Found problems: 85335
Find the unique 3 digit number $N=\underline{A}$ $\underline{B}$ $\underline{C},$ whose digits $(A, B, C)$ are all nonzero, with the property that the product $P=\underline{A}$ $\underline{B}$ $\underline{C}$ $\times$ $\underline{A}$ $\underline{B}$ $\times$ $\underline{A}$ is divisible by $1000$.
[i]Proposed by Kyle Lee[/i]
In one class in the school, number of abscent students is $\frac{1}{6}$ of number of students who were present. When teacher sent one student to bring chalk, number of abscent students was $\frac{1}{5}$ of number of students who were present. How many students are in that class?
Let $L(n)$ denote the least common multiple of $\{1, 2 . . . , n\}$.
(i) Prove that there exists a positive integer $k$ such that $L(k) = L(k + 1) = ... = L(k + 2000)$.
(ii) Find all $m$ such that $L(m + i) \ne L(m + i + 1)$ for all $i = 0, 1, 2$.
Let $n$ be a positive integer and $a_1, a_2, \dots , a_{2n}$ mutually distinct integers. Find all integers $x$ satisfying
\[(x - a_1) \cdot (x - a_2) \cdots (x - a_{2n}) = (-1)^n(n!)^2.\]
Suppose a function $ f: \mathbb R\to \mathbb R$ satisfies $ f(f(x)) \equal{} \minus{} x$ for all $ x\in \mathbb R$. Prove that $ f$ has infinitely many points of discontinuity.
In the coordinate plane, a point is called a $\text{lattice point}$ if both of its coordinates are integers. Let $A$ be the point $(12,84)$. Find the number of right angled triangles $ABC$ in the coordinate plane $B$ and $C$ are lattice points, having a right angle at vertex $A$ and whose incenter is at the origin $(0,0)$.
The numbers $p, 4p^2 + 1,$ and $6p^2 + 1$ are primes. Determine $p.$
Let $q{}$ be an arbitrary polynomial with complex coefficients which is not identically $0$ and $\Gamma_q =\{z : |q(z)| = 1\}$ be its contour line. Prove that for every point $z_0\in\Gamma_q$ there is a polynomial $p{}$ for which $|p(z_0)| = 1$ and $|p(z)|<1$ for any $z\in\Gamma_q\setminus\{z_0\}.$
Consider a checkered $3m\times 3m$ square, where $m$ is an integer greater than $1.$ A frog sits on the lower left corner cell $S$ and wants to get to the upper right corner cell $F.$ The frog can hop from any cell to either the next cell to the right or the next cell upwards.
Some cells can be [i]sticky[/i], and the frog gets trapped once it hops on such a cell. A set $X$ of cells is called [i]blocking[/i] if the frog cannot reach $F$ from $S$ when all the cells of $X$ are sticky. A blocking set is [i] minimal[/i] if it does not contain a smaller blocking set.[list=a][*]Prove that there exists a minimal blocking set containing at least $3m^2-3m$ cells.
[*]Prove that every minimal blocking set containing at most $3m^2$ cells.
For some positive integer $n$, $(1 + i) + (1 + i)^2 + (1 + i)^3 + ... + (1 + i)^n = (n^2 - 1)(1 - i)$, where $i = \sqrt{-1}$. Compute the value of $n$.
What is the area of the region in the coordinate plane defined by the inequality \[\left||x|-1\right|+\left||y|-1\right|\leq 1?\]
$\textbf{(A)}~4\qquad\textbf{(B)}~8\qquad\textbf{(C)}~10\qquad\textbf{(D)}~12\qquad\textbf{(E)}~15$
The equation of line $\ell_1$ is $24x-7y = 319$ and the equation of line $\ell_2$ is $12x-5y = 125$. Let $a$ be the number of positive integer values $n$ less than $2023$ such that for both $\ell_1$ and $\ell_2$ there exists a lattice point on that line that is a distance of $n$ from the point $(20,23)$. Determine $a$.
[i]Proposed by Christopher Cheng[/i]
[hide=Solution][i]Solution. [/i] $\boxed{6}$
Note that $(20,23)$ is the intersection of the lines $\ell_1$ and $\ell_2$. Thus, we only care about lattice points on the the two lines that are an integer distance away from $(20,23)$. Notice that $7$ and $24$ are part of the Pythagorean triple $(7,24,25)$ and $5$ and $12$ are part of the Pythagorean triple $(5,12,13)$. Thus, points on $\ell_1$ only satisfy the conditions when $n$ is divisible by $25$ and points on $\ell_2$ only satisfy the conditions when $n$ is divisible by $13$. Therefore, $a$ is just the number of positive integers less than $2023$ that are divisible by both $25$ and $13$. The LCM of $25$ and $13$ is $325$, so the answer is $\boxed{6}$.[/hide]
Find all integers $n \ge 3$ for which the polynomial
\[W(x) = x^n - 3x^{n-1} + 2x^{n-2} + 6\]
can be written as a product of two non-constant polynomials with integer coefficients.
Let be given a tetrahedron whose circumcenter is $O$. Draw diameters $AA_{1},BB_{1},CC_{1},DD_{1}$ of the circumsphere of $ABCD$. Let $A_{0},B_{0},C_{0},D_{0}$ be the centroids of triangle $BCD,CDA,DAB,ABC$. Prove that $A_{0}A_{1},B_{0}B_{1},C_{0}C_{1},D_{0}D_{1}$ are concurrent at a point, say, $F$. Prove that the line through $F$ and a midpoint of a side of $ABCD$ is perpendicular to the opposite side.
Given a square $ABCD,$ with $AB=1$ mark the midpoints $M$ and $N$ of $AB$ and $BC,$ respectively. A lasar beam shot from $M$ to $N,$ and the beam reflects of $BC,CD,DA,$ and comes back to $M.$ This path encloses a smaller area inside square $ABCD.$ Find this area.
A graph has $17$ points and each point has $4$ edges. Show that there are two points which are not joined and which are not both joined to the same point.
Let there be positive integers $a, c$. Positive integer $b$ is a divisor of $ac-1$. For a positive rational number $r$ which is less than $1$, define the set $A(r)$ as follows.
$$A(r) = \{m(r-ac)+nab| m, n \in \mathbb{Z} \}$$
Find all rational numbers $r$ which makes the minimum positive rational number in $A(r)$ greater than or equal to $\frac{ab}{a+b}$.
The sequence $(z_n)$ of complex numbers satisfies the following properties:
[list]
[*]$z_1$ and $z_2$ are not real.
[*]$z_{n+2}=z_{n+1}^2z_n$ for all integers $n\geq 1$.
[*]$\dfrac{z_{n+3}}{z_n^2}$ is real for all integers $n\geq 1$.
[*]$\left|\dfrac{z_3}{z_4}\right|=\left|\dfrac{z_4}{z_5}\right|=2$. [/list]
Find the product of all possible values of $z_1$.
Let $p$ be a polynomial with integer coefficients and let $a_1<a_2<\cdots <a_k$ be integers. Given that $p(a_i)\ne 0\forall\; i=1,2,\cdots, k$.
[list]
(a) Prove $\exists\; a\in \mathbb{Z}$ such that
\[ p(a_i)\mid p(a)\;\;\forall i=1,2,\dots ,k \]
(b) Does there exist $a\in \mathbb{Z}$ such that
\[ \prod_{i=1}^{k}p(a_i)\mid p(a) \][/list]
Given pairwise different real numbers $a,b,c,d,e$ such that
$$
\left\{ \begin{array}{ll}
ab + b = ac + a, \\
bc + c = bd + b, \\
cd + d = ce + c, \\
de + e = da + d.
\end{array} \right.
$$
Prove that $abcde=1$.
Siva has the following expression, which is missing operations:
$$\frac12 \,\, \_ \,\,\frac14 \,\, \_ \,\, \frac18 \,\, \_ \,\,\frac{1}{16} \,\, \_ \,\,\frac{1}{32}.$$ For each blank, he flips a fair coin: if it comes up heads, he fills it with a plus, and if it comes up tails, he fills it with a minus. Afterwards, he computes the value of the expression. He then repeats the entire process with a new set of coinflips and operations. If the probability that the positive difference between his computed values is greater than $\frac12$ can be expressed as $\frac{a}{b}$ for relatively prime positive integers $a$, $b$, then find $a + b$.
The products of the opposite sidelengths of a cyclic quadrilateral $ABCD$ are
equal. Let $B'$ be the reflection of $B$ about $AC$. Prove that the circle passing through $A,B', D$ touches $AC$
[b]p1.[/b] Ana, Bob, Cho, Dan, and Eve want to use a microwave. In order to be fair, they choose a random order to heat their food in (all orders have equal probability). Ana's food needs $5$ minutes to cook, Bob's food needs $7$ minutes, Cho's needs $1$ minute, Dan's needs $12$ minutes, and Eve's needs $5$ minutes. What is the expected number of minutes Bob has to wait for his food to be done?
[b]p2.[/b] $ABC$ is an equilateral triangle. $H$ lies in the interior of $ABC$, and points $X$, $Y$, $Z$ lie on sides $AB, BC, CA$, respectively, such that $HX\perp AB$, $HY \perp BC$, $HZ\perp CA$. Furthermore, $HX =2$, $HY = 3$, $HZ = 4$. Find the area of triangle $ABC$.
[b]p3.[/b] Amy, Ben, and Chime play a dice game. They each take turns rolling a die such that the $first$ person to roll one of his favorite numbers wins. Amy's favorite number is $1$, Ben's favorite numbers are $2$ and $3$, and Chime's are $4$, $5$, and $6$. Amy rolls first, Ben rolls second, and Chime rolls third. If no one has won after Chime's turn, they repeat the sequence until someone has won. What's the probability that Chime wins the game?
[b]p4.[/b] A point $P$ is chosen randomly in the interior of a square $ABCD$. What is the probability that the angle $\angle APB$ is obtuse?
[b]p5.[/b] Let $ABCD$ be the quadrilateral with vertices $A = (3, 9)$, $B = (1, 1)$, $C = (5, 3)$, and $D = (a, b)$, all of which lie in the first quadrant. Let $M$ be the midpoint of $AB$, $N$ the midpoint of $BC$, $O$ the midpoint of $CD$, and $P$ the midpoint of $AD$. If $MNOP$ is a square, find $(a, b)$.
[b]p6.[/b] Let $M$ be the number of positive perfect cubes that divide $60^{60}$. What is the prime factorization of $M$?
[b]p7.[/b] Given that $x$, $y$, and $z$ are complex numbers with $|x|=|y| =|z|= 1$, $x + y + z = 1$ and $xyz = 1$, find $|(x + 2)(y + 2)(z + 2)|$.
[b]p8.[/b] If $f(x)$ is a polynomial of degree $2008$ such that $f(m) = \frac{1}{m}$ for $m = 1, 2, ..., 2009$, find $f(2010)$.
[b]p9.[/b] A drunkard is randomly walking through a city when he stumbles upon a $2 \times 2$ sliding tile puzzle. The puzzle consists of a $2 \times 2$ grid filled with a blank square, as well as $3$ square tiles, labeled $1$, $2$, and $3$. During each turn you may fill the empty square by sliding one of the adjacent tiles into it. The following image shows the puzzle's correct state, as well as two possible moves you can make:
[img]https://cdn.artofproblemsolving.com/attachments/c/6/7ddd9305885523deeee2a530dc90505875d1cc.png[/img]
Assuming that the puzzle is initially in an incorrect (but solvable) state, and that the drunkard will make completely random moves to try and solve it, how many moves is he expected to make before he restores the puzzle to its correct state?
[b]p10.[/b] How many polynomials $p(x)$ exist such that the coeffients of $p(x)$ are a rearrangement of $\{0, 1, 2, .., deg \, p(x)\}$ and all of the roots of $p(x)$ are rational? (Note that the leading coefficient of $p(x)$ must be nonzero.)
PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
Let $a,b,c\geq 0$ and $a+b+c=1.$ Prove that$$\frac{a}{2a+1}+\frac{b}{3b+1}+\frac{c}{6c+1}\leq \frac{1}{2}.$$
[size=50](Marian Dinca)[/size]
Let $p>2$ be a prime number, $m>1$ and $n$ be positive integers such that $\frac {m^{pn}-1}{m^n-1}$ is a prime number. Show that: $$pn\mid (p-1)^n+1$$