Found problems: 85335
We call a positive integer [i]alternating[/i] if every two consecutive digits in its decimal representation are of different parity.
Find all positive integers $n$ such that $n$ has a multiple which is alternating.
Consider a $12$-card deck containing all four suits of $2, 3,$ and $4.$ A [i]double[/i] is defined as two cards directly next to each other in the deck, with the same value. Suppose we scan the deck left to right, and whenever we encounter a double, we remove all the cards up to that point (including the double). Let $N$ denote the number of times we have to remove cards. What is the expected value of $N$?
Find the prime factorization of $\textstyle\sum_{1\le i < j \le 100}ij$.
Let $b_m$ be numbers of factors $2$ of the number $m!$ (that is, $2^{b_m}|m!$ and $2^{b_m+1}\nmid m!$). Find the least $m$ such that $m-b_m = 1990$.
In a narrow alley of width $w$ a ladder of length $a$ is placed with its foot at point $P$ between the walls. Resting against one wall at $Q$, the distance $k$ above the ground makes a $45^\circ$ angle with the ground. Resting against the other wall at $R$, a distance $h$ above the ground, the ladder makes a $75^\circ$ angle with the ground. The width $w$ is equal to
$\textbf {(A) } a \qquad \textbf {(B) } RQ \qquad \textbf {(C) } k \qquad \textbf {(D) } \frac{h+k}{2} \qquad \textbf {(E) } h$
Equilateral $\triangle ABC$ has side length $600$. Points $P$ and $Q$ lie outside of the plane of $\triangle ABC$ and are on the opposite sides of the plane. Furthermore, $PA=PB=PC$, and $QA=QB=QC$, and the planes of $\triangle PAB$ and $\triangle QAB$ form a $120^{\circ}$ dihedral angle (The angle between the two planes). There is a point $O$ whose distance from each of $A,B,C,P$ and $Q$ is $d$. Find $d$.
Show that if the integers $a_1$; $\dots$ $a_m$ are nonzero and for each $k =0; 1; \dots ;n$ ($n < m - 1$),
$a_1 + a_22^k + a_33^k + \dots + a_mm^k = 0$; then the sequence $a_1, \dots, a_m$ contains at least $n+1$ pairs of consecutive terms having opposite signs.
[i]O. Musin[/i]
Find all functions $ f : Z\rightarrow Z$ for which we have $ f (0) \equal{} 1$ and $ f ( f (n)) \equal{} f ( f (n\plus{}2)\plus{}2) \equal{} n$, for every natural number $ n$.
Let $a$ be a positive integer. We say that a positive integer $b$ is [i]$a$-good[/i] if $\tbinom{an}{b}-1$ is divisible by $an+1$ for all positive integers $n$ with $an \geq b$. Suppose $b$ is a positive integer such that $b$ is $a$-good, but $b+2$ is not $a$-good. Prove that $b+1$ is prime.
On the side $AC$ of a triangle $ABC$, let a $K$ be a point such that $AK = 2KC$ and $\angle ABK = 2 \angle KBC$. Let $F$ be the midpoint of $AC$, $L$ be the projection of $A$ on $BK$. Prove that $FL \bot BC$.
Let $P(x)$ be a non-constant polynomial with integer coefficients. Prove that there is no function $T$ from the set of integers into the set of integers such that the number of integers $x$ with $T^n(x)=x$ is equal to $P(n)$ for every $n\geq 1$, where $T^n$ denotes the $n$-fold application of $T$.
[i]Proposed by Jozsef Pelikan, Hungary[/i]
A collection of regular polygons with sides of equal length is said to "fit" if, when arranged around a common vertex, they exactly complete the surrounding area of the point on the plane. For example, a square fits with two octagons. Determine all possible collections of regular polygons that fit.
We say that a set $S$ of positive integers is special when there exists a function $f : \mathbb{N}\to \mathbb{N}$ satisfying that:
$\bullet$ $f(k)\in S, \forall k\in\mathbb{N}$
$\bullet$ No integer $k$ with $2\le k \le 2021$ can be written as $\frac{af(a)}{bf(b)}$ with $a,b\in \mathbb{N}$
Find the smallest positive integer $n$ such that the set $S = \{ 1, 2021, 2021^2, \ldots , 2021^n \}$ is special or prove that no such integer exists.
Note: $\mathbb{N}$ represents the set $\{ 1, 2, 3, \ldots \}$
A 100 foot long moving walkway moves at a constant rate of 6 feet per second. Al steps onto the start of the walkway and stands. Bob steps onto the start of the walkway two seconds later and strolls forward along the walkway at a constant rate of 4 feet per second. Two seconds after that, Cy reaches the start of the walkway and walks briskly forward beside the walkway at a constant rate of 8 feet per second. At a certain time, one of these three persons is exactly halfway between the other two. At that time, find the distance in feet between the start of the walkway and the middle person.
A line with slope $2$ intersects a line with slope $6$ at the point $(40, 30)$. What is the distance between the $x$-intercepts of these two lines?
$\textbf{(A) }5\qquad\textbf{(B) }10\qquad\textbf{(C) }20\qquad\textbf{(D) }25\qquad\textbf{(E) }50$
Let $a, b, c, d$ be the roots of the quartic polynomial $f(x) = x^4 + 2x + 4$. Find the value of
$$\frac{a^2}{a^3 + 2} + \frac{b^2}{b^3 + 2} + \frac{c^2}{c^3 + 2} + \frac{d^2}{d^3 + 2}.$$
[hide=E stands for Euclid , L stands for Lobachevsky]they had two problem sets under those two names[/hide]
[b]E1.[/b] How many positive divisors does $72$ have?
[b]E2 / L2.[/b] Raymond wants to travel in a car with $3$ other (distinguishable) people. The car has $5$ seats: a driver’s seat, a passenger seat, and a row of $3$ seats behind them. If Raymond’s cello must be in a seat next to him, and he can’t drive, but every other person can, how many ways can everyone sit in the car?
[b]E3 / L3.[/b] Peter wants to make fruit punch. He has orange juice ($100\%$ orange juice), tropical mix ($25\%$ orange juice, $75\%$ pineapple juice), and cherry juice ($100\%$ cherry juice). If he wants his final mix to have $50\%$ orange juice, $10\%$ cherry juice, and $40\%$ pineapple juice, in what ratios should he mix the $3$ juices? Please write your answer in the form (orange):(tropical):(cherry), where the three integers are relatively prime.
[b]E4 / L4.[/b] Points $A, B, C$, and $D$ are chosen on a circle such that $m \angle ACD = 85^o$, $m\angle ADC = 40^o$,and $m\angle BCD = 60^o$. What is $m\angle CBD$?
[b]E5.[/b] $a, b$, and $c$ are positive real numbers. If $abc = 6$ and $a + b = 2$, what is the minimum possible value of $a + b + c$?
[b]E6 / L5.[/b] Circles $A$ and $B$ are drawn on a plane such that they intersect at two points. The centers of the two circles and the two intersection points lie on another circle, circle $C$. If the distance between the centers of circles $A$ and $B$ is $20$ and the radius of circle $A$ is $16$, what is the radius of circle $B$?
[b]E7.[/b] Point $P$ is inside rectangle $ABCD$. If $AP = 5$, $BP = 6$, and $CP = 7$, what is the length of $DP$?
[b]E8 / L6.[/b] For how many integers $n$ is $n^2 + 4$ divisible by $n + 2$?
[b]E9. [/b] How many of the perfect squares between $1$ and $10000$, inclusive, can be written as the sum of two triangular numbers? We define the $n$th triangular number to be $1 + 2 + 3 + ... + n$, where $n$ is a positive integer.
[b]E10 / L7.[/b] A small sphere of radius $1$ is sitting on the ground externally tangent to a larger sphere, also sitting on the ground. If the line connecting the spheres’ centers makes a $60^o$ angle with the ground, what is the radius of the larger sphere?
[b]E11 / L8.[/b] A classroom has $12$ chairs in a row and $5$ distinguishable students. The teacher wants to position the students in the seats in such a way that there is at least one empty chair between any two students. In how many ways can the teacher do this?
[b]E12 / L9.[/b] Let there be real numbers $a$ and $b$ such that $a/b^2 + b/a^2 = 72$ and $ab = 3$. Find the value of $a^2 + b^2$.
[b]E13 / L10.[/b] Find the number of ordered pairs of positive integers $(x, y)$ such that $gcd \, (x, y)+lcm \, (x, y) =x + y + 8$.
[b]E14 / L11.[/b] Evaluate $\sum_{i=1}^{\infty}\frac{i}{4^i}=\frac{1}{4} +\frac{2}{16} +\frac{3}{64} +...$
[b]E15 / L12.[/b] Xavier and Olivia are playing tic-tac-toe. Xavier goes first. How many ways can the game play out such that Olivia wins on her third move? The order of the moves matters.
[b]L1.[/b] What is the sum of the positive divisors of $100$?
[b]L13.[/b] Let $ABCD$ be a convex quadrilateral with $AC = 20$. Furthermore, let $M, N, P$, and $Q$ be the midpoints of $DA, AB, BC$, and $CD$, respectively. Let $X$ be the intersection of the diagonals of quadrilateral $MNPQ$. Given that $NX = 12$ and $XP = 10$, compute the area of $ABCD$.
[b]L14.[/b] Evaluate $(\sqrt3 + \sqrt5)^6$ to the nearest integer.
[b]L15.[/b] In Hatland, each citizen wears either a green hat or a blue hat. Furthermore, each citizen belongs to exactly one neighborhood. On average, a green-hatted citizen has $65\%$ of his neighbors wearing green hats, and a blue-hatted citizen has $80\%$ of his neighbors wearing blue hats. Each neighborhood has a different number of total citizens. What is the ratio of green-hatted to blue-hatted citizens in Hatland? (A citizen is his own neighbor.)
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
Let $ABCD$ be a convex quadrilateral with $\angle ABD = \angle BCD$, $AD = 1000$, $BD = 2000$, $BC = 2001$, and $DC = 1999$. Point $E$ is chosen on segment $DB$ such that $\angle ABD = \angle ECD$. Find $AE$.
For which real numbers $p$ does the equation $x^{2}+px+3p=0$ have integer solutions ?
Consider any rectangular table having finitely many rows and columns, with a real number $a(r, c)$ in the cell in row $r$ and column $c$. A pair $(R, C)$, where $R$ is a set of rows and $C$ a set of columns, is called a [i]saddle pair[/i] if the following two conditions are satisfied:
[list]
[*] $(i)$ For each row $r^{\prime}$, there is $r \in R$ such that $a(r, c) \geqslant a\left(r^{\prime}, c\right)$ for all $c \in C$;
[*] $(ii)$ For each column $c^{\prime}$, there is $c \in C$ such that $a(r, c) \leqslant a\left(r, c^{\prime}\right)$ for all $r \in R$.
[/list]
A saddle pair $(R, C)$ is called a [i]minimal pair[/i] if for each saddle pair $\left(R^{\prime}, C^{\prime}\right)$ with $R^{\prime} \subseteq R$ and $C^{\prime} \subseteq C$, we have $R^{\prime}=R$ and $C^{\prime}=C$. Prove that any two minimal pairs contain the same number of rows.
Find all pairs $(p,q)$ of positive primes such that the equation $3x^2 - px + q = 0$ has two distinct rational roots.
If $a_1,a_2,\cdots,a_{2013}\in[-2,2]$ and $a_1+a_2+\cdots+a_{2013}=0$ , find the maximum of $a^3_1+a^3_2+\cdots+a^3_{2013}$.
Let $r_1,r_2,\dots ,r_n$ be real numbers. Given $n$ reals $a_1,a_2,\dots ,a_n$ that are not all equal to $0$, suppose that inequality
\[r_1(x_1-a_1)+ r_2(x_2-a_2)+\dots + r_n(x_n-a_n)\leq\sqrt{x_1^2+ x_2^2+\dots + x_n^2}-\sqrt{a_1^2+a_2^2+\dots +a_n^2}\]
holds for arbitrary reals $x_1,x_2,\dots ,x_n$. Find the values of $r_1,r_2,\dots ,r_n$.
The set of points in $3$-dimensional coordinate space that lie in the plane $x+y+z=75$ whose coordinates satisfy the inequalities $$x-yz<y-zx<z-xy$$forms three disjoint convex regions. Exactly one of those regions has finite area. The area of this finite region can be expressed in the form $a\sqrt{b},$ where $a$ and $b$ are positive integers and $b$ is not divisible by the square of any prime. Find $a+b.$
Determine the smallest integer $k > 1$ such that there exist $k$ distinct primes whose squares sum to a power of $2$.