Found problems: 190
2016 ASDAN Math Tournament, 11
Ebeneezer is painting the edges of a cube. He wants to paint the edges so that the colored edges form a loop that does not intersect itself. For example, the loop should not look like a “figure eight” shape. If two colorings are considered equivalent if there is a rotation of the cubes so that the colored edges are the same, what is the number of possible edge colorings?
2016 ASDAN Math Tournament, 6
Compute all real solutions $(x,y)$ with $x\geq y$ that satisfy the pair of equations
\begin{align*}
xy&=5\\
\frac{x^2+y^2}{x+y}&=3.
\end{align*}
2016 CMIMC, 3
Let $ABC$ be a triangle. The angle bisector of $\angle B$ intersects $AC$ at point $P$, while the angle bisector of $\angle C$ intersects $AB$ at a point $Q$. Suppose the area of $\triangle ABP$ is 27, the area of $\triangle ACQ$ is 32, and the area of $\triangle ABC$ is $72$. The length of $\overline{BC}$ can be written in the form $m\sqrt n$ where $m$ and $n$ are positive integers with $n$ as small as possible. What is $m+n$?
2016 Macedonia JBMO TST, 1
Solve the following equation in the set of integers
$x_{1}^4 + x_{2}^4 +...+ x_{14}^4=2016^3 - 1$.
2016 ASDAN Math Tournament, 18
Compute the number of nonnegative integer triples $(x,y,z)$ which satisfy $4x+2y+z\leq36$.
2016 ASDAN Math Tournament, 10
Using the fact that
$$\sum_{n=1}^\infty\frac{1}{n^2}=\frac{\pi^2}{6},$$
compute
$$\int_0^1(\ln x)\ln(1-x)dx.$$
2016 ASDAN Math Tournament, 23
Find all quadruples of real numbers $(a,b,c,d)$ that satisfy the system of equations:
\begin{align*}
a+4b+8c+4d&=53\\
3a^2+4b^2+12c^2+2d^2&=159\\
9a^3+4b^3+18c^3+d^3&=477.
\end{align*}
2016 CMIMC, 2
Determine the value of the sum \[\left|\sum_{1\leq i<j\leq 50}ij(-1)^{i+j}\right|.\]
2016 CMIMC, 4
Kevin colors three distinct squares in a $3\times 3$ grid red. Given that there exist two uncolored squares such that coloring one of them would create a horizontal or vertical red line, find the number of ways he could have colored the original three squares.
2016 CMIMC, 1
David, when submitting a problem for CMIMC, wrote his answer as $100\tfrac xy$, where $x$ and $y$ are two positive integers with $x<y$. Andrew interpreted the expression as a product of two rational numbers, while Patrick interpreted the answer as a mixed fraction. In this case, Patrick's number was exactly double Andrew's! What is the smallest possible value of $x+y$?
2016 ASDAN Math Tournament, 8
Consider all fractions $\tfrac{a}{b}$ where $1\leq b\leq100$ and $0\leq a\leq b$. Of these fractions, let $\tfrac{m}{n}$ be the smallest fraction such that $\tfrac{m}{n}>\tfrac{2}{7}$. What is $\tfrac{m}{n}$?
2016 CMIMC, 2
Suppose that some real number $x$ satisfies
\[\log_2 x + \log_8 x + \log_{64} x = \log_x 2 + \log_x 16 + \log_x 128.\] Given that the value of $\log_2 x + \log_x 2$ can be expressed as $\tfrac{a\sqrt{b}}{c}$, where $a$ and $c$ are coprime positive integers and $b$ is squarefree, compute $abc$.
2016 ASDAN Math Tournament, 4
At a festival, Jing Jing plays a game where she must knock down ten targets with as few balls as possible. Every time Jing Jing knocks down a target, she can reuse the ball she just threw and does not have to pick up a new ball. Suppose that Jing Jing knocks down each target with a probability of $\tfrac{3}{4}$. Compute the expected number of balls that Jing Jing needs to knock down all ten targets.
2016 ASDAN Math Tournament, 9
An equilateral triangle $\triangle ABC$ with side length $3$ has center $O$. A circle is drawn centered at $O$ with radius $1$. Find the area of the region contained inside both the triangle and circle.
2016 CMIMC, 3
Let $S$ be the set containing all positive integers whose decimal representations contain only 3’s and 7’s, have at most 1998 digits, and have at least one digit appear exactly 999 times. If $N$ denotes the number of elements in $S$, find the remainder when $N$ is divided by 1000.
2016 ASDAN Math Tournament, 1
Pooh has an unlimited supply of $1\times1$, $2\times2$, $3\times3$, and $4\times4$ squares. What is the minimum number of squares he needs to use in order to fully cover a $5\times5$ with no $2$ squares overlapping?
2016 CMIMC, 10
Given $x_0\in\mathbb R$, $f,g:\mathbb R\to\mathbb R$, we define the $\emph{non-redundant binary tree}$ $T(x_0,f,g)$ in the following way:
[list=1]
[*]The tree $T$ initially consists of just $x_0$ at height $0$.
[*]Let $v_0,\dots,v_k$ be the vertices at height $h$. Then the vertices of height $h+1$ are added to $T$ by: for $i=0,1,\dots,k$, $f(v_i)$ is added as a child of $v_i$ if $f(v_i)\not\in T$, and $g(v_i)$ is added as a child of $v_i$ if $g(v_i)\not\in T$.
[/list]
For example, if $f(x)=x+1$ and $g(x)=x-1$, then the first three layers of $T(0,f,g)$ look like:
[asy]
size(100);
draw((-0.1,-0.2)--(-0.4,-0.8),EndArrow(size=3));
draw((0.1,-0.2)--(0.4,-0.8),EndArrow(size=3));
draw((-0.6,-1.2)--(-0.9,-1.8),EndArrow(size=3));
draw((0.6,-1.2)--(0.9,-1.8),EndArrow(size=3));
label("$0$",(0,0));
label("$1$",(-.5,-1));
label("$-1$",(.5,-1));
label("$2$",(-1,-2));
label("$-2$",(1,-2));[/asy]
If $f(x)=1024x-2047\lfloor x/2\rfloor$ and $g(x)=2x-3\lfloor x/2\rfloor+2\lfloor x/4\rfloor$, then how many vertices are in $T(2016,f,g)$?
2017 NIMO Problems, 4
For how many positive integers $100 < n \le 10000$ does $\lfloor \sqrt{n-100} \rfloor$ divide $n$?
[i]Proposed by Michael Tang[/i]
2016 CMIMC, 10
Denote by $F_0(x)$, $F_1(x)$, $\ldots$ the sequence of Fibonacci polynomials, which satisfy the recurrence $F_0(x)=1$, $F_1(x)=x$, and $F_n(x)=xF_{n-1}(x)+F_{n-2}(x)$ for all $n\geq 2$. It is given that there exist unique integers $\lambda_0$, $\lambda_1$, $\ldots$, $\lambda_{1000}$ such that \[x^{1000}=\sum_{i=0}^{1000}\lambda_iF_i(x)\] for all real $x$. For which integer $k$ is $|\lambda_k|$ maximized?
2016 ASDAN Math Tournament, 16
Let the notation $\underline{ABC}$ denote the number compromised of the digits $A$, $B$, and $C$ with $0\leq A,B,C\leq9$. That is, $\underline{ABC}=100A+10B+C$ and $\underline{CCAAC}=10000C+1000C+100A+10A+C$. Now, if $(\underline{ABC})^2=\underline{CCAAC}$, where $A$, $B$, and $C$ are distinct nonzero digits, find the $3$ digit number $\underline{ABC}$.
2016 ASDAN Math Tournament, 14
Suppose that $x,y,z$ are positive real numbers that satisfy
\begin{align*}
x+y+z&=xyz\\
\frac{x^2}{16(1+x^2)}=\frac{y^2}{25(1+y^2)}&=\frac{z^2}{36(1+z^2)}.
\end{align*}
Compute
$$\frac{x^2(1+x^2)^2}{x^2(1+z^2)^2}.$$
2016 CMIMC, 5
Determine the sum of the positive integers $n$ such that there exist primes $p,q,r$ satisfying $p^{n} + q^{2} = r^{2}$.
2016 ASDAN Math Tournament, 5
Find
$$\lim_{x\rightarrow0}\frac{\sin(x)-x}{x\cos(x)-x}.$$
2016 CMIMC, 2
Right isosceles triangle $T$ is placed in the first quadrant of the coordinate plane. Suppose that the projection of $T$ onto the $x$-axis has length $6$, while the projection of $T$ onto the $y$-axis has length $8$. What is the sum of all possible areas of the triangle $T$?
[asy]
import olympiad;
size(120);
defaultpen(linewidth(0.8));
pair A = (0.9,0.6), B = (1.7, 0.8), C = rotate(270, B)*A;
pair PAx = (A.x,0), PBx = (B.x,0), PAy = (0, A.y), PCy = (0, C.y);
draw(PAx--A--PAy^^PCy--C^^PBx--B, linetype("4 4"));
draw(rightanglemark(A,B,C,3));
draw(A--B--C--cycle);
draw((0,2)--(0,0)--(2,0),Arrows(size=8));
label("$6$",(PAx+PBx)/2,S);
label("$8$",(PAy+PCy)/2,W);
[/asy]
2016 ASDAN Math Tournament, 3
If $f(x)=e^xg(x)$, where $g(2)=1$ and $g'(2)=2$, find $f'(2)$.