to play the Game of Glory, seven friends have to form four teams which, obviously, may not have the same number of members. In how many different ways can they form these teams? (The order of people within teams is not important, but each person can only be part of one team)

Let $a,b,c\in\mathbb{N}$ prove that if there is a polynomial $P,Q,R\in\mathbb{C}[x]$, which have no common factors and satisfy $$P^a+Q^b=R^c$$ and $$\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}&gt;1.$$ [i](tatari/nightmare)[/i]