Found problems: 85335
In a triangle $ABC$, $D$ is the orthogonal projection of the incenter $I$ onto $BC$. Line $DI$ meets the incircle again at $E$. Line $AE$ intersects side $BC$ at point $F$. Suppose that the segment IO is parallel to $BC$, where $O$ is the circumcenter of $\triangle ABC$. If $R$ is the circumradius and $r$ the inradius of the triangle, prove that $EF=2(R-2r)$.
Two people, $A$ and $B$, play the following game: $A$ start choosing a positive integrer number and then, each player in it's turn, say a number due to the following rule:
If the last number said was odd, the player add $7$ to this number;
If the last number said was even, the player divide it by $2$.
The winner is the player that repeats the first number said. Find all numbers that $A$ can choose in order to win. Justify your answer.
From $25$ points in a plane, both of whose coordinates are integers of the set $\{-2,-1, 0, 1, 2\}$, some $17$ points are marked. Prove that there are three points on one line, one of them is the midpoint of two others.
A certain $4$ digit prime number has all prime digits. When any one of the digits is removed, the remaning three digits form a composite number in their initial order (i.e. if $1234$ were the answer, then $123$, $234$, $134$, and $124$ would have to be composite.) What is the largest possible value of this number?
Prove that for every given positive integer $k$, there exist infinitely many $n$, such that $2^{n}+3^{n}-1, 2^{n}+3^{n}-2,\ldots, 2^{n}+3^{n}-k$ are all composite numbers.
Given is a cyclic pentagon $ABCDE$, inscribed in a circle $k$. The line $CD$ intersects $AB$ and $AE$ in $X$ and $Y$ respectively. Segments $EX$ and $BY$ intersect again at $P$, and they intersect $k$ in $Q$ and $R$, respectively. Point $A'$ is reflection of $A$ across $CD$. The circles $(PQR)$ and $(A'XY)$ intersect at $M$ and $N$. Prove that $CM$ and $DN$ intersect on $(PQR)$.
Find the minimum possible value of
\[\left\lfloor \dfrac{a+b}{c}\right\rfloor+2 \left\lfloor \dfrac{b+c}{a}\right\rfloor+ \left\lfloor \dfrac{c+a}{b}\right\rfloor\]
where $a,b,c$ are the sidelengths of a triangle.
[i]Proposed by Nathan Ramesh
A convex polygon $P$ is placed inside a unit square $Q$. Prove that the perimeter of $P$ does not exceed $4$.
Find all positive integers $n$ such that for any integer $k$ there exists an integer $a$ for which $a^3+a-k$ is divisible by $n$.
[i]Warut Suksompong, Thailand[/i]
6) $G$ group, $G_{m}$ and $G_{n}$ commutative subgroups being the $m$ and $n$ th powers of the elements in $G$. Prove $G_{gcd(m,n)}$ is commutative.
Distinct points $A$ and $B$ are given on the plane. Consider all triangles $ABC$ in this plane on whose sides $BC,CA$ points $D,E$ respectively can be taken so that
(i) $\frac{BD}{BC}=\frac{CE}{CA}=\frac{1}{3}$;
(ii) points $A,B,D,E$ lie on a circle in this order.
Find the locus of the intersection points of lines $AD$ and $BE$.
$(POL 3)$ Given a polynomial $f(x)$ with integer coefficients whose value is divisible by $3$ for three integers $k, k + 1,$ and $k + 2$. Prove that $f(m)$ is divisible by $3$ for all integers $m.$
There are nonzero integers $a$, $b$, $r$, and $s$ such that the complex number $r+si$ is a zero of the polynomial $P(x) = x^3 - ax^2 + bx - 65$. For each possible combination of $a$ and $b$, let $p_{a,b}$ be the sum of the zeroes of $P(x)$. Find the sum of the $p_{a,b}$'s for all possible combinations of $a$ and $b$.
Let $M_n$ be the $n\times n$ matrix with entries as follows: for $1\leq i \leq n$, $m_{i,i}=10$; for $1\leq i \leq n-1, m_{i+1,i}=m_{i,i+1}=3$; all other entries in $M_n$ are zero. Let $D_n$ be the determinant of matrix $M_n$. Then $\displaystyle \sum_{n=1}^{\infty} \dfrac{1}{8D_n+1}$ can be represented as $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.
Note: The determinant of the $1\times 1$ matrix $[a]$ is $a$, and the determinant of the $2\times 2$ matrix $\left[ \begin{array}{cc} a & b \\ c & d \end{array} \right]=ad-bc$; for $n\geq 2$, the determinant of an $n\times n$ matrix with first row or first column $a_1\ a_2\ a_3 \dots\ a_n$ is equal to $a_1C_1 - a_2C_2 + a_3C_3 - \dots + (-1)^{n+1} a_nC_n$, where $C_i$ is the determinant of the $(n-1)\times (n-1)$ matrix found by eliminating the row and column containing $a_i$.
For any positive integer $n$, $a_n>0$, and $\sum_{j=1}^{n}a_j^3=\left(\sum_{j=1}^{n}a_j\right)^2$. Prove that $a_n=n$
Prove that there are infinitely many positive integers $ n$ such that $ n^{2} \plus{} 1$ has a prime divisor greater than $ 2n \plus{} \sqrt {2n}$.
[i]Author: Kestutis Cesnavicius, Lithuania[/i]
Hannusya, Petrus and Mykolka drew independently one isosceles triangle $ABC$, all angles of which are measured as a integer number of degrees. It turned out that the bases $AC$ of these triangles are equals and for each of them on the ray $BC$ there is a point $E$ such that $BE=AC$, and the angle $AEC$ is also measured by an integer number of degrees. Is it in necessary that:
a) all three drawn triangles are equal to each other?
b) among them there are at least two equal triangles?
Let $a_{1}, a_{2}, \cdots, a_{k}$ be relatively prime positive integers. Determine the largest integer which cannot be expressed in the form \[x_{1}a_{2}a_{3}\cdots a_{k}+x_{2}a_{1}a_{3}\cdots a_{k}+\cdots+x_{k}a_{1}a_{2}\cdots a_{k-1}\] for some nonnegative integers $x_{1}, x_{2}, \cdots, x_{k}$.
Let be given a triangle $ABC$ and points $D,M,N$ belong to $BC,AB,AC$, respectively. Suppose that $MD$ is parallel to $AC$ and $ND$ is parallel to $AB$. If $S_{\vartriangle BMD} = 9$ cm $^2, S_{\vartriangle DNC} = 25$ cm$^2$, compute $S_{\vartriangle AMN}$?
[b]p1.[/b] There are $2024$ apples in a very large basket. First, Julie takes away half of the apples in the basket; then, Diane takes away $202$ apples from the remaining bunch. How many apples remain in the basket?
[b]p2.[/b] The set of all permutations (different arrangements) of the letters in ”ABMC” are listed in alphabetical order. The first item on the list is numbered $1$, the second item is numbered $2$, and in general, the kth item on the list is numbered $k$. What number is given to ”ABMC”?
[b]p3.[/b] Daniel has a water bottle that is three-quarters full. After drinking $3$ ounces of water, the water bottle is three-fifths full. The density of water is $1$ gram per milliliter, and there are around $28$ grams per ounce. How many milliliters of water could the bottle fit at full capacity?
[b]p4.[/b] How many ways can four distinct $2$-by-$1$ rectangles fit on a $2$-by-$4$ board such that each rectangle is fully on the board?
[b]p5.[/b] Iris and Ivy start reading a $240$ page textbook with $120$ left-hand pages and $120$ right-hand pages. Iris takes $4$ minutes to read each page, while Ivy takes $5$ minutes to read a left-hand page and $3$ minutes to read a right-hand page. Iris and Ivy move onto the next page only when both sisters have completed reading. If a sister finishes reading a page first, the other sister will start reading three times as fast until she completes the page. How many minutes after they start reading will both sisters finish the textbook?
[b]p6.[/b] Let $\vartriangle ABC$ be an equilateral triangle with side length $24$. Then, let $M$ be the midpoint of $BC$. Define $P$ to be the set of all points $P$ such that $2PM = BC$. The minimum value of $AP$ can be expressed as $\sqrt{a}- b$, where $a$ and $b$ are positive integers. Find $a + b$.
[b]p7.[/b] Jonathan has $10$ songs in his playlist: $4$ rap songs and $6$ pop songs. He will select three unique songs to listen to while he studies. Let $p$ be the probability that at least two songs are rap, and let $q$ be the probability that none of them are rap. Find $\frac{p}{q}$ .
[b]p8.[/b] A number $K$ is called $6,8$-similar if $K$ written in base $6$ and $K$ written in base $8$ have the same number of digits. Find the number of $6,8$-similar values between $1$ and $1000$, inclusive.
[b]p9.[/b] Quadrilateral $ABCD$ has $\angle ABC = 90^o$, $\angle ADC = 120^o$, $AB = 5$, $BC = 18$, and $CD = 3$. Find $AD^2$.
[b]p10.[/b] Bob, Eric, and Raymond are playing a game. Each player rolls a fair $6$-sided die, and whoever has the highest roll wins. If players are tied for the highest roll, the ones that are tied reroll until one wins. At the start, Bob rolls a $4$. The probability that Eric wins the game can be expressed as $\frac{p}{q}$ where $p$ and $q$ are relatively prime positive integers. Find $p + q$.
[b]p11.[/b] Define the following infinite sequence $s$:
$$s = \left\{\frac92,\frac{99}{2^2},\frac{999}{2^3} , ... , \overbrace{\frac{999...999}{2^k}}^{k\,\,nines}, ...\right\}$$
The sum of the first $2024$ terms in $s$, denoted $S$, can be expressed as
$$S =\frac{5^a - b}{4}+\frac{1}{2^c},$$
where $a, b$, and $c$ are positive integers. Find $a + b + c$.
[b]p12.[/b] Andy is adding numbers in base $5$. However, he accidentally forgets to write the units digit of each number. If he writes all the consecutive integers starting at $0$ and ending at $50$ (base $10$) and adds them together, what is the difference between Andy’s sum and the correct sum? (Express your answer in base-$10$.)
[b]p13.[/b] Let $n$ be the positive real number such that the system of equations
$$y =\frac{1}{\sqrt{2024 - x^2}}$$
$$y =\sqrt{x^2 - n}$$
has exactly two real solutions for $(x, y)$: $(a, b)$ and $(-a, b)$. Then, $|a|$ can be expressed as $j\sqrt{k}$, where $j$ and $k$ are integers such that $k$ is not divisible by any perfect square other than $1$. Find $j · k$.
[b]p14.[/b] Nakio is playing a game with three fair $4$-sided dice. But being the cheater he is, he has secretly replaced one of the three die with his own $4$-sided die, such that there is a $1/2$ chance of rolling a $4$, and a $1/6$ chance to roll each number from $1$ to $3$. To play, a random die is chosen with equal probability and rolled. If Nakio guesses the number that is on the die, he wins. Unfortunately for him, Nakio’s friends have an anti-cheating mechanism in place: when the die is picked, they will roll it three times. If each roll lands on the same number, that die is thrown out and one of the two unused dice is chosen instead with equal probability.
If Nakio always guesses $4$, the probability that he wins the game can be expressed as $\frac{m}{n}$ , where $m$ and $n$ are relatively prime. Find $m + n$.
[b]p15.[/b] A particle starts in the center of a $2$m-by-$2$m square. It moves in a random direction such that the angle between its direction and a side of the square is a multiple of $30^o$. It travels in that direction at $1$ m/s, bouncing off of the walls of the square. After a minute, the position of the particle is recorded.
The expected distance from this point to the start point can be written as $$\frac{1}{a}\left(b - c\sqrt{d}\right),$$ where $a$ and $b$ are relatively prime, and d is not divisible by any perfect square. Find $a + b + c + d$.
PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
If
\[
\sum_{n=1}^{\infty}\frac{\frac11 + \frac12 + \dots + \frac 1n}{\binom{n+100}{100}} = \frac pq
\]
for relatively prime positive integers $p,q$, find $p+q$.
[i]Proposed by Michael Kural[/i]
Suppose a table with one row and infinite columns. We call each $1\times1$ square a [i]room[/i]. Let the table be finite from left. We number the rooms from left to $\infty$. We have put in some rooms some coins (A room can have more than one coin.). We can do $2$ below operations:
$a)$ If in $2$ adjacent rooms, there are some coins, we can move one coin from the left room $2$ rooms to right and delete one room from the right room.
$b)$ If a room whose number is $3$ or more has more than $1$ coin, we can move one of its coins $1$ room to right and move one other coin $2$ rooms to left.
$i)$ Prove that for any initial configuration of the coins, after a finite number of movements, we cannot do anything more.
$ii)$ Suppose that there is exactly one coin in each room from $1$ to $n$. Prove that by doing the allowed operations, we cannot put any coins in the room $n+2$ or the righter rooms.
The monic polynomial $P(x) = x^n + a_{n-1}x^{n-1} +...+ a_0$ of degree $n > 1$ has $n$ distinct negative roots. Prove that $a_1P(1) > 2n^2a_o$
Let $n$ be a given number greater than 2. We consider the set $V_n$ of all the integers of the form $1 + kn$ with $k = 1, 2, \ldots$ A number $m$ from $V_n$ is called indecomposable in $V_n$ if there are not two numbers $p$ and $q$ from $V_n$ so that $m = pq.$ Prove that there exist a number $r \in V_n$ that can be expressed as the product of elements indecomposable in $V_n$ in more than one way. (Expressions which differ only in order of the elements of $V_n$ will be considered the same.)
There are a) $2022$, b) $2023$ plates placed around a round table and on each of them there is one coin. Alice and Bob are playing a game that proceeds in rounds indefinitely as follows. In each round, Alice first chooses a plate on which there is at least one coin. Then Bob moves one coin from this plate to one of the two adjacent plates, chosen by him. Determine whether it is possible for Bob to select his moves so that, no matter how Alice selects her moves, there are never more than two coins on any plate.