Found problems: 158
2017 ASDAN Math Tournament, 13
Let $S_1$ be a square of side length $3$. For $i=2,3,4,\dots$, inscribe a square $S_i$ inside $S_{i-1}$ such that the sides of the inner square form four $30^\circ-60^\circ-90^\circ$ triangles with the outer square. Compute the total sum
$$\sum_{i=1}^\infty\text{area}(S_i).$$
2017 ASDAN Math Tournament, 7
Compute
$$\lim_{t\rightarrow0}\int_0^t\frac{x^2+4x+4}{\sqrt{t^2-x^2}}dx.$$
2017 CMIMC Combinatorics, 1
Robert colors each square in an empty 3 by 3 grid either red or green. Find the number of colorings such that no row or column contains more than one green square.
2017 ASDAN Math Tournament, 5
Let $\alpha$ and $\beta$ be the two roots of $x^2+2017x+k$. What is the sum of the possible values of $k$ so that the lines
\begin{align*}
y&=2\alpha x+2017^2\\
y&=3\alpha x+2017^3
\end{align*}
are perpendicular?
2017 ASDAN Math Tournament, 2
Let $f(x)=x^2$ and let $g(x)=x+1$. Let $h(x)=f(g(x))$. Compute $h'(1)$.
2017 ASDAN Math Tournament, 25
Consider the sequence $\{a_n\}$ defined so that $a_n$ is the leftmost digit of $2^n$. The first few terms of this sequence are $1,2,4,8,1,3,6,\dots$. For how many $0\le n\le100000$ is $a_n=1$? If $C$ is the correct answer and $A$ is your answer, then your score will be rounded up from $\max\left(0,25-\tfrac{1}{6}\sqrt{|A-C|}\right)$.
2017 ASDAN Math Tournament, 10
Let $\zeta=e^{2\pi i/36}$. Compute
$$\prod_{\stackrel{a=1}{\gcd(a,36)=1}}^{35}(\zeta^a-2).$$
2017 CMIMC Combinatorics, 3
Annie stands at one vertex of a regular hexagon. Every second, she moves independently to one of the two vertices adjacent to her, each with equal probability. Determine the probability that she is at her starting position after ten seconds.
2017 ASDAN Math Tournament, 1
Compute
$$\int_0^6\frac{x-3}{x^2-6x-7}dx.$$
2017 India National Olympiad, 2
Suppose $n \ge 0$ is an integer and all the roots of $x^3 +
\alpha x + 4 - ( 2 \times 2016^n) = 0$ are integers. Find all possible values of $\alpha$.
2017 ASDAN Math Tournament, 2
Two distinct positive factors of $144$ are selected at random. What is the probability that their product is greater than $144$?
2017 CMI B.Sc. Entrance Exam, 3
Let $p(x)$ be a polynomial of degree strictly less than $100$ and such that it does not have $(x^3-x)$ as a factor. If $$\frac{d^{100}}{dx^{100}}\bigg(\frac{p(x)}{x^3-x}\bigg)=\frac{f(x)}{g(x)}$$ for some polynomials $f(x)$ and $g(x)$ then find the smallest possible degree of $f(x)$.
2017 ASDAN Math Tournament, 1
What is the surface area of a cube with volume $64$?
2017 CMIMC Geometry, 5
Two circles $\omega_1$ and $\omega_2$ are said to be $\textit{orthogonal}$ if they intersect each other at right angles. In other words, for any point $P$ lying on both $\omega_1$ and $\omega_2$, if $\ell_1$ is the line tangent to $\omega_1$ at $P$ and $\ell_2$ is the line tangent to $\omega_2$ at $P$, then $\ell_1\perp \ell_2$. (Two circles which do not intersect are not orthogonal.)
Let $\triangle ABC$ be a triangle with area $20$. Orthogonal circles $\omega_B$ and $\omega_C$ are drawn with $\omega_B$ centered at $B$ and $\omega_C$ centered at $C$. Points $T_B$ and $T_C$ are placed on $\omega_B$ and $\omega_C$ respectively such that $AT_B$ is tangent to $\omega_B$ and $AT_C$ is tangent to $\omega_C$. If $AT_B = 7$ and $AT_C = 11$, what is $\tan\angle BAC$?
2017 ASDAN Math Tournament, 14
What are the last two digits of $2017^{2017}$?
2017 ASDAN Math Tournament, 3
What is the remainder when $2^{1023}$ is divided by $1023$?
2017 CMIMC Algebra, 5
The set $S$ of positive real numbers $x$ such that
\[ \left\lfloor\frac{2x}{5}\right\rfloor + \left\lfloor\frac{3x}{5}\right\rfloor + 1 = \left\lfloor x\right\rfloor \]
can be written as $S = \bigcup_{j = 1}^{\infty} I_{j}$, where the $I_{i}$ are disjoint intervals of the form $[a_{i}, b_{i}) = \{x \, | \, a_i \leq x < b_i\}$ and $b_{i} \leq a_{i+1}$ for all $i \geq 1$. Find $\sum_{i=1}^{2017} (b_{i} - a_{i})$.
2017 ASDAN Math Tournament, 5
A $\textit{shuffle}$ is a permutation of the integers $1,2,3,4,5$. More formally, a shuffle is a function $f:\{1,2,3,4,5\}\rightarrow\{1,2,3,4,5\}$ such that if $i\neq j$ then $f(i)\neq f(j)$. For example, $12345\mapsto23154$ denotes a shuffle $f$ so that $f(1)=2$, $f(2)=3$, $f(3)=1$, $f(4)=5$, and $f(5)=4$. A shuffle can be repeated some number of times to obtain another shuffle. For example, if $f$ is the shuffle $12345\mapsto23154$ from above, then repeating $f$ twice gives the shuffle $g(x)=f(f(x))$ which is $12345\mapsto31245$. How many shuffles are there that, when repeated $6$ times, give the shuffle $12345\mapsto12345$?
2017 CMIMC Number Theory, 1
There exist two distinct positive integers, both of which are divisors of $10^{10}$, with sum equal to $157$. What are they?
2017 ASDAN Math Tournament, 19
How many ways can you tile a $2\times5$ rectangle with $2\times1$ dominoes of $4$ different colors if no two dominoes of the same color may be adjacent?
2017 CMIMC Team, 5
We have four registers, $R_1,R_2,R_3,R_4$, such that $R_i$ initially contains the number $i$ for $1\le i\le4$. We are allowed two operations:
[list]
[*] Simultaneously swap the contents of $R_1$ and $R_3$ as well as $R_2$ and $R_4$.
[*] Simultaneously transfer the contents of $R_2$ to $R_3$, the contents of $R_3$ to $R_4$, and the contents of $R_4$ to $R_2$. (For example if we do this once then $(R_1,R_2,R_3,R_4)=(1,4,2,3)$.)
[/list]
Using these two operations as many times as desired and in whatever order, what is the total number of possible outcomes?
2017 ASDAN Math Tournament, 18
Find the sum of all integers $0\le a \le124$ so that $a^3-2$ is a multiple of $125$.
2017 Romania Team Selection Test, P2
Let $n$ be a positive integer, and let $S_n$ be the set of all permutations of $1,2,...,n$. let $k$ be a non-negative integer, let $a_{n,k}$ be the number of even permutations $\sigma$ in $S_n$ such that $\sum_{i=1}^{n}|\sigma(i)-i|=2k$ and $b_{n,k}$ be the number of odd permutations $\sigma$ in $S_n$ such that $\sum_{i=1}^{n}|\sigma(i)-i|=2k$. Evaluate $a_{n,k}-b_{n,k}$.
[i]* * *[/i]
2017 ASDAN Math Tournament, 2
Circles $A,B,C$ are externally tangent. Let $P$ be the tangent point between circles $A$ and $C$, and $Q$ be the tangent point between circles $B$ and $C$. Let $r_C$ be the radius of circle $C$. If the chord connecting $P$ and $Q$ has length $r_C\sqrt{2}$ and the radii of circles $A$ and $B$ are $4$ and $7$, respectively, what is the radius of circle $C$?
2017 CMIMC Individual Finals, 3
Let $n=2017$ and $x_1,\dots,x_n$ be boolean variables. An \emph{$7$-CNF clause} is an expression of the form $\phi_1(x_{i_1})+\dots+\phi_7(x_{i_7})$, where $\phi_1,\dots,\phi_7$ are each either the function $f(x)=x$ or $f(x)=1-x$, and $i_1,i_2,\dots,i_7\in\{1,2,\dots,n\}$. For example, $x_1+(1-x_1)+(1-x_3)+x_2+x_4+(1-x_3)+x_{12}$ is a $7$-CNF clause. What's the smallest number $k$ for which there exist $7$-CNF clauses $f_1,\dots,f_k$ such that \[f(x_1,\dots,x_n):=f_1(x_1,\dots,x_n)\cdots f_k(x_1,\dots,x_n)\] is $0$ for all values of $(x_1,\dots,x_n)\in\{0,1\}^n$?