We have $n$ positive integers greater than $1$ and less than $10000$ such that neither of them is prime but any two of them are relative prime. Find the maximum value of $n $.

$M$ is a postive real and $f:[0,\infty)\to[0,M]$ is a continuous function such that $$\int_0^\infty (1+x)f(x) dx<\infty$$ Then, prove the following inequality. $$\left(\int_0^\infty f(x) dx \right)^2 \leq 4M \int_0^\infty x f(x) dx$$ (@below, Thank you. I fixed.)

At the Rice Mathematics Tournament, 80% of contestants wear blue jeans, 70% wear tennis shoes, and 80% of those who wear blue jeans also wear tennis shoes. What fraction of people wearing tennis shoes are wearing blue jeans?

[b]p1.[/b] Let $f$ be a function such that $f(x + y) = f(x) + f(y)$ for all $x$ and $y$. Assume $f(5) = 9$. Compute $f(2015)$. [b]p2.[/b] There are six cards, with the numbers $2, 2, 4, 4, 6, 6$ on them. If you pick three cards at random, what is the probability that you can make a triangles whose side lengths are the chosen numbers? [b]p3. [/b]A train travels from Berkeley to San Francisco under a tunnel of length $10$ kilometers, and then returns to Berkeley using a bridge of length $7$ kilometers. If the train travels at $30$ km/hr underwater and 60 km/hr above water, what is the train’s average speed in km/hr on the round trip? [b]p4.[/b] Given a string consisting of the characters A, C, G, U, its reverse complement is the string obtained by first reversing the string and then replacing A’s with U’s, C’s with G’s, G’s with C’s, and U’s with A’s. For example, the reverse complement of UAGCAC is GUGCUA. A string is a palindrome if it’s the same as its reverse. A string is called self-conjugate if it’s the same as its reverse complement. For example, UAGGAU is a palindrome and UAGCUA is self-conjugate. How many six letter strings with just the characters A, C, G (no U’s) are either palindromes or self-conjugate? [b]p5.[/b] A scooter has $2$ wheels, a chair has $6$ wheels, and a spaceship has $11$ wheels. If there are $10$ of these objects, with a total of $50$ wheels, how many chairs are there? [b]p6.[/b] How many proper subsets of $\{1, 2, 3, 4, 5, 6\}$ are there such that the sum of the elements in the subset equal twice a number in the subset? [b]p7.[/b] A circle and square share the same center and area. The circle has radius $1$ and intersects the square on one side at points $A$ and $B$. What is the length of $\overline{AB}$ ? [b]p8. [/b]Inside a circle, chords $AB$ and $CD$ intersect at $P$ in right angles. Given that $AP = 6$, $BP = 12$ and $CD = 15$, find the radius of the circle. [b]p9.[/b] Steven makes nonstandard checkerboards that have $29$ squares on each side. The checkerboards have a black square in every corner and alternate red and black squares along every row and column. How many black squares are there on such a checkerboard? [b]p10.[/b] John is organizing a race around a circular track and wants to put $3$ water stations at $9$ possible spots around the track. He doesn’t want any $2$ water stations to be next to each other because that would be inefficient. How many ways are possible? [b]p11.[/b] In square $ABCD$, point $E$ is chosen such that $CDE$ is an equilateral triangle. Extend $CE$ and $DE$ to $F$ and $G$ on $AB$. Find the ratio of the area of $\vartriangle EFG$ to the area of $\vartriangle CDE$. [b]p12.[/b] Let $S$ be the number of integers from $2$ to $8462$ (inclusive) which does not contain the digit $1,3,5,7,9$. What is $S$? [b]p13.[/b] Let x, y be non zero solutions to $x^2 + xy + y^2 = 0$. Find $\frac{x^{2016} + (xy)^{1008} + y^{2016}}{(x + y)^{2016}}$ . [b]p14.[/b] A chess contest is held among $10$ players in a single round (each of two players will have a match). The winner of each game earns $2$ points while loser earns none, and each of the two players will get $1$ point for a draw. After the contest, none of the $10$ players gets the same score, and the player of the second place gets a score that equals to $4/5$ of the sum of the last $5$ players. What is the score of the second-place player? [b]p15.[/b] Consider the sequence of positive integers generated by the following formula $a_1 = 3$, $a_{n+1} = a_n + a^2_n$ for $n = 2, 3, ...$ What is the tens digit of $a_{1007}$? [b]p16.[/b] Let $(x, y, z)$ be integer solutions to the following system of equations $x^2z + y^2z + 4xy = 48$ $x^2 + y^2 + xyz = 24$ Find $\sum x + y + z$ where the sum runs over all possible $(x, y, z)$. [b]p17.[/b] Given that $x + y = a$ and $xy = b$ and $1 \le a, b \le 50$, what is the sum of all a such that $x^4 + y^4 - 2x^2y^2$ is a prime squared? [b]p18.[/b] In $\vartriangle ABC$, $M$ is the midpoint of $\overline{AB}$, point $N$ is on side $\overline{BC}$. Line segments $\overline{AN}$ and $\overline{CM}$ intersect at $O$. If $AO = 12$, $CO = 6$, and $ON = 4$, what is the length of $OM$? [b]p19.[/b] Consider the following linear system of equations. $1 + a + b + c + d = 1$ $16 + 8a + 4b + 2c + d = 2$ $81 + 27a + 9b + 3c + d = 3$ $256 + 64a + 16b + 4c + d = 4$ Find $a - b + c - d$. [b]p20.[/b] Consider flipping a fair coin $ 8$ times. How many sequences of coin flips are there such that the string HHH never occurs? PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

A circle is inscribed in a triangle with side lengths $8$, $13$, and $17$. Let the segments of the side of length $8$, made by a point of tangency, be $r$ and $s$, with $r<s$. What is the ratio $r:s$? ${{ \textbf{(A)}\ 1:3 \qquad\textbf{(B)}\ 2:5 \qquad\textbf{(C)}\ 1:2 \qquad\textbf{(D)}\ 2:3 }\qquad\textbf{(E)}\ 3:4 } $

Each day for four days, Linda traveled for one hour at a speed that resulted in her traveling one mile in an integer number of minutes. Each day after the first, her speed decreased so that the number of minutes to travel one mile increased by 5 minutes over the preceding day. Each of the four days, her distance traveled was also an integer number of miles. What was the total number of miles for the four trips? $\textbf{(A) }10\qquad\textbf{(B) }15\qquad\textbf{(C) }25\qquad\textbf{(D) }50\qquad\textbf{(E) }82$

$n$ cars are racing. At first they have a particular order. At each moment a car may overtake another car. No two overtaking actions occur at the same time, and except moments a car is passing another, the cars always have an order. A set of overtaking actions is called "small" if any car overtakes at most once. A set of overtaking actions is called "complete" if any car overtakes exactly once. If $F$ is the set of all possible orders of the cars after a small set of overtaking actions and $G$ is the set of all possible orders of the cars after a complete set of overtaking actions, prove that \[\mid F\mid=2\mid G\mid\] (20 points) [i]Proposed by Morteza Saghafian[/i]

Seven cubes, whose volumes are $1$, $8$, $27$, $64$, $125$, $216$, and $343$ cubic units, are stacked vertically to form a tower in which the volumes of the cubes decrease from bottom to top. Except for the bottom cube, the bottom face of each cube lies completely on top of the cube below it. What is the total surface area of the tower (including the bottom) in square units? $\textbf{(A) } 644 \qquad \textbf{(B) } 658 \qquad \textbf{(C) } 664 \qquad \textbf{(D) } 720 \qquad \textbf{(E) } 749$

For some positive integer $n,$ Elmo writes down the equation \[x_1+x_2+\dots+x_n=x_1+x_2+\dots+x_n.\] Elmo inserts at least one $f$ to the left side of the equation and adds parentheses to create a valid functional equation. For example, if $n=3,$ Elmo could have created the equation \[f(x_1+f(f(x_2)+x_3))=x_1+x_2+x_3.\] Cookie Monster comes up with a function $f: \mathbb{Q}\to\mathbb{Q}$ which is a solution to Elmo's functional equation. (In other words, Elmo's equation is satisfied for all choices of $x_1,\dots,x_n\in\mathbb{Q})$. Is it possible that there is no integer $k$ (possibly depending on $f$) such that $f^k(x)=x$ for all $x$? [i]Srinivas Arun[/i]

Let $ABC$ be a triangle with incenter $I$ such that $AB=20$ and $AC=19$. Point $P \neq A$ lies on line $AB$ and point $Q \neq A$ lies on line $AC$. Suppose that $IA=IP=IQ$ and that line $PQ$ passes through the midpoint of side $BC$. Suppose that $BC=\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $100m+n$. [i]Proposed by Ankit Bisain[/i]

[b]i.)[/b] Let $ABC$ be a triangle with $AB = 12$ and $AC = 16.$ Suppose $M$ is the midpoint of side $BC$ and points $E$ and $F$ are chosen on sides $AC$ and $AB$, respectively, and suppose that lines $EF$ and $AM$ intersect at $G.$ If $AE = 2 \cdot AF$ then find the ratio \[ \frac{EG}{GF} \] [b]ii.)[/b] Let $E$ be a point external to a circle and suppose that two chords $EAB$ and $EDC$ meet at angle of $40^{\circ}.$ If $AB = BC = CD$ find the size of angle $ACD.$

Given is an acute triangle $ABC$. $M$ is an arbitrary point at the side $AB$ and $N$ is midpoint of $AC$. The foots of the perpendiculars from $A$ to $MC$ and $MN$ are points $P$ and $Q$. Prove that center of the circumcircle of triangle $PQN$ lies on the fixed line for all points $M$ from the side $AB$.

Let $ b$ be an even positive integer for which there exists a natural number n such that $ n>1$ and $ \frac{b^n\minus{}1}{b\minus{}1}$ is a perfect square. Prove that $ b$ is divisible by 8.

For any set $S$ of five points in the plane, no three of which are collinear, let $M(S)$ and $m(S)$ denote the greatest and smallest areas, respectively, of triangles determined by three points from $S$. What is the minimum possible value of $M(S)/m(S)$ ?

A right-angled triangle $ABC$ is given. Its catheus $AB$ is the base of a regular triangle $ADB$ lying in the exterior of $ABC$, and its hypotenuse $AC$ is the base of a regular triangle $AEC$ lying in the interior of $ABC$. Lines $DE$ and $AB$ meet at point $M$. The whole configuration except points $A$ and $B$ was erased. Restore the point $M$.

Find all pairs $(m, n)$ of positive integers $m$ and $n$ for which one has $$\sqrt{ m^2 - 4} < 2\sqrt{n} - m < \sqrt{ m^2 - 2}$$

This year, some contestants at the Memorial Contest ABC are friends with each other (friendship is always mutual). For each contestant $X$, let $t(X)$ be the total score that this contestant achieved in previous years before this contest. It is known that the following statements are true: $1)$ For any two friends $X'$ and $X''$, we have $t(X') \neq t(X''),$ $2)$ For every contestant $X$, the set $\{ t(Y) : Y \text{ is a friend of } X \}$ consists of consecutive integers. The organizers want to distribute the contestants into contest halls in such a way that no two friends are in the same hall. What is the minimal number of halls they need?

Which of the following sets of $ x$-values satisfy the inequality $ 2x^2 \plus{} x < 6?$ $ \textbf{(A)}\ \minus{} 2 < x < \frac{3}{2} \qquad \textbf{(B)}\ x > \frac32 \text{ or }x < \minus{} 2 \qquad \textbf{(C)}\ x < \frac32 \qquad \textbf{(D)}\ \frac32 < x < 2 \qquad \textbf{(E)}\ x < \minus{} 2$

Goldfish are sold at $ 15$ cents each. The rectangular coordinate graph showing the cost of $ 1$ to $ 12$ goldfish is: $ \textbf{(A)}\ \text{a straight line segment} \qquad \\ \textbf{(B)}\ \text{a set of horizontal parallel line segments}\qquad \\ \textbf{(C)}\ \text{a set of vertical parallel line segments}\qquad \\ \textbf{(D)}\ \text{a finite set of distinct points}\qquad \textbf{(E)}\ \text{a straight line}$

Find all prime number $p$, such that there exist an infinite number of positive integer $n$ satisfying the following condition: $p|n^{ n+1}+(n+1)^n.$ (September 29, 2012, Hohhot)

Let $f:[0,1]\rightarrow [0,1]$ a continuous function in $0$ and in $1$, which has one-side limits in any point and $f(x-0)\le f(x)\le f(x+0),\ (\forall)x\in (0,1)$. Prove that: a)for the set $A=\{x\in [0,1]\ |\ f(x)\ge x\}$, we have $\sup A\in A$. b)there is $x_0\in [0,1]$ such that $f(x_0)=x_0$. [i]Mihai Piticari[/i]

Let $n,s,$ and $t$ be positive integers and $0<\lambda<1.$ A simple graph on $n$ vertices with at least $\lambda n^2$ edges is given. We say that $(x_1,\ldots,x_s,y_1,\ldots,y_t)$ is a [i]good intersection[/i] if letters $x_i$ and $y_j$ denote not necessarily distinct vertices and every $x_iy_j$ is an edge of the graph $(1\leq i\leq s,$ $1\leq j\leq t).$ Prove that the number of good insertions is at least $\lambda^{st}n^{s+t}.$ [i]Proposed by Kada Williams, Cambridge[/i]

If $ n$ is a real number, then the simultaneous system $ nx \plus{} y \equal{} 1$ $ ny \plus{} z \equal{} 1$ $ x \plus{} nz \equal{} 1$ has no solution if and only if $ n$ is equal to $ \textbf{(A)}\ \minus{}1 \qquad \textbf{(B)}\ 0 \qquad \textbf{(C)}\ 1 \qquad \textbf{(D)}\ 0 \text{ or } 1 \qquad \textbf{(E)}\ \frac12$

Show that if $2 + 2\sqrt{28n^2 + 1}$ is an integer, then it is a square (for $n$ an integer).

Find all $f(x)\in \mathbb Z (x)$ that satisfies the following condition, with the lowest degree. [b]Condition[/b]: There exists $g(x),h(x)\in \mathbb Z (x)$ such that $$f(x)^4+2f(x)+2=(x^4+2x^2+2)g(x)+3h(x)$$.