Olympiad problems

CNCM Online Round 3, 7

Tags: CNCM

A subset of the positive integers $S$ is said to be a \emph{configuration} if 200 $\notin S$ and for all nonnegative integers $x$, $x \in S$ if and only if both 2$x\in S$ and $\left \lfloor{\frac{x}{2}}\right \rfloor\in S$. Let the number of subsets of $\{1, 2, 3, \dots, 130\}$ that are equal to the intersection of $\{1, 2, 3, \dots, 130\}$ with some configuration $S$ equal $k$. Compute the remainder when $k$ is divided by 1810. [i]Proposed Hari Desikan (HariDesikan)[/i]

CNCM Online Round 2, 4

Tags: CNCM , Problem Discussion

On a chessboard with $6$ rows and $9$ columns, the Slow Rook is placed in the bottom-left corner and the Blind King is placed on the top-left corner. Then, $8$ Sleeping Pawns are placed such that no two Sleeping Pawns are in the same column, no Sleeping Pawn shares a row with the Slow Rook or the Blind King, and no Sleeping Pawn is in the rightmost column. The Slow Rook can move vertically or horizontally $1$ tile at a time, the Slow Rook cannot move into any tile containing a Sleeping Pawn, and the Slow Rook takes the shortest path to reach the Blind King. How many ways are there to place the Sleeping Pawns such that the Slow Rook moves exactly $15$ tiles to get to the space containing the Blind King? Proposed by Harry Chen (Extile)

CNCM Online Round 2, 5

Tags: CNCM , Problem Discussion

Consider a regular $n$-gon of side length 1. For each of its vertices, a circle of radius one is drawn centered at that vertex. The resulting figure, consisting of the polygon and the $n$ circles, partitions the plane into $f(n)$ finite, bounded regions. Find $$\sum_{n=3}^{25} f(n).$$ The first term corresponding to $i=3$ is shown; each of the various colors corresponds to a distinct region with $f(3)=10$. Note that the lines corresponding to the polygon are treated no differently than the arcs corresponding to the circles in counting regions. Proposed by Hari Desikan (HariDesikan)

CNCM Online Round 3, 1

Tags: CNCM , v4913 orz

Harry, who is incredibly intellectual, needs to eat carrots $C_1, C_2, C_3$ and solve [i]Daily Challenge[/i] problems $D_1, D_2, D_3$. However, he insists that carrot $C_i$ must be eaten only after solving [i]Daily Challenge[/i] problem $D_i$. In how many satisfactory orders can he complete all six actions? [i]Proposed by Albert Wang (awang2004)[/i]

CNCM Online Round 2, 7

Tags: CNCM , Problem Discussion

A circle is centered at point $O$ in the plane. Distinct pairs of points $A, B$ and $C, D$ are diametrically opposite on this circle. Point $P$ is chosen on line segment $AD$ such that line $BP$ hits the circle again at $M$ and line $AC$ at $X$ such that $M$ is the midpoint of $PX$. Now, the point $Y \neq X$ is taken for $BX = BY, CD \parallel XY$. IF $\angle PYB = 10^{\circ}$, find the measure of $\angle XCM$. Proposed by Albert Wang (awang11)

CNCM Online Round 2, 1

Tags: CNCM , Problem Discussion

Adi the Baller is shooting hoops, and makes a shot with probability $p$. He keeps shooting hoops until he misses. The value of $p$ that maximizes the chance that he makes between 35 and 69 (inclusive) buckets can be expressed as $\frac{1}{\sqrt[b]{a}}$ for a prime $a$ and positive integer $b$. Find $a+b$. Proposed by Minseok Eli Park (wolfpack)

CNCM Online Round 2, 2

Tags: CNCM , Problem Discussion

There is a rectangle $ABCD$ such that $AB=12$ and $BC=7$. $E$ and $F$ lie on sides $AB$ and $CD$ respectively such that $\frac{AE}{EB} = 1$ and $\frac{CF}{FD} = \frac{1}{2}$. Call $X$ the intersection of $AF$ and $DE$. What is the area of pentagon $BCFXE$? Proposed by Minseok Eli Park (wolfpack)

CNCM Online Round 2, 3

Tags: CNCM , Problem Discussion

An ordered pair $(n,p)$ is [i]juicy[/i] if $n^{2} \equiv 1 \pmod{p^{2}}$ and $n \equiv -1 \pmod{p}$ for positive integer $n$ and odd prime $p$. How many juicy pairs exist such that $n,p \leq 200$? Proposed by Harry Chen (Extile)

CNCM Online Round 3, 5

Tags: CNCM

How many positive integers $N$ less than $10^{1000}$ are such that $N$ has $x$ digits when written in base ten and $\frac{1}{N}$ has $x$ digits after the decimal point when written in base ten? For example, 20 has two digits and $\frac{1}{20}$= 0.05 has two digits after the decimal point, so $20$ is a valid N. [i]Proposed by Hari Desikan (HariDesikan)[/i]

CNCM Online Round 3, 4

Tags: CNCM

Hari is obsessed with cubics. He comes up with a cubic with leading coefficient 1, rational coefficients and real roots $0 < a < b < c < 1$. He knows the following three facts: $P(0) = -\frac{1}{8}$, the roots form a geometric progression in the order $a,b,c$, and \[ \sum_{k=1}^{\infty} (a^k + b^k + c^k) = \dfrac{9}{2} \] The value $a + b + c$ can be expressed as $\frac{m}{n}$, where $m,n$ are relatively prime positive integers. Find $m + n$. [i]Proposed by Akshar Yeccherla (TopNotchMath)[/i]

CNCM Online Round 2, 6

Tags: CNCM , Problem Discussion

Let $S$ be the set of all ordered pairs $(x,y)$ of integer solutions to the equation $$6x^2+y^2+6x=3xy+6y+x^2y.$$ $S$ contains a unique ordered pair $(a,b)$ with a maximal value of $b$. Compute $a+b$. Proposed by Kenan Hasanaliyev (claserken)

CNCM Online Round 3, 6

Tags: CNCM

Triangle $ABC$ has side lengths $AB=13, BC=14,$ and $CA=15$. Let $\Gamma$ denote the circumcircle of $\triangle ABC$. Let $H$ be the orthocenter of $\triangle ABC$. Let $AH$ intersect $\Gamma$ at a point $D$ other than $A$. Let $BH$ intersect $AC$ at $F$ and $\Gamma$ at point $G$ other than $B$. Suppose $DG$ intersects $AC$ at $X$. Compute the greatest integer less than or equal to the area of quadrilateral $HDXF$. [i]Proposed by Kenan Hasanaliyev (claserken)[/i]

CNCM Online Round 3, 2

Tags: CNCM

Consider rectangle $ABCD$ with $AB = 6$ and $BC = 8$. Pick points $E, F, G, H$ such that the angles $\angle AEB, \angle BFC, \angle CGD, \angle AHD$ are all right. What is the largest possible area of quadrilateral $EFGH$? [i]Proposed by Akshar Yeccherla (TopNotchMath)[/i]

CNCM Online Round 3, 3

Tags: CNCM

Let $a_1 = 1$ and $a_{n+1} = a_n \cdot p_n$ for $n \geq 1$ where $p_n$ is the $n$th prime number, starting with $p_1 = 2$. Let $\tau(x)$ be equal to the number of divisors of $x$. Find the remainder when $$\sum_{n=1}^{2020} \sum_{d \mid a_n} \tau (d)$$ is divided by 91 for positive integers $d$. Recall that $d|a_n$ denotes that $d$ divides $a_n$. [i]Proposed by Minseok Eli Park (wolfpack)[/i]