Found problems: 116
2018 ASDAN Math Tournament, 10
Quadrilateral $ABCD$ has the property that $AD = BD = CD$ and $\angle ADB < \angle CDB$. Let the circumcircle of $ABD$ be $O$. $O$ intersects $BC$ at $E$ and $CD$ at $F$. Next, extend $AB$ and $CD$ to intersect at a point $G$. Suppose that $BE = 1$, $EF = 3$, and $F D = 4$. Compute the perimeter of $\vartriangle ADG$.
2018 ASDAN Math Tournament, 3
Compute $ax^{2018}+by^{2018}$, given that there exist real $a$, $b$, $x$, and $y$ which satisfy the following four equations:
\begin{align*}
ax^{2014}+by^{2014}&=6\\
ax^{2015}+by^{2015}&=7\\
ax^{2016}+by^{2016}&=3\\
ax^{2017}+by^{2017}&=50.
\end{align*}
2018 ASDAN Math Tournament, 3
The integers $a, b,$ and $c$ form a strictly increasing geometric sequence. Suppose that $abc = 216$. What is the maximum possible value of $a + b + c$?
2018 ASDAN Math Tournament, 5
In the expansion of $(x + b)^{2018}$, the coefficients of $x^2$ and $x^3$ are equal. Compute $b$.
2018 ASDAN Math Tournament, 4
What is the remainder when $13^{16} + 17^{12}$ is divided by $221$?
MOAA Team Rounds, 2018.9
Quadrilateral $ABCD$ with $AC = 800$ is inscribed in a circle, and $E, W, X, Y, Z$ are the midpoints of segments $BD$, $AB$, $BC$, $CD$, $DA$, respectively. If the circumcenters of $EW Z$ and $EXY$ are $O_1$ and $O_2$, respectively, determine $O_1O_2$.
2018 Turkey Team Selection Test, 2
Find all $f:\mathbb{R}\to\mathbb{R}$ surjective functions such that
$$f(xf(y)+y^2)=f((x+y)^2)-xf(x) $$ for all real numbers $x,y$.
2019 Belarus Team Selection Test, 3.1
Determine all pairs $(n, k)$ of distinct positive integers such that there exists a positive integer $s$ for which the number of divisors of $sn$ and of $sk$ are equal.
2018 ISI Entrance Examination, 4
Let $f:(0,\infty)\to\mathbb{R}$ be a continuous function such that for all $x\in(0,\infty)$, $$f(2x)=f(x)$$
Show that the function $g$ defined by the equation $$g(x)=\int_{x}^{2x} f(t)\frac{dt}{t}~~\text{for}~x>0$$ is a constant function.
MOAA Team Rounds, 2018.5
Mr. DoBa likes to listen to music occasionally while he does his math homework. When he listens to classical music, he solves one problem every $3$ minutes. When he listens to rap music, however, he only solves one problem every $5$ minutes. Mr. DoBa listens to a playlist comprised of $60\%$ classical music and $40\%$ rap music. Each song is exactly $4$ minutes long. Suppose that the expected number of problems he solves in an hour does not depend on whether or not Mr. DoBa is listening to music at any given moment, and let $m$ the average number of problems Mr. DoBa solves per minute when he is not listening to music. Determine the value of $1000m$.
2018 CMIMC Algebra, 9
Suppose $a_0,a_1,\ldots, a_{2018}$ are integers such that \[(x^2-3x+1)^{1009} = \sum_{k=0}^{2018}a_kx^k\] for all real numbers $x$. Compute the remainder when $a_0^2 + a_1^2 + \cdots + a_{2018}^2$ is divided by $2017$.
Russian TST 2019, P1
Determine all pairs $(n, k)$ of distinct positive integers such that there exists a positive integer $s$ for which the number of divisors of $sn$ and of $sk$ are equal.
2018 ASDAN Math Tournament, 1
Moor has $3$ different shirts, labeled $T, E,$ and $A$. Across $5$ days, the only days Moor can wear shirt $T$ are days $2$ and $5$. How many different sequences of shirts can Moor wear across these $5$ days?
2019 Hong Kong TST, 1
Determine all pairs $(n, k)$ of distinct positive integers such that there exists a positive integer $s$ for which the number of divisors of $sn$ and of $sk$ are equal.
2019 Estonia Team Selection Test, 9
Determine all pairs $(n, k)$ of distinct positive integers such that there exists a positive integer $s$ for which the number of divisors of $sn$ and of $sk$ are equal.
2018 ASDAN Math Tournament, 2
Aurick throws $2$ fair $6$-sided dice labeled with the integers from $1$ through $6$. What is the probability that the sum of the rolls is a multiple of $3$?
MOAA Team Rounds, 2018.7
For a positive integer $k$, define the $k$-[i]pop[/i] of a positive integer $n$ as the infinite sequence of integers $a_1, a_2, ...$ such that $a_1 = n$ and $$a_{i+1}= \left\lfloor \frac{a_i}{k} \right\rfloor , i = 1, 2, ..$$
where $ \lfloor x\rfloor $ denotes the greatest integer less than or equal to $x$. Furthermore, define a positive integer $m$ to be $k$-[i]pop avoiding[/i] if $k$ does not divide any nonzero term in the $k$-pop of $m$. For example, $14$ is 3-pop avoiding because $3$ does not divide any nonzero term in the $3$-pop of $14$, which is $14, 4, 1, 0, 0, ....$ Suppose that the number of positive integers less than $13^{2018}$ which are $13$-pop avoiding is equal to N. What is the remainder when $N$ is divided by $1000$?
2018 ISI Entrance Examination, 1
Find all pairs $(x,y)$ with $x,y$ real, satisfying the equations $$\sin\bigg(\frac{x+y}{2}\bigg)=0~,~\vert x\vert+\vert y\vert=1$$
2018 ASDAN Math Tournament, 2
In 3D coordinate space, $O$ is the origin, $A$ lies on the positive $x$-axis, $B$ lies on the positive $y$-axis, and $C$ lies on the positive $z$-axis such that $BO = 2AO$ and $CO = 3AO$. Suppose that a unit cube with sides parallel to the axes can be inscribed within tetrahedron $ABCO$. Compute $AO + BO + CO$.
2018 Bulgaria JBMO TST, Source
For real numbers $a$ and $b$, define
$$f(a,b) = \sqrt{a^2+b^2+26a+86b+2018}.$$
Find the smallest possible value of the expression $$f(a, b) + f (a,-b) + f(-a, b) + f (-a, -b).$$
2019 Estonia Team Selection Test, 9
Determine all pairs $(n, k)$ of distinct positive integers such that there exists a positive integer $s$ for which the number of divisors of $sn$ and of $sk$ are equal.
2018 ASDAN Math Tournament, 6
Sam and Ben are each flipping fair coins. If Sam flips a single coin until he gets a tails, and Ben flips $10$ coins in total, what is the probability Sam and Ben get the same number of heads?
2018 MOAA, 1
In $\vartriangle ABC$, $AB = 3$, $BC = 5$, and $CA = 6$. Points $D$ and $E$ are chosen such that $ACDE$ is a square which does not overlap with $\vartriangle ABC$. The length of $BD$ can be expressed in the form $\sqrt{m + n\sqrt{p}}$, where $m$, $n$, and $p$ are positive integers and $p$ is not divisible by the square of a prime. Determine the value of $m + n + p$.
2018 ASDAN Math Tournament, 2
Given that $\sec x+\tan x=2018$, compute $\csc x+\cot x$.
2018 MOAA, 7
For a positive integer $k$, define the $k$-[i]pop[/i] of a positive integer $n$ as the infinite sequence of integers $a_1, a_2, ...$ such that $a_1 = n$ and $$a_{i+1}= \left\lfloor \frac{a_i}{k} \right\rfloor , i = 1, 2, ..$$
where $ \lfloor x\rfloor $ denotes the greatest integer less than or equal to $x$. Furthermore, define a positive integer $m$ to be $k$-[i]pop avoiding[/i] if $k$ does not divide any nonzero term in the $k$-pop of $m$. For example, $14$ is 3-pop avoiding because $3$ does not divide any nonzero term in the $3$-pop of $14$, which is $14, 4, 1, 0, 0, ....$ Suppose that the number of positive integers less than $13^{2018}$ which are $13$-pop avoiding is equal to N. What is the remainder when $N$ is divided by $1000$?