Practice (EndingDigits,TheDivideByNineMethod,MODBasic)

2187

Let $\gamma_i$ and $\overline{\gamma_i}$ be the 10 zeros of $x^{10}+(13x-1)^{10}$, where $i=1, 2, 3, 4, 5$. Compute $$\frac{1}{\gamma_1 \overline{\gamma_1}}+\frac{1}{\gamma_2 \overline{\gamma_2}}+\cdots+\frac{1}{\gamma_5 \overline{\gamma_5}}$$

2188

If complex numbers $z_1, z_2, z_3$ satisfy $$ \left\{ \begin{array}{l} |z_1|=|z_2|=|z_3|=1\\ \\ \displaystyle\frac{z_1}{z_2}+\frac{z_2}{z_3}+\frac{z_3}{z_1}=1 \end{array} \right. $$ Compute $|az_1 +bz_2+cz_3|$ where $a, b, c$ are three given real numbers.

2189

Show that $$\sin\frac{\pi}{2n+1}\cdot\sin\frac{2\pi}{2n+1}\cdots\sin\frac{n\pi}{2n+1}=\frac{\sqrt{2n+1}}{2^n}$$

2191

A spider has one sock and one shoe for each of its eight legs. In how many different orders can the spider put on its socks and shoes, assuming that, on each leg, the sock must be put on before the shoe.

2192

Find the smallest positive integer $n$ such that the remainder is always $1$ when $n$ is divided by $2$, $3$, $4$, $5$, or $6$. In addition, $n$ must be a multiple of $7$.

2193

There are four masses all whose weights are all integers. The total weight of these masses is 40$g$. If it is possible to measure any integer weight between 1$g$ and 40$g$ using some combinations of these masses, what are their weights respectively?

2194

Show that any positive integer can be expressed as a sum of integers which are some power of 3.

2195

Let {$a_n$} be a sequence with $a_1=1$. If for any $n > 1$, $a_n$ equals one plus twice of the sum of all the previous terms, express $a_n$ in terms of $n$.

2196

Let $x$ and $y$ be real numbers such that $$2 < \frac{x-y}{x+y} < 5$$ If $\frac{x}{y}$ is an integer, what is its value?

2197

In how many different ways can 900 be expressed as the product of two (possibly equal) positive integers? Regard $m\cdot n$ and $n\cdot m$ as the same.

2198

A binary palindrome is a positive integer whose standard base 2 (binary) representation is a palindrome (reads the same backward or forward). (Leading zeros are not permitted in the standard representation.) For example, 2015 is a binary palindrome, because in base 2 it is 11111011111. How many positive integers less than 2015 are binary palindromes?

2199

What is the area of region bounded by the graphs of $y=|x+2| -|x-2|$ and $y=|x+1|-|x-3|$?

2200

How many distinct positive integers can be expressed in the form $ABCD-DCBA$, where $ABCD$ and $DCBA$ are 4-digit positive integers? (Here $A, B, C$ and $D$ are digits, possible equal)

2201

In the diagram below, how many different routes are there from point $M$ to point $P$ using only the ling segments shown? A route is not allowed to intersect itself, not even at a single point.

2202

In baseball, a player's batting average is the number of hits divided by the number of at bats, rounded to three decimal places. Danielle's batting average is $0.399$. What is the fewest number of at bets that Danielle could have?

2203

Let $n$ be a positive integer. In $n$-dimensional space, consider the $2^n$ points whose coordinates are all $\pm 1$. Imagine placing an $n$-dimensional ball of radius 1 center at each of the $2^n$ points. let $B_n$ be the largest $n$-dimensional ball centered at the origin that does not intersect the interior of any of the original $2^n$ balls. What is the smallest value of $n$ such that $B_n$ contains a point with a coordinate greater than 2?

2204

Say that a rational number is special if its decimal expression is of the form $0.\overline{abcdef}$, where $a, b, c, d, e$ and $f$ are digits (possibly equal) that include each of the digits $2, 0, 1$, and $5$ at least once (in some order). How many special rational numbers are there?

2205

Among all pairs of real numbers $(x, y)$ such that $\sin\sin x=\sin\sin y$ with $-10\pi \le x, y \le 10\pi$. Oleg randomly selected a pair $(X, Y)$. Compute the probability that $X = Y$.

2206

Let $A=(2,0)$, $B=(0,2)$, $C=(-2,0)$, and $D=(0, -2)$. Compute the greatest possible value of the product $PA\cdot PB\cdot PC\cdot PD$, where $P$ is a point on the circle $x^2 + y^2=9$.

2207

A $\textit{permutation}$ of a finite set is a one-to-one function from the set onto itself. A $\textit{cycle}$ in a permutation $P$ is a nonempty sequence of distinct items $x_1, \cdots, x_n$ such that $P(x_1)=x_2$, $P(x_2)=x_3$, $\cdots$, $P(x_n)=x_1$. Note that we allow the 1-cycle $x_1$ where $P(x_1)=x_1$ and the 2-cycle $x_1, x_2$ where $P(x_1)=x_2$ and $P(x_2)=x_1$. Every permutation of a finite site splits the set into a finite number of disjoint cycles. If this number equal to 2, then the permutation is called $\textit{bi-cyclic}$. Computer the number of bi-cyclic permutation of the 7-element set formed by letters "PROBLEMS".

2208

Joel selected an acute angle $x$ (strictly between 0 and 90 degrees) and wrote the value of $\sin x$, $\cos x$, and $\tan x$ on three different cards. Then he gave those cards to three students, Malvina, Paulina, and Georgian, one card to each, and asked them to figure out which trigonometric function (sin, cos, tan) produced their cards. Even after sharing the values on their cards with each other, only Malvian was able to surly identify which function produced the value on her card. Compute the sum of all possible values that Joel wrote on Marlvina's card.

2209

Let $C$ be a three-dimensional cube with edge length 1. There are 8 equilateral triangles whose vertices are vertices of $C$. The 8 planes that contain these 8 equilateral triangles divide $C$ into several non-overlapping regions. Find the volume of the region that contains the center of $C$.

2210

Let $z_1$, $z_2$, $z_3$, and $z_4$ be the four distinct complex solutions of the equation $$z^4-6z^2+8z+1=-4(z^3-z+2)i$$ Find the sum of the six pairwise distance between $z_1, z_2, z_3$ and $z_4$.

2211

An ant begins at a vertex of a convex regular icosahedron (a figure with 20 triangular faces and 12 vertices). The ant moves along one edge at a time. Each time, the ant reaches a vertex, it randomly choose to next walk along any of the edges extending from that vertex (including the edge it just arrived from). Find the probability that after walking along exactly six (not necessarily distinct) edges, the ant finds itself at its starting vertex.

2212

Let $S$ be the sum of all distinct real solutions of the equation $$\sqrt{x+2015}=x^2-2015$$ Compute $\lfloor 1/S \rfloor$.

Practice (EndingDigits,TheDivideByNineMethod,MODBasic)

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