[b]p1.[/b] Let $a$ be an integer. How many fractions $\frac{a}{100}$ are greater than $\frac17$ and less than $\frac13$ ?. [b]p2.[/b] Justin Bieber invited Justin Timberlake and Justin Shan to eat sushi. There were $5$ different kinds of fish, $3$ different rice colors, and $11$ different sauces. Justin Shan insisted on a spicy sauce. If the probability of a sushi combination that pleased Justin Shan is $6/11$, then how many non-spicy sauces were there? [b]p3.[/b] A palindrome is any number that reads the same forward and backward (for example, $99$ and $50505$ are palindromes but $2020$ is not). Find the sum of all three-digit palindromes whose tens digit is $5$. [b]p4.[/b] Isaac is given an online quiz for his chemistry class in which he gets multiple tries. The quiz has $64$ multiple choice questions with $4$ choices each. For each of his previous attempts, the computer displays Isaac's answer to that question and whether it was correct or not. Given that Isaac is too lazy to actually read the questions, the maximum number of times he needs to attempt the quiz to guarantee a $100\%$ can be expressed as $2^{2^k}$. Find $k$. [b]p5.[/b] Consider a three-way Venn Diagram composed of three circles of radius $1$. The area of the entire Venn Diagram is of the form $\frac{a}{b}\pi +\sqrt{c}$ for positive integers $a$, $b$, $c$ where $a$, $b$ are relatively prime. Find $a+b+c$. (Each of the circles passes through the center of the other two circles) [b]p6.[/b] The sum of two four-digit numbers is $11044$. None of the digits are repeated and none of the digits are $0$s. Eight of the digits from $1-9$ are represented in these two numbers. Which one is not? [b]p7.[/b] Al wants to buy cookies. He can buy cookies in packs of $13$, $15$, or $17$. What is the maximum number of cookies he can not buy if he must buy a whole number of packs of each size? [b]p8.[/b] Let $\vartriangle ABC$ be a right triangle with base $AB = 2$ and hypotenuse $AC = 4$ and let $AD$ be a median of $\vartriangle ABC$. Now, let $BE$ be an altitude in $\vartriangle ABD$ and let $DF$ be an altitude in $\vartriangle ADC$. The quantity $(BE)^2 - (DF)^2$ can be expressed as a common fraction $\frac{a}{b}$ in lowest terms. Find $a + b$. [b]p9.[/b] Let $P(x)$ be a monic cubic polynomial with roots $r$, $s$, $t$, where $t$ is real. Suppose that $r + s + 2t = 8$, $2rs + rt + st = 12$ and $rst = 9$. Find $|P(2)|$. [b]p10.[/b] Let S be the set $\{1, 2,..., 21\}$. How many $11$-element subsets $T$ of $S$ are there such that there does not exist two distinct elements of $T$ such that one divides the other? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

If $\left|x\right|-x+y=42$ and $x+\left|y\right|+y=24$, then what is the value of $x+y$? Express your answer in simplest terms. $\text{(A) }-4\qquad\text{(B) }\frac{26}{5}\qquad\text{(C) }6\qquad\text{(D) }10\qquad\text{(E) }18$

Let $X$ be a point lying in the interior of the acute triangle $ABC$ such that \begin{align*} \sphericalangle BAX = 2\sphericalangle XBA \ \ \ \ \hbox{and} \ \ \ \ \sphericalangle XAC = 2\sphericalangle ACX. \end{align*} Denote by $M$ the midpoint of the arc $BC$ of the circumcircle $(ABC)$ containing $A$. Prove that $XM=XA$.

A [i]complete [/i] number is a $9$ digit number that contains each of the digits $1$ to $9$ exactly once. The [i]difference [/i] number of a number $N$ is the number you get by taking the differences of consecutive digits in $N$ and then stringing these digits together. For instance, the [i]difference [/i] number of $25143$ is equal to $3431$. The [i]complete [/i] number $124356879$ has the additional property that its [i]difference [/i] number, $12121212$, consists of digits alternating between $1$ and $2$. Determine all $a$ with $3 \le a \le 9$ for which there exists a [i]complete [/i] number $N$ with the additional property that the digits of its [i]difference[/i] number alternate between $1 $ and $a$.

Let $M$ be a set of 999 points in the plane with the property: For any 3 distinct points in $M$ we can choose two of them, such that the distance between them is less than $1$. a)Prove that there exists a disc of radius not greater than 1 that covers at least 500 points in $M$. b)Is it true that there always exists a disc of radius not greater than 1 that covers at least 501 points in $M$?

A cube of side $n$ is cut into $n^3$ unit cubes, and m of these cubes are marked so that the centers of any three marked cubes do not form a right-angled triangle with legs parallel to sides of the cube. Find the maximum possible value of $m$.

Prove that for any nonisosceles triangle $l_1^2>\sqrt3 S>l_2^2$, where $l_1, l_2$ are the greatest and the smallest bisectors of the triangle and $S$ is its area.

The circle $\Omega$ with center $O$ is circumscribed to acute triangle $ABC$. Let $P$ be a point on the circumscribed circle of $OBC$, such that $P$ is inside $ABC$ and is different from $B$ and $C$. Bisectors of angles $BPA$ and $CPA$ intersect the sides $AB$ and $AC$ in points $E$ and $F.$ Prove that the incenters of triangles $PEF, PCA$ and $PBA$ are collinear.

Let $ABCDE$ be a regular pentagon, and let $F$ be a point on $\overline{AB}$ with $\angle CDF=55^\circ$. Suppose $\overline{FC}$ and $\overline{BE}$ meet at $G$, and select $H$ on the extension of $\overline{CE}$ past $E$ such that $\angle DHE=\angle FDG$. Find the measure of $\angle GHD$, in degrees. [i]Proposed by David Stoner[/i]

Let $P(x)$ be a polynomial with integer coefficients for which there exists a positive integer n such that the real parts of all roots of $P(x)$ are less than $n- \frac{1}{2}$ , polynomial $x-n+1$ does not divide $P(x)$, and $P(n)$ is a prime number. Prove that the polynomial $P(x)$ is irreducible (over $Z[x]$).

A region in space is determined by a sphere with center $O$ and radius $R$, and a cone with vertex $O$ which intersects the sphere in a circle of radius $r$. Find the maximum volume of a cylinder contained in this region, having the same axis as the cone.

Determine the integral part of $A$, where $A =\frac{1}{672}+\frac{1}{673}+... +\frac{1}{2014}$

A cone-shaped mountain has its base on the ocean floor and has a height of $ 8000$ feet. The top $ \frac{1}{8}$ of the volume of the mountain is above water. What is the depth of the ocean at the base of the mountain, in feet? $ \textbf{(A)}\ 4000 \qquad \textbf{(B)}\ 2000(4\minus{}\sqrt{2}) \qquad \textbf{(C)}\ 6000 \qquad \textbf{(D)}\ 6400 \qquad \textbf{(E)}\ 7000$

There is a sequence with $a(2)=0$, $a(3)=1$ and $a(n)=a\left(\left\lfloor\dfrac n2\right\rfloor\right)+a\left(\left\lceil\dfrac n2\right\rceil\right)$ for $n\geq 4$. Find $a(2014)$. [Note that $\left\lfloor\dfrac n2\right\rfloor$ and $\left\lceil\dfrac n2\right\rceil$ denote the floor function (largest integer $\leq\tfrac n2$) and the ceiling function (smallest integer $\geq\tfrac n2$), respectively.]

If the sum of the slope and the $y$-intercept of a line is $3$, then through which point is the line guaranteed to pass?

Suppose that $(f_{n})_{n=1}^{\infty}$ is a sequence of continuous functions on the interval $[0,1]$ such that $$\int_{0}^{1}f_{m}(x)f_{n}(x) dx= \begin{cases} 1& \text{if}\;n=m\\ 0 & \text{if} \;n\ne m \end{cases}$$ and $\sup\{|f_{n}(x)|: x\in [0,1]\, \text{and}\, n=1,2,\dots\}< \infty$. Show that there exists no subsequence $(f_{n_{k}})$ of $(f_{n})$ such that $\lim_{k\to \infty}f_{n_{k}}(x)$ exist for all $x\in [0,1]$.

Let $n$ be an integer greater than $3$. Prove that all the roots of the polynomial $P(x) = x^n - 5x^{n-1} + 12x^{n-2}- 15x^{n-3} + a_{n-4}x^{n-4} +...+ a_0$ cannot be both real and positive.

A bus that moves along a 100 km route is equipped with a computer, which predicts how much more time is needed to arrive at its final destination. This prediction is made on the assumption that the average speed of the bus in the remaining part of the route is the same as that in the part already covered. Forty minutes after the departure of the bus, the computer predicts that the remaining travelling time will be 1 hour. And this predicted time remains the same for the next 5 hours. Could this possibly occur? If so, how many kilometers did the bus cover when these 5 hours passed? (Average speed is the number of kilometers covered divided by the time it took to cover them.)

For geometric series $(a_n)$ with all items real, if $S_{10}=10,S_{30}=70$, then $S_{40}=$ $\text{(A)}150\qquad\text{(B)}-200\qquad\text{(C)}150\text{ or }-200\qquad\text{(D)}-50\text{ or }400$ Note: $S_n=\sum_{i=1}^{n}a_i$.

$ 20 $ discs of radius $ 1 $ are bounded by a circle of radius $ 10. $ Show that in the interior of this circle is sufficient space to insert $ 7 $ discs of radius $ \frac{1}{3} $ that doesn't touch any other disc. [i]Flavian Georgescu[/i]

Find all possible pairs of real numbers $ (x, y) $ that satisfy the equalities $ y ^ 2- [x] ^ 2 = 2001 $ and $ x ^ 2 + [y] ^ 2 = 2001 $.

Prove that for every nonnegative integer $n$, the number $7^{7^{n}}+1$ is the product of at least $2n+3$ (not necessarily distinct) primes.

Let $\alpha$ be a root of $x^6-x-1$, and call two polynomials $p$ and $q$ with integer coefficients $\textit{equivalent}$ if $p(\alpha)\equiv q(\alpha)\pmod3$. It is known that every such polynomial is equivalent to exactly one of $0,1,x,x^2,\ldots,x^{727}$. Find the largest integer $n<728$ for which there exists a polynomial $p$ such that $p^3-p-x^n$ is equivalent to $0$.

Prove that from a set of ten distinct two-digit numbers, it is always possible to find two disjoint subsets whose members have the same sum.

Let $ABCD$ be a convex quadrilateral. The common external tangents to circles $(ABC)$ and $(ACD)$ meet at point $E$, the common external tangents to circles $(ABD)$ and $(BCD)$ meet at point $F$. Let $F$ lie on $AC$, prove that $E$ lies on $BD$.