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 $1000$.

Let $\triangle ABC$ be a triangle and $\omega$ its circumcircle. The exterior angle bisector of $\angle BAC$ intersects $\omega$ at point $D$. Let $X$ be the foot of the altitude from $C$ to $AD$ and let $F$ be the intersection of the internal angle bisector of $\angle BAC$ and $BC$. Show that $BX$ bisects segment $AF$.

For a positive integer $n$ and any real number $c$, define $x_k$ recursively by : \[ x_0=0,x_1=1 \text{ and for }k\ge 0, \;x_{k+2}=\frac{cx_{k+1}-(n-k)x_k}{k+1} \] Fix $n$ and then take $c$ to be the largest value for which $x_{n+1}=0$. Find $x_k$ in terms of $n$ and $k,\; 1\le k\le n$.

Each side of a convex quadrilateral $ ABCD $ is divided into three equal parts; a straight line is drawn through the dividing points of sides $ AB $ and $ AD $ that lie closer to vertex $ A $, and similarly for vertices $ B $, $ C $, $ D $. Prove that the center of gravity of the quadrilateral formed by the drawn lines coincides with the center of gravity of quadrilateral $ ABCD $.

Let $a_{ij}, i = 1, 2, \dots, m$ and $j = 1, 2, \dots, n$ be positive real numbers. Prove that \[ \sum_{i = 1}^m \left( \sum_{j = 1}^n \frac{1}{a_{ij}} \right)^{-1} \le \left( \sum_{j = 1}^n \left( \sum_{i = 1}^m a_{ij} \right)^{-1} \right)^{-1} \]

Given $a \in R^{+}$ and an integer $n > 4$ determine all n-tuples ($x_1, ...,x_n$) of positive real numbers that satisfy the following system of equations: $\begin {cases} x_1x_2(3a-2x_3) = a^3\\ x_2x_3(3a-2x_4) = a^3\\ ...\\ x_{n-2}x_{n-1}(3a-2x_n) = a^3\\ x_{n-1}x_n(3a-2x_1) = a^3 \\ x_nx_1(3a-2x_2) = a^3 \end {cases}$ .

The 2010 positive numbers $a_1, a_2, \ldots , a_{2010}$ satisfy the inequality $a_ia_j \le i+j$ for all distinct indices $i, j$. Determine, with proof, the largest possible value of the product $a_1a_2\ldots a_{2010}$.

[b]p1.[/b] Velociraptor $A$ is located at $x = 10$ on the number line and runs at $4$ units per second. Velociraptor $B$ is located at $x = -10$ on the number line and runs at $3$ units per second. If the velociraptors run towards each other, at what point do they meet? [b]p2.[/b] Let $n$ be a positive integer. There are $n$ non-overlapping circles in a plane with radii $1, 2, ... , n$. The total area that they enclose is at least $100$. Find the minimum possible value of $n$. [b]p3.[/b] How many integers between $1$ and $50$, inclusive, are divisible by $4$ but not $6$? [b]p4.[/b] Let $a \star b = 1 + \frac{b}{a}$. Evaluate $((((((1 \star 1) \star 1) \star 1) \star 1) \star 1) \star 1) \star 1$. [b]p5.[/b] In acute triangle $ABC$, $D$ and $E$ are points inside triangle $ABC$ such that $DE \parallel BC$, $B$ is closer to $D$ than it is to $E$, $\angle AED = 80^o$ , $\angle ABD = 10^o$ , and $\angle CBD = 40^o$. Find the measure of $\angle BAE$, in degrees. [b]p6. [/b]Al is at $(0, 0)$. He wants to get to $(4, 4)$, but there is a building in the shape of a square with vertices at $(1, 1)$, $(1, 2)$, $(2, 2)$, and $(2, 1)$. Al cannot walk inside the building. If Al is not restricted to staying on grid lines, what is the shortest distance he can walk to get to his destination? [b]p7. [/b]Point $A = (1, 211)$ and point $B = (b, 2011)$ for some integer $b$. For how many values of $b$ is the slope of $AB$ an integer? [b]p8.[/b] A palindrome is a number that reads the same forwards and backwards. For example, $1$, $11$ and $141$ are all palindromes. How many palindromes between $1$ and 1000 are divisible by $11$? [b]p9.[/b] Suppose $x, y, z$ are real numbers that satisfy: $$x + y - z = 5$$ $$y + z - x = 7$$ $$z + x - y = 9$$ Find $x^2 + y^2 + z^2$. [b]p10.[/b] In triangle $ABC$, $AB = 3$ and $AC = 4$. The bisector of angle $A$ meets $BC$ at $D$. The line through $D$ perpendicular to $AD$ intersects lines $AB$ and $AC$ at $F$ and $E$, respectively. Compute $EC - FB$. (See the following diagram.) [img]https://cdn.artofproblemsolving.com/attachments/2/7/e26fbaeb7d1f39cb8d5611c6a466add881ba0d.png[/img] [b]p11.[/b] Bob has a six-sided die with a number written on each face such that the sums of the numbers written on each pair of opposite faces are equal to each other. Suppose that the numbers $109$, $131$, and $135$ are written on three faces which share a corner. Determine the maximum possible sum of the numbers on the three remaining faces, given that all three are positive primes less than $200$. [b]p12.[/b] Let $d$ be a number chosen at random from the set $\{142, 143, ..., 198\}$. What is the probability that the area of a rectangle with perimeter $400$ and diagonal length $d$ is an integer? [b]p13.[/b] There are $3$ congruent circles such that each circle passes through the centers of the other two. Suppose that $A, B$, and $C$ are points on the circles such that each circle has exactly one of $A, B$, or $C$ on it and triangle $ABC$ is equilateral. Find the ratio of the maximum possible area of $ABC$ to the minimum possible area of $ABC$. (See the following diagram.) [img]https://cdn.artofproblemsolving.com/attachments/4/c/162554fcc6aa21ce3df3ce6a446357f0516f5d.png[/img] [b]p14.[/b] Let $k$ and $m$ be constants such that for all triples $(a, b, c)$ of positive real numbers, $$\sqrt{ \frac{4}{a^2}+\frac{36}{b^2}+\frac{9}{c^2}+\frac{k}{ab} }=\left| \frac{2}{a}+\frac{6}{b}+\frac{3}{c}\right|$$ if and only if $am^2 + bm + c = 0$. Find $k$. [b]p15.[/b] A bored student named Abraham is writing $n$ numbers $a_1, a_2, ..., a_n$. The value of each number is either $1, 2$, or $3$; that is, $a_i$ is $1, 2$ or $3$ for $1 \le i \le n$. Abraham notices that the ordered triples $$(a_1, a_2, a_3), (a_2, a_3, a_4), ..., (a_{n-2}, a_{n-1}, a_n), (a_{n-1}, a_n, a_1), (a_n, a_1, a_2)$$ are distinct from each other. What is the maximum possible value of $n$? Give the answer n, along with an example of such a sequence. Write your answer as an ordered pair. (For example, if the answer were $5$, you might write $(5, 12311)$.) PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $ABC$ be a non-isosceles triangle with circumcircle $\omega$ and let $H, M$ be orthocenter and midpoint of $AB$ respectively. Let $P,Q$ be points on the arc $AB$ of $\omega$ not containing $C$ such that $\angle ACP=\angle BCQ < \angle ACQ$.Let $R,S$ be the foot of altitudes from $H$ to $CQ,CP$ respectively. Prove that thé points $P,Q,R,S$ are concyclic and $M$ is the center of this circle.