This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

AND:
OR:
NO:

Found problems: 158

2017 ASDAN Math Tournament, 4
Tags: 2017 , Algebra Test
What is the maximum possible value for the sum of the squares of the roots of $x^4+ax^3+bx^2+cx+d$ where $a$, $b$, $c$, and $d$ are $2$, $0$, $1$, and $7$ in some order?

2017 CMIMC Combinatorics, 7
Tags: 2017 , combinatorics
Given a finite set $S \subset \mathbb{R}^3$, define $f(S)$ to be the mininum integer $k$ such that there exist $k$ planes that divide $\mathbb{R}^3$ into a set of regions, where no region contains more than one point in $S$. Suppose that \[M(n) = \max\{f(S) : |S| = n\} \text{ and } m(n) = \min\{f(S) : |S| = n\}.\] Evaluate $M(200) \cdot m(200)$.

2017 CMIMC Algebra, 4
Tags: 2017 , algebra
It is well known that the mathematical constant $e$ can be written in the form $e = \tfrac{1}{0!}+\tfrac{1}{1!}+\tfrac{1}{2!}+\cdots$. With this in mind, determine the value of \[\sum_{j=3}^\infty\dfrac{j}{\lfloor\frac j2\rfloor!}.\] Express your answer in terms of $e$.

2017 CMIMC Combinatorics, 5
Tags: 2017 , combinatorics
Emily draws six dots on a piece of paper such that no three lie on a straight line, then draws a line segment connecting each pair of dots. She then colors five of these segments red. Her coloring is said to be $\emph{red-triangle-free}$ if for every set of three points from her six drawn points there exists an uncolored segment connecting two of the three points. In how many ways can Emily color her drawing such that it is red-triangle-free?

2017 ASDAN Math Tournament, 2
Tags: 2017 , Algebra Tiebreaker
Let $a$ and $b$ be positive integers that satisfy $ab-7a-11b+13=0$. What is the minimum possible value of $a+b$?

2017 CMIMC Geometry, 8
Tags: 2017 , geometry
In triangle $ABC$ with $AB=23$, $AC=27$, and $BC=20$, let $D$ be the foot of the $A$ altitude. Let $\mathcal{P}$ be the parabola with focus $A$ passing through $B$ and $C$, and denote by $T$ the intersection point of $AD$ with the directrix of $\mathcal P$. Determine the value of $DT^2-DA^2$. (Recall that a parabola $\mathcal P$ is the set of points which are equidistant from a point, called the $\textit{focus}$ of $\mathcal P$, and a line, called the $\textit{directrix}$ of $\mathcal P$.)

2017 CMI B.Sc. Entrance Exam, 2
Tags: CMI , Chennai Mathematical Institute , B.Sc , math , CS , 2017 , 3D geometry
Let $L$ be the line of intersection of the planes $~x+y=0~$ and $~y+z=0$. [b](a)[/b] Write the vector equation of $L$, i.e. find $(a,b,c)$ and $(p,q,r)$ such that $$L=\{(a,b,c)+\lambda(p,q,r)~~\vert~\lambda\in\mathbb{R}\}$$ [b](b)[/b] Find the equation of a plane obtained by $x+y=0$ about $L$ by $45^{\circ}$.

2017 ASDAN Math Tournament, 26
Tags: 2017 , Guts Round
A lattice point is a coordinate pair $(a,b)$ where both $a,b$ are integers. What is the number of lattice points $(x,y)$ that satisfy $\tfrac{x^2}{2017}+\tfrac{2y^2}{2017}<1$ and $y\equiv2x\pmod{7}$? Let $C$ be the actual answer, $A$ be the answer you submit, and $D=|A-C|$. Your score will be rounded up from $\max(0,25-e^{D/100})$.

2017 CMIMC Algebra, 2
Tags: 2017 , algebra , function
For nonzero real numbers $x$ and $y$, define $x\circ y = \tfrac{xy}{x+y}$. Compute \[2^1\circ \left(2^2\circ \left(2^3\circ\cdots\circ\left(2^{2016}\circ 2^{2017}\right)\right)\right).\]

2017 CMIMC Algebra, 1
Tags: algebra , 2017
The residents of the local zoo are either rabbits or foxes. The ratio of foxes to rabbits in the zoo is $2:3$. After $10$ of the foxes move out of town and half the rabbits move to Rabbitretreat, the ratio of foxes to rabbits is $13:10$. How many animals are left in the zoo?

2017 CMIMC Geometry, 10
Tags: Euler , geometry , 2017
Suppose $\triangle ABC$ is such that $AB=13$, $AC=15$, and $BC=14$. It is given that there exists a unique point $D$ on side $\overline{BC}$ such that the Euler lines of $\triangle ABD$ and $\triangle ACD$ are parallel. Determine the value of $\tfrac{BD}{CD}$. (The $\textit{Euler}$ line of a triangle $ABC$ is the line connecting the centroid, circumcenter, and orthocenter of $ABC$.)

2017 ASDAN Math Tournament, 3
Tags: 2017 , Calculus Tiebreaker
Compute $$\int_0^1\frac{x^{2017}-1}{\log x}dx.$$

2017 CMIMC Algebra, 7
Tags: 2017 , algebra , system of equations
Let $a$, $b$, and $c$ be complex numbers satisfying the system of equations \begin{align*}\dfrac{a}{b+c}+\dfrac{b}{c+a}+\dfrac{c}{a+b}&=9,\\\dfrac{a^2}{b+c}+\dfrac{b^2}{c+a}+\dfrac{c^2}{a+b}&=32,\\\dfrac{a^3}{b+c}+\dfrac{b^3}{c+a}+\dfrac{c^3}{a+b}&=122.\end{align*} Find $abc$.

2017 CMIMC Number Theory, 7
Tags: 2017 , number theory , calculus , derivative
The $\textit{arithmetic derivative}$ $D(n)$ of a positive integer $n$ is defined via the following rules: [list] [*] $D(1) = 0$; [*] $D(p)=1$ for all primes $p$; [*] $D(ab)=D(a)b+aD(b)$ for all positive integers $a$ and $b$. [/list] Find the sum of all positive integers $n$ below $1000$ satisfying $D(n)=n$.

2017 CMIMC Number Theory, 4
Tags: 2017 , number theory
Let $a_1, a_2, a_3, a_4, a_5$ be positive integers such that $a_1, a_2, a_3$ and $a_3, a_4, a_5$ are both geometric sequences and $a_1, a_3, a_5$ is an arithmetic sequence. If $a_3 = 1575$, find all possible values of $\vert a_4 - a_2 \vert$.

2017 ASDAN Math Tournament, 2
Tags: 2017 , Guts Round
Let $5$ and $13$ be lengths of two sides of a right triangle. Compute the sum of all possible lengths of the third side.

2017 CMIMC Geometry, 1
Tags: geometry , 2017 , angle bisector
Let $ABC$ be a triangle with $\angle BAC=117^\circ$. The angle bisector of $\angle ABC$ intersects side $AC$ at $D$. Suppose $\triangle ABD\sim\triangle ACB$. Compute the measure of $\angle ABC$, in degrees.

2017 CMIMC Number Theory, 5
Tags: 2017 , number theory , greatest common divisor
One can define the greatest common divisor of two positive rational numbers as follows: for $a$, $b$, $c$, and $d$ positive integers with $\gcd(a,b)=\gcd(c,d)=1$, write \[\gcd\left(\dfrac ab,\dfrac cd\right) = \dfrac{\gcd(ad,bc)}{bd}.\] For all positive integers $K$, let $f(K)$ denote the number of ordered pairs of positive rational numbers $(m,n)$ with $m<1$ and $n<1$ such that \[\gcd(m,n)=\dfrac{1}{K}.\] What is $f(2017)-f(2016)$?

2017 CMIMC Geometry, 4
Tags: 2017 , geometry , analytic geometry
Let $\mathcal S$ be the sphere with center $(0,0,1)$ and radius $1$ in $\mathbb R^3$. A plane $\mathcal P$ is tangent to $\mathcal S$ at the point $(x_0,y_0,z_0)$, where $x_0$, $y_0$, and $z_0$ are all positive. Suppose the intersection of plane $\mathcal P$ with the $xy$-plane is the line with equation $2x+y=10$ in $xy$-space. What is $z_0$?

2017 CMIMC Geometry, 2
Tags: 2017 , geometry
Triangle $ABC$ has an obtuse angle at $\angle A$. Points $D$ and $E$ are placed on $\overline{BC}$ in the order $B$, $D$, $E$, $C$ such that $\angle BAD=\angle BCA$ and $\angle CAE=\angle CBA$. If $AB=10$, $AC=11$, and $DE=4$, determine $BC$.

2017 ASDAN Math Tournament, 27
Tags: 2017 , Guts Round
How many primes between $2$ and $2^{30}$ are $1$ more than a multiple of $2017$? If $C$ is the correct answer and $A$ is your answer, then your score will be rounded up from $\max(0,25-15|\ln\tfrac{A}{C}|)$.

2017 ASDAN Math Tournament, 8
Tags: 2017 , Algebra Test
Consider the sequence of real numbers $a_n$ satisfying the recurrence $$a_na_{n+2}-a_{n+1}^2-(n+1)a_na_{n+1}=0.$$ Given that $a_1=1$ and $a_2=2018$, compute $$\frac{a_{2018}\cdot a_{2016}}{a_{2017}^2}.$$

2017 ASDAN Math Tournament, 6
Tags: 2017 , Calculus Test
Compute $$\lim_{x\rightarrow0}\frac{\sqrt[5]{\cos x}-\sqrt[3]{\cos x}}{x^2}.$$

2017 CMIMC Geometry, 9
Tags: 2017 , geometry , cyclic quadrilateral
Let $\triangle ABC$ be an acute triangle with circumcenter $O$, and let $Q\neq A$ denote the point on $\odot (ABC)$ for which $AQ\perp BC$. The circumcircle of $\triangle BOC$ intersects lines $AC$ and $AB$ for the second time at $D$ and $E$ respectively. Suppose that $AQ$, $BC$, and $DE$ are concurrent. If $OD=3$ and $OE=7$, compute $AQ$.

2017 CMIMC Number Theory, 6
Tags: 2017 , number theory
Find the largest positive integer $N$ satisfying the following properties: [list] [*]$N$ is divisible by $7$; [*]Swapping the $i^{\text{th}}$ and $j^{\text{th}}$ digits of $N$ (for any $i$ and $j$ with $i\neq j$) gives an integer which is $\textit{not}$ divisible by $7$. [/list]