Olympiad problems

2017 ASDAN Math Tournament, 9

Compute $$\int_0^4\frac{x^4-4x+4}{1+2017^{x-2}}dx.$$

2017 CMIMC Computer Science, 10

Tags: 2017 , computer science

How many distinct spanning trees does the graph below have? Recall that a $\emph{spanning tree}$ of a graph $G$ is a subgraph of $G$ that is a tree and containing all the vertices of $G$. [center][img]http://i.imgur.com/NMF12pE.png[/img][/center]

2017 CMI B.Sc. Entrance Exam, 1

Tags: CMI , Chennai Mathematical Institute , B.Sc , mathematics , computer science , 2017 , calculus

Answer the following questions : [b](a)[/b] Evaluate $~~\lim_{x\to 0^{+}} \Big(x^{x^x}-x^x\Big)$ [b](b)[/b] Let $A=\frac{2\pi}{9}$, i.e. $40$ degrees. Calculate the following $$1+\cos A+\cos 2A+\cos 4A+\cos 5A+\cos 7A+\cos 8A$$ [b](c)[/b] Find the number of solutions to $$e^x=\frac{x}{2017}+1$$

2017 CMIMC Geometry, 7

Tags: 2017 , geometry

Two non-intersecting circles, $\omega$ and $\Omega$, have centers $C_\omega$ and $C_\Omega$ respectively. It is given that the radius of $\Omega$ is strictly larger than the radius of $\omega$. The two common external tangents of $\Omega$ and $\omega$ intersect at a point $P$, and an internal tangent of the two circles intersects the common external tangents at $X$ and $Y$. Suppose that the radius of $\omega$ is $4$, the circumradius of $\triangle PXY$ is $9$, and $XY$ bisects $\overline{PC_\Omega}$. Compute $XY$.

2017 ASDAN Math Tournament, 23

Tags: 2017 , Guts Round

Ben creates an $8\times8$ grid of coins, where each coin faces heads with probability $\tfrac{1}{2}$, and tails with probability $\tfrac{1}{2}$. Ben then makes a series of moves; each move consists of selecting a coin in the grid and flipping over all coins in the same row and column as the selected coin. Suppose that in Ben’s current grid of coins, it is possible to make a series of moves so that all coins in the grid are heads, and that Ben will make the fewest number of moves to do so. What is the expected number of moves that Ben makes?

2017 CMIMC Combinatorics, 9

Tags: 2017 , combinatorics

At a conference, six people place their name badges in a hat, which is shaken up; one badge is then distributed to each person such that each distribution is equally likely. Each turn, every person who does not yet have their own badge finds the person whose badge they have and takes that person's badge. For example, if Alice has Bob's badge and Bob has Charlie's badge, Alice would have Charlie's badge after a turn. Compute the probability that everyone will eventually end up with their own badge.

2017 ASDAN Math Tournament, 24

Tags: 2017 , Guts Round

Consider all rational numbers of the form $\tfrac{p}{q}$ where $p,q$ are relatively prime positive integers less than or equal to $8$, and plot them on the $xy$-plane, where $\tfrac{p}{q}$ corresponds to point $(p,q)$. Arrange the rationals in increasing order $\{P_1,P_2,\dots,P_n\}$ and form a polygon by connecting points $P_i$ and $P_{i+1}$ for $1\le i<n$ and connecting both $P_1$ and $P_n$ to the origin. What is the area of the polygon?

2017 CMIMC Algebra, 6

Tags: 2017 , algebra

Suppose $P$ is a quintic polynomial with real coefficients with $P(0)=2$ and $P(1)=3$ such that $|z|=1$ whenever $z$ is a complex number satisfying $P(z) = 0$. What is the smallest possible value of $P(2)$ over all such polynomials $P$?

2017 CMIMC Individual Finals, 2

Tags: 2017 , Tiebreaker , combinatorics

Kevin likes drawing. He takes a large piece of paper and draws on it every rectangle with positive integer side lengths and perimeter at most 2017, with no two rectangles overlapping. Compute the total area of the paper that is covered by a rectangle.

2017 ASDAN Math Tournament, 9

Tags: 2017 , Geometry Test

Triangle $ABC$ is isosceles with $AC=BC=25$ and $AB=10$. Let $O$ be the orthocenter of $\triangle ABC$, the intersection of the three altitudes of $\triangle ABC$. Reflect $O$ across $AB$ to a point $D$, and extend $CB$ and $AD$ to intersect at point $E$. Compute the area of $\triangle ABE$.

2017 CMI B.Sc. Entrance Exam, 5

Tags: CMI , Chennai Mathematical Institute , B.Sc , CS , 2017 , Coloring

Each integer is colored with exactly one of $3$ possible colors -- black, red or white -- satisfying the following two rules : the negative of a black number must be colored white, and the sum of two white numbers (not necessarily distinct) must be colored black. [b](a)[/b] Show that, the negative of a white number must be colored black and the sum of two black numbers must be colored white. [b](b)[/b] Determine all possible colorings of the integers that satisfy these rules.

2017 Kazakhstan NMO, Problem 1

Tags: Kazakhstan , geometry , 2017 , projective geometry

The non-isosceles triangle $ABC$ is inscribed in the circle ω. The tangent to this circle at the point $C$ intersects the line $AB$ at the point $D$. Let the bisector of the angle $CDB$ intersect the segments $AC$ and $BC$ at the points $K$ and $L$, respectively. On the side $AB$, the point $M$ is taken such that $AK / BL = AM / BM$. Let the perpendiculars from the point $M$ to the lines $KL$ and $DC$ intersect the lines $AC$ and $DC$ at the points $P$ and $Q$, respectively. Prove that the angle $CQP$ is half of the angle $ACB$.

2017 CMIMC Individual Finals, 2

Tags: 2017 , Tiebreaker , number theory

Find the smallest three-digit divisor of the number \[1\underbrace{00\ldots 0}_{100\text{ zeros}}1\underbrace{00\ldots 0}_{100\text{ zeros}}1.\]

2017 CMI B.Sc. Entrance Exam, 6

Tags: CMI , Chennai Mathematical Institute , B.Sc , math , CS , 2017 , geometry

You are given a regular hexagon. We say that a square is inscribed in the hexagon if it can be drawn in the interior such that all the four vertices lie on the perimeter of the hexagon. [b](a)[/b] A line segment has its endpoints on opposite edges of the hexagon. Show that, it passes through the centre of the hexagon if and only if it divides the two edges in the same ratio. [b](b)[/b] Suppose, a square $ABCD$ is inscribed in the hexagon such that $A$ and $C$ are on the opposite sides of the hexagon. Prove that, centre of the square is same as that of the hexagon. [b](c)[/b] Suppose, the side of the hexagon is of length $1$. Then find the length of the side of the inscribed square whose one pair of opposite sides is parallel to a pair of opposite sides of the hexagon. [b](d)[/b] Show that, up to rotation, there is a unique way of inscribing a square in a regular hexagon.

2017 CMIMC Computer Science, 1

Tags: 2017 , computer science

What is the minimum number of times you have to take your pencil off the paper to draw the following figure (the dots are for decoration)? You must lift your pencil off the paper after you're done, and this is included in the number of times you take your pencil off the paper. You're not allowed to draw over an edge twice. [center][img]http://i.imgur.com/CBGmPmv.png[/img][/center]

2017 ASDAN Math Tournament, 8

Tags: 2017 , Calculus Test

Compute $$\int_0^1\frac{2xe^x-1}{2x^2e^x+2}dx.$$

2017 CMIMC Algebra, 9

Tags: 2017 , algebra

Define a sequence $\{a_{n}\}_{n=1}^{\infty}$ via $a_{1} = 1$ and $a_{n+1} = a_{n} + \lfloor \sqrt{a_{n}} \rfloor$ for all $n \geq 1$. What is the smallest $N$ such that $a_{N} > 2017$?

2017 CMIMC Computer Science, 3

Tags: 2017 , computer science

In the following list of numbers (given in their binary representations), each number appears an even number of times, except for one number that appears exactly three times. Find the number that appears exactly three times. Leave the answer in its binary representation. \begin{tabular}{cccccc} 010111 & 000001 & 100000 & 011000 & 110101 & 100001 \\ 010100 & 011111 & 111001 & 010001 & 010100 & 101100 \\ 010001 & 011011 & 011111 & 011011 & 100000 & 000001 \\ 110011 & 001000 & 111101 & 100001 & 101100 & 110011 \\ 111111 & 011000 & 001000 & 101000 & 111111 & 101000 \\ 010111 & 100011 & 111001 & 100011 & 110101 & 011111 \\ 100000 & 010100 & 010001 & 101100 & 010111 & 011011 \\ 011000 & 111101 & 111111 & 100001 & 101000 & 100011 \\ 011011 & 010111 & 110011 & 111111 & 000001 & 010001 \\ 101000 & 111001 & 010100 & 110101 & 011000 & 110101 \\ 001000 & 000001 & 100000 & 111101 & 100011 & 001000 \\ 111001 & 110011 & 100001 & 011111 & 101100 \end{tabular}

2017 ASDAN Math Tournament, 5

Tags: 2017 , Geometry Test

Regular hexagon $ABCDEF$ has side length $2$. Line segment $BD$ is drawn, and circle $O$ is inscribed inside the pentagon $ABDEF$ such that $O$ is tangent to $AF$, $BD$, and $EF$. Compute the radius of $O$.

2017 CMIMC Combinatorics, 8

Tags: 2017 , combinatorics

Andrew generates a finite random sequence $\{a_n\}$ of distinct integers according to the following criteria: [list] [*] $a_0 = 1$, $0 < |a_n| < 7$ for all $n$, and $a_i \neq a_j$ for all $i < j$. [*] $a_{n+1}$ is selected uniformly at random from the set $\{a_n - 1, a_n + 1, -a_n\}$, conditioned on the above rule. The sequence terminates if no element of the set satisfies the first condition. [/list] For example, if $(a_0, a_1) = (1, 2)$, then $a_2$ would be chosen from the set $\{-2,3\}$, each with probability $\tfrac12$. Determine the probability that there exists an integer $k$ such that $a_k = 6$.

2017 ASDAN Math Tournament, 3

Tags: 2017 , Geometry Test

Line segment $AB$ has length $10$. A circle centered at $A$ has radius $5$, and a circle centered at $B$ has radius $5\sqrt{3}$. What is the area of the intersection of the two circles?

2017 CMIMC Individual Finals, 3

Tags: 2017 , Tiebreaker , combinatorics

In a certain game, the set $\{1, 2, \dots, 10\}$ is partitioned into equally-sized sets $A$ and $B$. In each of five consecutive rounds, Alice and Bob simultaneously choose an element from $A$ or $B$, respectively, that they have not yet chosen; whoever chooses the larger number receives a point, and whoever obtains three points wins the game. Determine the probability that Alice is guaranteed to win immediately after the set is initially partitioned.

2017 CMIMC Computer Science, 8

Tags: 2017 , computer science

We have a collection of $1720$ balls, half of which are black and half of which are white, aligned in a straight line. Our task is to make the balls alternating in color along the line. The following greedy algorithm accomplishes that task for $2n$ balls: \begin{tabular}{l} 1: \textbf{FOR} $i$ \textbf{IN} $[2,3,\dots,2n]$ \\ 2: $\quad$ \textbf{IF} balls $i-1$ and $i$ have the same color: \\ 3: $\quad\quad$ $j\gets$ smallest index greater than $i$ for which balls $i-1$ and $j$ have different colors \\ 4: $\quad\quad$ swap balls $i$ and $j$ \end{tabular} Given a configuration $C$ of our $1720$ balls, let $\hat{\sigma}(C)$ denote the number of swaps the greedy algorithm takes, and let $\sigma(C)$ denote the minimum number of swaps actually necessary to perform the task. Find the maximum value over all configurations $C$ of $\hat{\sigma}(C)-\sigma(C)$.

2017 ASDAN Math Tournament, 7

Tags: 2017 , Guts Round

Point $C$ is chosen on the arc of a semicircle with diameter $AB$. The two circles with diameters of $AC$ and $BC$ intersect again at point $D$. If $DA=20$ and $DB=16$, compute the length of $DC$.

2017 ASDAN Math Tournament, 1

Tags: 2017 , Discrete Math Test

Clara and Nick each randomly and independently pick an integer between $0$ and $2017$, inclusive. What is the probability that the two integers they pick sum to an even number?