The roots of $x^3 + 2x^2 - x + 3$ are $p, q,$ and $r.$ What is the value of \[(p^2 + 4)(q^2 + 4)(r^2 + 4)?\] $\textbf{(A) } 64 \qquad \textbf{(B) } 75 \qquad \textbf{(C) } 100 \qquad \textbf{(D) } 125 \qquad \textbf{(E) } 144$

Let $a,b,c$ be positive numbers such that $a+b+c \geq \dfrac 1a + \dfrac 1b + \dfrac 1c$. Prove that \[ a+b+c \geq \frac 3{abc}. \]

Let $f: \mathbb{R}^2 \to \mathbb{R}$ be a function satisfying the following property: If $A, B, C \in \mathbb{R}^2$ are the vertices of an equilateral triangle with sides of length $1$, then $$f(A) + f(B) + f(C) = 0.$$ Show that $f(x) = 0$ for all $x \in \mathbb{R}^2$. [i]Proposed by Ilir Snopce[/i]

Given a regular hexagon $ABCDEF$, let point $P$ be the intersection of lines $BC$ and $DE$, and let point $Q$ be the intersection of lines $AP$ and $CD$. If the area of $\triangle QEP$ is equal to $72$, find the area of regular hexagon $ABCDEF$. [i]Proposed by [b]DeToasty3[/b][/i]

Prove it is possible to find $2^{2021}$ different pairs of positive integers $(a_i,b_i)$ such that: $$ \frac{1}{a_ib_i}+\frac{1}{a_2b_2} + \ldots + \frac{1}{a_{2^{2021}}b_{2^{2021}}} = 1 $$ $$ a_1+a_2 +\ldots a_{2^{2021}} +b_1+b_2 + \ldots +b_{2^{2021}} = 3^{2022} $$ [b]Note: [/b]Pairs $(a,b)$ and $(c,d)$ are different if $a \neq c$ or $b \neq d$

Let $ P(x)$ be a quadratic polynomial with real coefficients satisfying \[x^2 \minus{} 2x \plus{} 2 \le P(x) \le 2x^2 \minus{} 4x \plus{} 3\] for all real numbers $ x$, and suppose $ P(11) \equal{} 181$. Find $ P(16)$.

For $n ,p \in N^*$ , $ 1 \le p \le n$, we define $$ R_n^p = \sum_{k=0}^p (p-k)^n(-1)^k C_{n+1}^k $$ Show that: $R_n^{n-p+1} =R_n^p$ .

Find a $n\in\mathbb{N}$ such that for all primes $p$, $n$ is divisible by $p$ if and only if $n$ is divisible by $p-1$.

Consider the polynomial $x^3-3x^2+10$. Let $a, b, c$ be its roots. Compute $a^2b^2c^2 + a^2b^2 + b^2c^2 + c^2a^2 + a^2 + b^2 + c^2$.

A two-digit number $\overline{ab}$ is [i]self-loving[/i] if $a$ and $b$ are relatively prime positive integers and $\overline{ab}$ is divisible by $a+b$. How many self-loving numbers are there? [i]Proposed by Anthony Yang and Andy Xu[/i]

Two not necessarily equal non-intersecting wooden disks, one gray and one black, are glued to a plane. An infinite angle with one gray side and one black side can be moved along the plane so that the disks remain outside the angle, while the colored sides of the angle are tangent to the disks of the same color (the tangency points are not the vertices). Prove that it is possible to draw a ray in the angle, starting from the vertex of the angle and such that no matter how the angle is positioned, the ray passes through some fixed point of the plane. (Egor Bakaev, Ilya Bogdanov, Pavel Kozhevnikov, Vladimir Rastorguev) (Junior version [url=https://artofproblemsolving.com/community/c6h2094701p15140671]here[/url]) [hide=note]There was a mistake in the text of the problem 3, we publish here the correct version. The solutions were estimated according to the text published originally.[/hide]

Let $d$ be a positive integer. Show that for every integer $S$, there exists an integer $n>0$ and a sequence of $n$ integers $\epsilon_1, \epsilon_2,..., \epsilon_n$, where $\epsilon_i = \pm 1$ (not necessarily dependent on each other) for all integers $1\le i\le n$, such that $S=\sum_{i=1}^{n}{\epsilon_i(1+id)^2}$.

The Binomial Expansion is valid for exponents that are not integers. That is, for all real numbers $ x, y,$ and $ r$ with $ |x| > |y|,$ \[ (x \plus{} y)^r \equal{} x^r \plus{} rx^{r \minus{} 1}y \plus{} \frac {r(r \minus{} 1)}2x^{r \minus{} 2}y^2 \plus{} \frac {r(r \minus{} 1)(r \minus{} 2)}{3!}x^{r \minus{} 3}y^3 \plus{} \cdots \] What are the first three digits to the right of the decimal point in the decimal representation of $ \left(10^{2002} \plus{} 1\right)^{10/7}?$

Show that in a triangle the altitude of the longest side is at most as long as it the the sum of the lengths of the perpendicular segments drawn from any point on the longest side on the other two sides.

The number obtained by taking the last two digits of $5^{2024}$ in the same order is:

Evaluate: $ \sum\limits_{k\equal{}1}^\infty \frac{1}{k\sqrt{k\plus{}2}\plus{}(k\plus{}2)\sqrt{k}}$

Given two (identical) polygonal domains in the Euclidean plane, it is not possible in general to superpose the two using only translations and rotations. Prove that this can however be achieved by splitting one of the domains into a finite number of polygonal subdomains which then fit together, via translations and rotations in the plane, to recover the other domain.

Find the number of ordered pairs of integers $(x, y)$ such that $2167$ divides $3x^2 + 27y^2 + 2021$ with $0 \le x, y \le 2166$.

The equation $(x+a) (x+b) = 9$ has a root $a+b$. Prove that $ab\le 1$.

Count the number $N$ of all sets $A:=\{x_1,x_2,x_3,x_4\}$ of non-negative integers satisfying $$x_1+x_2+x_3+x_4=36$$ in at least four different ways. [i]Proposed by Eugene J. Ionaşcu[/i]

A circle of length $10$ is inscribed in a convex polygon with perimeter $15$. What part of the area of this polygon is occupied by the resulting circle?

Let $n\ge 3$ be an integer. Consider $n$ points in the plane, no three lying on the same line, and a squirrel in each one of them. Alex wants to give hazelnuts to the squirrels, so he proceeds as follows: for each convex polygon with vertices among the n points, he identifies the squirrels which lie on its sides or in its interior and gives to each of these squirrels q hazelnuts, where q is their number. At the end of the process, a squirrel with the least number of hazelnuts is called unlucky. Determine the maximum number of hazelnuts an unlucky squirrel can get. ([i]proposed by Cristi Savesku[/i])

Let $ABCD$ be a parallelogram. Let $X \notin AC $ lie inside $ABCD$ so that $\angle AXB = \angle CXD = 90^ {\circ}$ and $\Omega$ denote the circumcircle of $AXC$. Consider a diameter $EF$ of $\Omega$ and assume neither $E, \ X, \ B$ nor $F, \ X, \ D$ are collinear. Let $K \neq X$ be an intersection point of circumcircles of $BXE$ and $DXF$ and $L \neq X$ be such point on $\Omega$ so that $\angle KXL = 90^{\circ}$. Prove that $AB = KL$.

A maths teacher has $10$ cards with the numbers $1$ to $10$ on them, one number per card. She places these cards in some order in a line next to each other on the table. The students come to the table, one at a time. The student whose turn it is goes once through the line of cards from left to right and removes every card she encounters that is (at that moment) the lowest card on the table. This continues till all cards are removed from the table. For example, if the line is in order $3$, $1$, $4$, $5$, $8,$ $6$, $9$, $10$, $2$, $7$ from left to right, the first student takes cards $1$ and $2$. Then the second student comes who, in our example, takes the cards $3$, $4$, $5$, $6$, and $7$. The third student then takes the cards $8$, $9$, and $10$. Let $A$ be the number of sequences of cards that the teacher can choose so that exactly nine students get a turn to pick cards. Let $B$ be the number of sequences of cards that the teacher can choose so that exactly two students get a turn to pick cards. Prove that $A = B$.

The LHS Math Team is going to have a Secret Santa event! Nine members are going to participate, and each person must give exactly one gift to a specific recipient so that each person receives exactly one gift. But to make it less boring, no pairs of people can just swap gifts. The number of ways to assign who gives gifts to who in the Secret Santa Exchange with these constraints is $N$. Find the remainder when $N$ is divided by $1000$.