Given a positive integer $m$, let $d(m)$ be the number of positive divisors of $m$. Determine all positive integers $n$ such that $d(n) +d(n+ 1) = 5$.

Find all ordered triples $ (x, y, z)$ of real numbers such that \[ 5 \left(x \plus{} \frac{1}{x} \right) \equal{} 12 \left(y \plus{} \frac{1}{y} \right) \equal{} 13 \left(z \plus{} \frac{1}{z} \right),\] and \[ xy \plus{} yz \plus{} zy \equal{} 1.\]

Consider the real numbers $ a\ne 0,b,c$ such that the function $ f(x) \equal{} ax^2 \plus{} bx \plus{} c$ satisfies $ |f(x)|\le 1$ for all $ x\in [0,1]$. Find the greatest possible value of $ |a| \plus{} |b| \plus{} |c|$.

Aisling and Brendan take alternate moves in the following game. Before the game starts, the number $x = 2023$ is written on a piece of paper. Aisling makes the first move. A move from a positive integer $x$ consists of replacing $x$ either with $x + 1$ or with $x/p$ where $p$ is a prime factor of $x$. The winner is the first player to write $x = 1$. Determine whether Aisling or Brendan has a winning strategy for this game.

Let $ C_1,C_2 $ be two distinct concentric circles, and $ BA $ be a diameter of $ C_1. $ Choose the points $ M,N $ on $ C_1,C_2, $ respectively, but not on the line $ BA. $ [b]a)[/b] Show that there are unique points $ P,Q $ on $ MA,MB, $ respectively, so that $ N $ is the middle of $ PQ. $ [b]b)[/b] Prove that the value $ AP^2+BQ^2 $ does not depend on $ M,N. $ [i]Mihai Piticari, Mihail Bălună[/i]

Given an acute triangle find a point inside the triangle such that the sum of the distances from this point to the three vertices is the least.

Find all positive integers $a$, $b$ and $c$ such that $ab$ is a square, and \[a+b+c-3\sqrt[3]{abc}=1.\] [i]Proposed by usjl[/i]

How many square yards of carpet are required to cover a rectangular floor that is $12$ feet long and $9$ feet wide? (There are 3 feet in a yard.) $\textbf{(A) }12\qquad\textbf{(B) }36\qquad\textbf{(C) }108\qquad\textbf{(D) }324\qquad \textbf{(E) }972$

Determine all three integers $(x, y, z)$ that are solutions of the system $$x + y -z = 6$$ $$x^3 + y^3 -z^3 = 414$$

Let be $n\geq 3$ fixed positive integer.Let be real numbers $a_1,a_2,...,a_n,b_1,b_2,...,b_n$ such that satisfied this conditions: [b]$i)$[/b] $ $ $a_n\geq a_{n-1}$ and $b_n\geq b_{n-1}$ [b]$ii)$[/b] $ $ $0<a_1\leq b_1\leq a_2\leq b_2\leq ... \leq a_{n-1}\leq b_{n-1}$ [b]$iii)$[/b] $ $ $a_1+a_2+...+a_n=b_1+b_2+...+b_n$ [b]$iv)$[/b] $ $ $a_{1}\cdot a_2\cdot ...\cdot a_n=b_1\cdot b_2\cdot ...\cdot b_n$ Show that $a_i=b_i$ for all $i=1,2,...,n$

Given is a triangle $ABC$, its circumcircle $\omega$, and a circle $k$ that touches $\omega$ from the outside, and also touches rays $AB$ and $AC$ in $P$ and $Q$, respectively. Prove that the $A$-excenter of $\triangle ABC$ is the midpoint of $\overline{PQ}$.

Find the smallest positive integer for which when we move the last right digit of the number to the left, the remaining number be $\frac 32$ times of the original number.

In the convex quadrilateral $ABCD$ we have that $\angle BCD = \angle ADC \ge 90 ^o$. The bisectors of $\angle BAD$ and $\angle ABC$ intersect in $M$. Prove that if $M \in CD$, then $M$ is the middle of $CD$.

Square $ABCD$ has side length $2$. For each $0 \leq r \leq 2$, point $P_r$ is on side $\overline{AB}$ with $AP_r = r$, and square $\Sigma_r$ is constructed with diagonal $\overline{DP_r}$. Let region $\mathcal{R}$ be the set of all points that are in both $\Sigma_0$ and $\Sigma_2$, but not in $\Sigma_r$ for at least one value of $r$. Find the area of the convex hull of $\mathcal{R}$. [i]Proposed by Justin Hsieh[/i]

Nine skiers left the start line in turn and covered the distance, each at their own constant speed. Could it turn out that each skier participated in exactly four overtakes? (In each overtaking, exactly two skiers participate - the one who is overtaking, and the one who is being overtaken.)

A tree has $10$ pounds of apples at dawn. Every afternoon, a bird comes and eats $x$ pounds of apples. Overnight, the amount of food on the tree increases by $10\%$. What is the maximum value of $x$ such that the bird can sustain itself indefinitely on the tree without the tree running out of food?

Find all positive real numbers $\lambda$ such that every sequence $a_1, a_2, \ldots$ of positive real numbers satisfying \[ a_{n+1}=\lambda\cdot\frac{a_1+a_2+\ldots+a_n}{n} \] for all $n\geq 2024^{2024}$ is bounded. [i]Remark:[/i] A sequence $a_1,a_2,\ldots$ of positive real numbers is \emph{bounded} if there exists a real number $M$ such that $a_i<M$ for all $i=1,2,\ldots$

Two circles intersect at two points $A$ and $B$. A line $\ell$ which passes through the point $A$ meets the two circles again at the points $C$ and $D$, respectively. Let $M$ and $N$ be the midpoints of the arcs $BC$ and $BD$ (which do not contain the point $A$) on the respective circles. Let $K$ be the midpoint of the segment $CD$. Prove that $\measuredangle MKN = 90^{\circ}$.

$2019$ point grasshoppers sit on a line. At each move one of the grasshoppers jumps over another one and lands at the point the same distance away from it. Jumping only to the right, the grasshoppers are able to position themselves so that some two of them are exactly $1$ mm apart. Prove that the grasshoppers can achieve the same, jumping only to the left and starting from the initial position. (Sergey Dorichenko)

For a subset $ S$ of vertices of graph $ G$, let $ \Lambda(S)$ be the subset of all edges of $ G$ such that at least one of their ends is in $ S$. Suppose that $ G$ is a graph with $ m$ edges. Let $ d^*: V(G)\longrightarrow\mathbb N\cup\{0\}$ be a function such that a) $ \sum_{u}d^*(u)\equal{}m$. b) For each subset $ S$ of $ V(G)$: \[ \sum_{u\in S}d^*(u)\leq|\Lambda(S)|\] Prove that we can give directions to edges of $ G$ such that for each edge $ e$, $ d^\plus{}(e)\equal{}d^*(e)$.

A subset of the set $A={1,2,\dots ,n}$ is called [i]connected[/i], if it consists of one number or a certain amount of consecutive numbers. Find the greatest $k$ (defined as a function of $n$) for which there exists $k$ different subsets $A_1,A_2,…,A_k$ of $A$ the intersection of each two of which is a [i]connected[/i] set.

[b]p1.[/b] Suppose that $a \oplus b = ab - a - b$. Find the value of $$((1 \oplus 2) \oplus (3 \oplus 4)) \oplus 5.$$ [b]p2.[/b] Neethin scores a $59$ on his number theory test. He proceeds to score a $17$, $23$, and $34$ on the next three tests. What score must he achieve on his next test to earn an overall average of $60$ across all five tests? [b]p3.[/b] Consider a triangle with side lengths $28$ and $39$. Find the number of possible integer lengths of the third side. [b]p4.[/b] Nithin is thinking of a number. He says that it is an odd two digit number where both of its digits are prime, and that the number is divisible by the sum of its digits. What is the sum of all possible numbers Nithin might be thinking of? [b]p5.[/b] Dora sees a fire burning on the dance floor. She calls her friends to warn them to stay away. During the first pminute Dora calls Poonam and Serena. During the second minute, Poonam and Serena call two more friends each, and so does Dora. This process continues, with each person calling two new friends every minute. How many total people would know of the fire after $6$ minutes? [b]p6.[/b] Charlotte writes all the positive integers $n$ that leave a remainder of $2$ when $2018$ is divided by $n$. What is the sum of the numbers that she writes? [b]p7.[/b] Consider the following grid. Stefan the bug starts from the origin, and can move either to the right, diagonally in the positive direction, or upwards. In how many ways can he reach $(5, 5)$? [img]https://cdn.artofproblemsolving.com/attachments/9/9/b9fdfdf604762ec529a1b90d663e289b36b3f2.png[/img] [b]p8.[/b] Let $a, b, c$ be positive numbers where $a^2 + b^2 + c^2 = 63$ and $2a + 3b + 6c = 21\sqrt7$. Find $\left( \frac{a}{c}\right)^{\frac{a}{b}} $. [b]p9.[/b] What is the sum of the distinct prime factors of $12^5 + 12^4 + 1$? [b]p10.[/b] Allen starts writing all permutations of the numbers $1$, $2$, $3$, $4$, $5$, $6$ $7$, $8$, $9$, $10$ on a blackboard. At one point he writes the permutation $9$, $4$, $3$, $1$, $2$, $5$, $6$, $7$, $8$, $10$. David points at the permutation and observes that for any two consecutive integers $i$ and $i+1$, all integers that appear in between these two integers in the permutation are all less than $i$. For example, $4$ and $5$ have only the numbers $3$, $1$, $2$ in between them. How many of the $10!$ permutations on the board satisfy this property that David observes? [b]p11.[/b] (Estimation) How many positive integers less than $2018$ can be expressed as the sum of $3$ square numbers? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Two operations are allowed: (i) to write two copies of number $1$; (ii) to replace any two identical numbers $n$ by $(n + 1)$ and $(n - 1)$. Find the minimal number of operations that required to produce the number $2005$ (at the beginning there are no numbers). [i](8 points)[/i]

The number of ounces of water needed to reduce $ 9$ ounces of shaving lotion containing $ 50\%$ alcohol to a lotion containing $ 30\%$ alcohol is: $ \textbf{(A)}\ 3 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}\ 6 \qquad\textbf{(E)}\ 7$

Let $k$ be a positive integer. A sequence $a_0,a_1,...,a_n,n>0$ of positive integers satisfies the following conditions: $(i)$ $a_0=a_n=1$; $(ii)$ $2\leq a_i\leq k$ for each $i=1,2,...,n-1$; $(iii)$For each $j=2,3,...,k$, the number $j$ appears $\phi(j)$ times in the sequence $a_0,a_1,...,a_n$, where $\phi(j)$ is the number of positive integers that do not exceed $j$ and are coprime to $j$; $(iv)$For any $i=1,2,...,n-1$, $\gcd(a_i,a_{i-1})=1=\gcd(a_i,a_{i+1})$, and $a_i$ divides $a_{i-1}+a_{i+1}$. Suppose there is another sequence $b_0,b_1,...,b_n$ of integers such that $\frac{b_{i+1}}{a_{i+1}}>\frac{b_i}{a_i}$ for all $i=0,1,...,n-1$. Find the minimum value of $b_n-b_0$.