Practice (90/1000)
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The number $21!=51,090,942,171,709,440,000$ has over $60,000$ positive integer divisors. One of them is chosen at random. What is the probability that it is odd?
Call a positive integer $monotonous$ if it is a one-digit number or its digits, when read from left to right, form either a strictly increasing or a strictly decreasing sequence. For example, $3$, $23578$, and $987620$ are monotonous, but $88$, $7434$, and $23557$ are not. How many monotonous positive integers are there?
When each of $702$, $787$, and $855$ is divided by the positive integer $m$, the remainder is always the positive integer $r$. When each of $412$, $722$, and $815$ is divided by the positive integer $n$, the remainder is always the positive integer $s \neq r$. Fine $m+n+r+s$.
For a positive integer $n$, let $d_n$ be the units digit of $1 + 2 + \dots + n$. Find the remainder when \[\sum_{n=1}^{2017} d_n\]is divided by $1000$.
A pyramid has a triangular base with side lengths $20$, $20$, and $24$. The three edges of the pyramid from the three corners of the base to the fourth vertex of the pyramid all have length $25$. The volume of the pyramid is $m\sqrt{n}$, where $m$ and $n$ are positive integers, and $n$ is not divisible by the square of any prime. Find $m+n$.
A rational number written in base eight is $\underline{a} \underline{b} . \underline{c} \underline{d}$, where all digits are nonzero. The same number in base twelve is $\underline{b} \underline{b} . \underline{b} \underline{a}$. Find the base-ten number $\underline{a} \underline{b} \underline{c}$.
A circle is circumscribed around an isosceles triangle whose two congruent angles have degree measure $x$. Two points are chosen independently and uniformly at random on the circle, and a chord is drawn between them. The probability that the chord intersects the triangle is $\frac{14}{25}$. Find the difference between the largest and smallest possible values of $x$.
For nonnegative integers $a$ and $b$ with $a + b \leq 6$, let $T(a, b) = \binom{6}{a} \binom{6}{b} \binom{6}{a + b}$. If $S$ denotes the sum of all $T(a, b)$, where $a$ and $b$ are nonnegative integers with $a + b \leq 6$. Find $S$.
Let $z_1 = 18 + 83i$, $z_2 = 18 + 39i,$ and $z_3 = 78 + 99i,$ where $i = \sqrt{-1}$. Let $z$ be the unique complex number with the properties that $\frac{z_3 - z_1}{z_2 - z_1} \cdot \frac{z - z_2}{z - z_3}$ is a real number and the imaginary part of $z$ is the greatest possible. Find the real part of $z$.
Solution
Let $a > 1$ and $x > 1$ satisfy $\log_a(\log_a(\log_a 2) + \log_a 24 - 128) = 128$ and $\log_a(\log_a x) = 256$. Find the remainder when $x$ is divided by $1000$.
Let $x$ be an integer and $p$ is a prime divisor of $(x^6 + x^5 + \cdots + 1)$. Show that $p=7$ or $p\equiv 1\pmod{7}$.
Find the number of positive integers less than or equal to $2017$ whose base-three representation contains no digit equal to $0$.
Find the sum of all positive integers $n$ such that $\sqrt{n^2+85n+2017}$ is an integer.
A special deck of cards contains $49$ cards, each labeled with a number from $1$ to $7$ and colored with one of seven colors. Each number-color combination appears on exactly one card. Sharon will select a set of eight cards from the deck at random. Given that she gets at least one card of each color and at least one card with each number, find the probability that Sharon can discard one of her cards and $\textit{still}$ have at least one card of each color and at least one card with each number.
Find the minimum possible value of \[\frac{a}{b^3+4}+\frac{b}{c^3+4}+\frac{c}{d^3+4}+\frac{d}{a^3+4}\]given that $a$, $b$, $c$, $d$ are nonnegative real numbers such that $a+b+c+d=4$.
Prove that there are infinitely many distinct pairs $(a,b)$ of relatively prime positive integers $a > 1$ and $b > 1$ such that $(a^b + b^a)$ is divisible by $(a + b)$.
Consider the equation \[\left(3x^3 + xy^2 \right) \left(x^2y + 3y^3 \right) = (x-y)^7.\]
(a) Prove that there are infinitely many pairs $(x,y)$ of positive integers satisfying the equation.
(b) Describe all pairs $(x,y)$ of positive integers satisfying the equation.
Let $ABC$ be an equilateral triangle and let $P$ be a point on its circumcircle. Let lines $PA$ and $BC$ intersect at $D$; let lines $PB$ and $CA$ intersect at $E$; and let lines $PC$ and $AB$ intersect at $F$. Prove that the area of triangle $DEF$ is twice the area of triangle $ABC$.
Let $O$ and $H$ be the circumcenter and the orthocenter of an acute triangle $ABC$. Points $M$ and $D$ lie on side $BC$ such that $BM = CM$ and $\angle BAD = \angle CAD$. Ray $MO$ intersects the circumcircle of triangle $BHC$ in point $N$. Prove that $\angle ADO = \angle HAN$.
Let $P_1, \ldots, P_{2n}$ be $2n$ distinct points on the unit circle $x^2 + y^2 = 1$ other than $(1,0)$. Each point is colored either red or blue, with exactly $n$ of them red and exactly $n$ of them blue. Let $R_1, \ldots, R_n$ be any ordering of the red points. Let $B_1$ be the nearest blue point to $R_1$ traveling counterclockwise around the circle starting from $R_1$. Then let $B_2$ be the nearest of the remaining blue points to $R_2$ traveling counterclockwise around the circle from $R_2$, and so on, until we have labeled all the blue points $B_1, \ldots, B_n$. Show that the number of counterclockwise arcs of the form $R_i \rightarrow B_i$ that contain the point $(1,0)$ is independent of the way we chose the ordering $R_1, \ldots, R_n$ of the red points.
Let $f(x)=x^3 -x^2 -13x+24$. Find three pairs of $(x,y)$ such that if $y=f(x)$, then $x=f(y)$.
Each point of a circle is colored either red or blue.
(a) Prove that there always exists an isosceles triangle inscribed in this circle such that all its vertices are colored the same.
(b) Does there always exist an equilateral triangle inscribed in this circle such that all its vertices are colored the same?
The function $f$ satisfies $f(0)=0$, $f(1)=1$, and $f(\frac{x+y}{2})=\frac{f(x)+f(y)}{2}$ for all $x,y\in\mathbb{R}$. Show that $f(x)=x$ for all rational numbers $x$.
Let $f$ be a function such that $$ \sqrt {x - \sqrt { x + f(x) } } = f(x) , $$for $x > 1$. In that domain, $f(x)$ has the form $\frac{a+\sqrt{cx+d}}{b},$ where $a,b,c,d$ are integers and $a,b$ are relatively prime. Find $a+b+c+d.$
Find the number of possible arrangements in Fisher Random Chess. The diagram below is one possible arrangement.
In a legal arrangement, the White's position must satisfy the following criteria:
- Eight pawns must be in the $2^{nd}$ row. (The same as regular chess)
- Two bishops must be in opposite colored squares (e.g. $b1$ and $e1$ in the above diagram)
- King must locate between two rooks (e.g. in the diagram above, King is at $c1$ and two rooks are at $a1$ and $g1$)
The Black's position will be mirroring to the White's.