Prove that every positive integer which is not a member of the infinite set below is equal to the sum of two or more distinct numbers of the set \[\{ 3,-2, 2^{2}3,-2^{3}, \cdots, 2^{2k}3,-2^{2k+1}, \cdots \}=\{3,-2, 12,-8, 48,-32, 192, \cdots \}.\]

[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].

Heather compares the price of a new computer at two different stores. Store A offers $ 15\%$ off the sticker price followed by a $ \$90$ rebate, and store B offers $ 25\%$ off the same sticker price with no rebate. Heather saves $ \$15$ by buying the computer at store A instead of store B. What is the sticker price of the computer, in dollars? $ \textbf{(A)}\ 750 \qquad \textbf{(B)}\ 900 \qquad \textbf{(C)}\ 1000 \qquad \textbf{(D)}\ 1050 \qquad \textbf{(E)}\ 1500$

Victor shuffles a standard 54-card deck then flips over cards one at a time onto a pile stopping after the first ace. However, if he ever reveals a joker he discards the entire pile, including the joker, and starts a new pile; for example, if the sequence of cards is 2-3-Joker-A, the pile ends with one card in it. Find the expected number of cards in the end pile.

We call a subset $S$ of vertices of graph $G$, $2$-dominating, if and only if for every vertex $v\notin S,v\in G$, $v$ has at least two neighbors in $S$. prove that every $r$-regular $(r\ge3)$ graph has a $2$-dominating set with size at most $\frac{n(1+\ln(r))}{r}$.(15 points) time of this exam was 3 hours

The sum of two angles of a triangle is $\frac{6}{5}$ of a right angle, and one of these two angles is $30 ^\circ$ larger than the other. What is the degree measure of the largest angle in the triangle? $ \textbf{(A)}\ 69 \qquad \textbf{(B)}\ 72 \qquad \textbf{(C)}\ 90 \qquad \textbf{(D)}\ 102 \qquad \textbf{(E)}\ 108 $

$P$ is a monic polynomial of odd degree greater than one such that there exists a function $f : \mathbb{R} \rightarrow \mathbb{N}$ such that for each $x \in \mathbb{R}$ ,\[f(P(x))=P(f(x))\] (a) Prove that there are a finite number of natural numbers in range of $f$. (b) Prove that if $f$ is not constant then the equation $P(x)-x=0$ has at least two real solutions. (c) For each natural $n>1$ prove that there exists a function $f : \mathbb{R} \rightarrow \mathbb{N}$ and a monic polynomial of odd degree greater than one $P$ such that for each $x \in \mathbb{R}$ ,\[f(P(x))=P(f(x))\] and range of $f$ contains exactly $n$ different numbers. Time allowed for this problem was 105 minutes.

Given an arbitrary set of $2k+1$ integers $\{a_1,a_2,...,a_{2k+1}\}$. We make a new set $$ \{(a_1+a_2)/2, (a_2+a_3)/2, (a_{2k}+a_{2k+1})/2, (a_{2k+1}+a_1)/2\}$$ and a new one, according to the same rule, and so on... Prove that if we obtain integers only, the initial set consisted of equal integers only.

There are two circles with a common center $ O $ and a point $ A $. Construct a circle with center $ A $ intersecting the given circles at points $ M $ and $ N $ such that the line $ MN $ passes through point $ O $.

Show that $1 + 11^{11} + 111^{111} + 1111^{1111} +...+ 1111111111^{1111111111}$ is divisible by $100$.

Given an acute triangle $ABC$ . It's altitudes $AA_1 , BB_1$ and $CC_1$ intersect at a point $H$ , the orthocenter of $\vartriangle ABC$. Let the lines $B_1C_1$ and $AA_1$ intersect at a point $K$, point $M$ be the midpoint of the segment $AH$. Prove that the circumscribed circle of $\vartriangle MKB_1$ touches the circumscribed circle of $\vartriangle ABC$ if and only if $BA1 = 3A1C$. (Bondarenko Mykhailo)

In quadrilateral $ABCD$ with incenter $I$, points $W,X,Y,Z$ lie on sides $AB, BC,CD,DA$ with $AZ=AW$, $BW=BX$, $CX=CY$, $DY=DZ$. Define $T=\overline{AC}\cap\overline{BD}$ and $L=\overline{WY}\cap\overline{XZ}$. Let points $O_a,O_b,O_c,O_d$ be such that $\angle O_aZA=\angle O_aWA=90^\circ$ (and cyclic variants), and $G=\overline{O_aO_c}\cap\overline{O_bO_d}$. Prove that $\overline{IL}\parallel\overline{TG}$. [i]Neal Yan[/i]

Let $X_1$, $X_2$, ..., $X_{2012}$ be chosen independently and uniformly at random from the interval $(0,1]$. In other words, for each $X_n$, the probability that it is in the interval $(a,b]$ is $b-a$. Compute the probability that $\lceil\log_2 X_1\rceil+\lceil\log_4 X_2\rceil+\cdots+\lceil\log_{1024} X_{2012}\rceil$ is even. (Note: For any real number $a$, $\lceil a \rceil$ is defined as the smallest integer not less than $a$.)

Professor Conway collects a total of $58$ midterms from the two sections of his introductory linear algebra course. He notices that the number of midterms from the smaller section is equal to the product of the digits of the number of midterms from his larger section. Assuming that every student handed in a midterm, how many students are there in the smaller section?

In equilateral triangle $ABC$ with side length $2$, let the parabola with focus $A$ and directrix $BC$ intersect sides $AB$ and $AC$ at $A_1$ and $A_2$, respectively. Similarly, let the parabola with focus $B$ and directrix $CA$ intersect sides $BC$ and $BA$ at $B_1$ and $B_2$, respectively. Finally, let the parabola with focus $C$ and directrix $AB$ intersect sides $CA$ and $C_B$ at $C_1$ and $C_2$, respectively. Find the perimeter of the triangle formed by lines $A_1A_2$, $B_1B_2$, $C_1C_2$.

Let $BC$ be a chord not passing through the center of a circle $\omega$. Point $A$ varies on the major arc $BC$. Let $E$ and $F$ be the projection of $B$ onto $AC$, and of $C$ onto $AB$ respectively. The tangents to the circumcircle of $\vartriangle AEF$ at $E, F$ intersect at $P$. (a) Prove that $P$ is independent of the choice of $A$. (b) Let $H$ be the orthocenter of $\vartriangle ABC$, and let $T$ be the intersection of $EF$ and $BC$. Prove that $TH \perp AP$.