Found problems: 85335
Ann and Drew have purchased a mysterious slot machine; each time it is spun, it chooses a random positive integer such that $k$ is chosen with probability $2^{-k}$ for every positive integer $k$, and then it outputs $k$ tokens. Let $N$ be a fixed integer. Ann and Drew alternate turns spinning the machine, with Ann going first. Ann wins if she receives at least $N$ total tokens from the slot machine before Drew receives at least $M=2^{2018}$ total tokens, and Drew wins if he receives $M$ tokens before Ann receives $N$ tokens. If each person has the same probability of winning, compute the remainder when $N$ is divided by $2018$.
[i]Proposed by Brandon Wang[/i]
Find the number of solutions to the equation:
$6\{x\}^3 + \{x\}^2 + \{x\} + 2x = 2018. $
With {x} we denote the fractional part of the number x.
In a two-dimensional cave with a parallel floor and ceiling, two stalactites of lengths $16$ and $36$ hang perpendicularly from the ceiling, while two stalagmites of heights $25$ and $49$ grow perpendicularly from the ground. If the tips of these four structures form the vertices of a square in some order, compute the height of the cave.
Three different pairs of shoes are placed in a row so that no left shoe is next to a right shoe from a different pair. In how many ways can these six shoes be lined up?
$
\textbf{(A) }60\qquad
\textbf{(B) }72\qquad
\textbf{(C) }90\qquad
\textbf{(D) }108\qquad
\textbf{(E) }120\qquad
$
Emmy goes to buy radishes at the market. Radishes are sold in bundles of $3$ for $\$5$and bundles of $5$ for $\$7$. What is the least number of dollars Emmy needs to buy exactly $100$ radishes?
Let $a, b, c$ be positive real numbers with $abc = 1$. Prove that $1 + \frac{3}{a+b+c}\ge \frac{6}{ab+bc+ca}$
A $3\times 1$ paper rectangle is folded twice to give a square side $1$. The square is folded along a diagonal to give a right-angled triangle. A needle is driven through an interior point of the triangle, making $6$ holes in the paper. The paper is then unfolded. Where should the point be in order to maximise the smallest distance between any two holes?
Let $n = p_1p_2 \dots p_k$ be the product of distinct primes $p_1, p_2, \dots , p_k$, with $k > 1$. Find all $n$ such that $n$ is multiple of $p_1 - 1, p_2 - 1, \dots , p_k - 1$.
TB1. Two not necessarily distinct positive integers $a,b$ are randomly chosen from the set $\{1,2,\ldots, 20\}$. Find the expected value of the number of distinct prime factors of $ab$.
[i]Proposed by Harry Kim[/i]
TB2. Square $ABCD$ has side length $15$. Let $E$ and $F$ be points on $AD$ and $BC$ respectively such that $AE = 5$ and $BF = 5$. Find the area of intersection between triangles $\triangle{AFC}$ and $\triangle{BED}$.
[i]Proposed by Andy Xu[/i]
TB3. If $x$ and $y$ satisfy $$\frac{1}{x}+\frac{1}{y} = 2$$ $$\frac{x}{y}+\frac{y}{x} = 3$$ find $xy$.
[i]Proposed by Harry Kim and Andy Xu[/i]
Let $ABCDEF$ be a convex hexagon such that $BCEF$ is a parallelogram and $ABF$ an equilateral triangle. Given that $BC = 1, AD = 3, CD+DE = 2$, compute the area of $ABCDEF$
The smallest sum one could get by adding three different numbers from the set $ \{7,25,\minus{}1,12,\minus{}3 \}$ is
\[ \textbf{(A)}\ \minus{}3 \qquad
\textbf{(B)}\ \minus{}1 \qquad
\textbf{(C)}\ 3 \qquad
\textbf{(D)}\ 5 \qquad
\textbf{(E)}\ 21
\]
Let $ABC$ be a triangle with incenter $I$. Let $U$, $V$ and $W$ be the intersections of the angle bisectors of angles $A$, $B$, and $C$ with the incircle, so that $V$ lies between $B$ and $I$, and similarly with $U$ and $W$. Let $X$, $Y$, and $Z$ be the points of tangency of the incircle of triangle $ABC$ with $BC$, $AC$, and $AB$, respectively. Let triangle $UVW$ be the [i]David Yang triangle[/i] of $ABC$ and let $XYZ$ be the [i]Scott Wu triangle[/i] of $ABC$. Prove that the David Yang and Scott Wu triangles of a triangle are congruent if and only if $ABC$ is equilateral.
[i]Proposed by Owen Goff[/i]
Planes $\alpha,\beta,\gamma,\delta$ are tangent to the circumsphere of a tetrahedron $ABCD$ at points $A,B,C,D$, respectively. Line $p$ is the intersection of $\alpha$ and $\beta$, and line $q$ is the intersection of $\gamma$ and $\delta$. Prove that if lines $p$ and $CD$ meet, then lines $q$ and $AB$ lie on a plane.
Let $\{a, b, c, d, e, f, g, h\}$ be a permutation of $\{1, 2, 3, 4, 5, 6, 7, 8\}$. What is the probability that $\overline{abc} +\overline{def}$ is even?
On a circle are arranged $100$ baskets, each containing at least one candy. The total number of candies is $780$. Asad and Sevinch make moves alternatingly, with Asad going first. On one move, Asad takes all the candies from $9$ consecutive non-empty baskets, while Sevinch takes all the candies from a single non-empty basket that has at least one empty neighboring basket. Prove that Asad can take overall at least $700$ candies, regardless of the initial distribution of candies and Sevinch's actions.
[i] Shubin Yakov, Russia [/i]
Find the least positive integer $n$ ($n\geq 3$), such that among any $n$ points (no three are collinear) in the plane, there exist three points which are the vertices of a non-isoscele triangle.
Circle $\omega$ is inscribed in the $\vartriangle ABC$, with center $I$. Using only a ruler, divide segment $AI$ in half.
(Grigory Filippovsky)
For any permutation $p$ of the set $\{1,2,...,n\}$, let us denote $d(p) = |p(1)-1|+|p(2)-2|+...+|p(n)-n|$. Let $i(p)$ be the number of inversions of $p$, i.e. the number of pairs $1 \le i < j \le n$ with $p(i) > p(j)$. Prove that $d(p)\le 2i(p)$$.
An ordered pair \((k,n)\) of positive integers is [i]good[/i] if there exists an ordered quadruple \((a,b,c,d)\) of positive integers such that \(a^3+b^k=c^3+d^k\) and \(abcd=n\). Prove that there exist infinitely many positive integers \(n\) such that \((2022,n)\) is not good but \((2023,n)\) is good.
[i]Proposed by Luke Robitaille[/i]
Give a triangle $ABC$ with heights $h_a = 3$ cm, $h_b = 7$ cm and $h_c = d$ cm, where $d$ is an integer. Determine $d$.
$O$ is a point inside triangle $ABC$ such that $OA=OB+OC$. Suppose $B',C'$ be midpoints of arcs $\overarc{AOC}$ and $AOB$. Prove that circumcircles $COC'$ and $BOB'$ are tangent to each other.
Two circles are said to be [i]orthogonal[/i] if they intersect in two points, and their tangents at either point of intersection are perpendicular. Two circles $\omega_1$ and $\omega_2$ with radii $10$ and $13$, respectively, are externally tangent at point $P$. Another circle $\omega_3$ with radius $2\sqrt2$ passes through $P$ and is orthogonal to both $\omega_1$ and $\omega_2$. A fourth circle $\omega_4$, orthogonal to $\omega_3$, is externally tangent to $\omega_1$ and $\omega_2$. Compute the radius of $\omega_4$.
Given positive integer $m,n$, color the points of the regular $(2m+2n)$-gon in black and white, $2m$ in black and $2n$ in white.
The [i]coloring distance[/i] $d(B,C) $ of two black points $B,C$ is defined as the smaller number of white points in the two paths linking the two black points.
The [i]coloring distance[/i] $d(W,X) $ of two white points $W,X$ is defined as the smaller number of black points in the two paths linking the two white points.
We define the matching of black points $\mathcal{B}$ : label the $2m$ black points with $A_1,\cdots,A_m,B_1,\cdots,B_m$ satisfying no $A_iB_i$ intersects inside the gon.
We define the matching of white points $\mathcal{W}$ : label the $2n$ white points with $C_1,\cdots,C_n,D_1,\cdots,D_n$ satisfying no $C_iD_i$ intersects inside the gon.
We define $P(\mathcal{B})=\sum^m_{i=1}d(A_i,B_i), P(\mathcal{W} )=\sum^n_{j=1}d(C_j,D_j) $.
Prove that: $\max_{\mathcal{B}}P(\mathcal{B})=\max_{\mathcal{W}}P(\mathcal{W})$
Prove the following inequality: $tan \, 1>\frac{3}{2}$.
Given a triangle $A_1 A_2 A_3$ and a point $P$ inside. Let $B_i$ be a point on the side opposite to $A_i$ for $i=1,2,3$, and let $C_i$ and $D_i$ be the midpoints of $A_i B_i$ and $P B_i$, respectively. Prove that the triangles $C_1 C_2 C_3$ and $D_1 D_2 D_3$ have equal area.