Let $a_1, a_2, a_3, . . .$ be a sequence of positive real numbers that satisfies $a_1 = 1$ and $a^2_{n+1} + a_{n+1} = a_n$ for every natural number $n$. Prove that $a_n \ge \frac{1}{n}$ for every natural number $n$.

Let $\triangle ABC$ have its vertices at $A(0, 0), B(7, 0), C(3, 4)$ in the Cartesian plane. Construct a line through the point $(6-2\sqrt 2, 3-\sqrt 2)$ that intersects segments $AC, BC$ at $P, Q$ respectively. If $[PQC] = \frac{14}3$, what is $|CP|+|CQ|$? [i](Source: China National High School Mathematics League 2021, Zhejiang Province, Problem 9)[/i]

Find the polynomial $P(x)$ with real coefficients such that $P(2)=12$ and $P(x^2)=x^2(x^2+1)P(x)$ for each $x\in\mathbb{R}$.

Jean writes five tests and achieves the marks shown on the graph. What is her average mark on these five tests? [asy] draw(origin -- (0, 10.1)); for(int i = 0; i < 11; ++i) { draw((0, i) -- (10.5, i)); label(string(10*i), (0, i), W); } filldraw((1, 0) -- (1, 8) -- (2, 8) -- (2, 0) -- cycle, black); filldraw((3, 0) -- (3, 7) -- (4, 7) -- (4, 0) -- cycle, black); filldraw((5, 0) -- (5, 6) -- (6, 6) -- (6, 0) -- cycle, black); filldraw((7, 0) -- (7, 9) -- (8, 9) -- (8, 0) -- cycle, black); filldraw((9, 0) -- (9, 8) -- (10, 8) -- (10, 0) -- cycle, black); label("Test Marks", (5, 0), S); label(rotate(90)*"Marks out of 100", (-2, 5), W); [/asy] $\textbf{(A)}\ 74 \qquad \textbf{(B)}\ 76 \qquad \textbf{(C)}\ 70 \qquad \textbf{(D)}\ 64 \qquad \textbf{(E)}\ 79$

Let $x, y$, and $z$ be three not necessarily real numbers that satisfy the following system of equations: $x^3 -4 = (2y +1)^2$ $y^3 -4 = (2z +1)^2$ $z^3 -4 = (2x +1)^2$. Find the greatest possible real value of $(x -1)(y -1)(z -1)$.

There are $n$ dots on the plane such that no three dots are collinear. Each dot is assigned a $0$ or a $1$. Each pair of dots is connected by a line segment. If the endpoints of a line segment are two dots with the same number, then the segment is assigned a $0$. Otherwise, the segment is assigned a $1$. Find all $n$ such that it is possible to assign $0$'s and $1$'s to the $n$ dots in a way that the corresponding line segments are assigned equally many $0$'s as $1$'s.

We say that the set of step lengths $D\subset \mathbb{Z}_+=\{1,2,\ldots\}$ is [i]excellent[/i] if it has the following property: If we split the set of integers into two subsets $A$ and $\mathbb{Z}\setminus{A}$, at least other set contains element $a-d,a,a+d$ (i.e. $\{a-d,a,a+d\} \subset A$ or $\{a-d,a,a+d\}\in \mathbb{Z}\setminus A$ from some integer $a\in \mathbb{Z},d\in D$.) For example the set of one element $\{1\}$ is not excellent as the set of integer can be split into even and odd numbers, and neither of these contains three consecutive integer. Show that the set $\{1,2,3,4\}$ is excellent but it has no proper subset which is excellent.

[b]p1.[/b] What is $7\%$ of one half of $11\%$ of $20000$ ? [b]p2.[/b] Three circles centered at $A, B$, and $C$ are tangent to each other. Given that $AB = 8$, $AC = 10$, and $BC = 12$, find the radius of circle $ A$. [b]p3. [/b]How many positive integer values of $x$ less than $2012$ are there such that there exists an integer $y$ for which $\frac{1}{x} +\frac{2}{2y+1} =\frac{1}{y}$ ? [b]p4. [/b]The positive difference between $ 8$ and twice $x$ is equal to $11$ more than $x$. What are all possible values of $x$? [b]p5.[/b] A region in the coordinate plane is bounded by the equations $x = 0$, $x = 6$, $y = 0$, and $y = 8$. A line through $(3, 4)$ with slope $4$ cuts the region in half. Another line going through the same point cuts the region into fourths, each with the same area. What is the slope of this line? [b]p6.[/b] A polygon is composed of only angles of degrees $138$ and $150$, with at least one angle of each degree. How many sides does the polygon have? [b]p7.[/b] $M, A, T, H$, and $L$ are all not necessarily distinct digits, with $M \ne 0$ and $L \ne 0$. Given that the sum $MATH +LMT$, where each letter represents a digit, equals $2012$, what is the average of all possible values of the three-digit integer $LMT$? [b]p8. [/b]A square with side length $\sqrt{10}$ and two squares with side length $\sqrt{7}$ share the same center. The smaller squares are rotated so that all of their vertices are touching the sides of the larger square at distinct points. What is the distance between two such points that are on the same side of the larger square? [b]p9.[/b] Consider the sequence $2012, 12012, 20120, 20121, ...$. This sequence is the increasing sequence of all integers that contain “$2012$”. What is the $30$th term in this sequence? [b]p10.[/b] What is the coefficient of the $x^5$ term in the simplified expansion of $(x +\sqrt{x} +\sqrt[3]{x})^{10}$ ? PS. You had better use hide for answers.

On each day, more than half of the inhabitants of Évora eats [i]sericaia[/i] as dessert. Show that there is a group of 10 inhabitants of Évora such that, on each of the last 2010 days, at least one of the inhabitants ate [i]sericaia[/i] as dessert.

Let $A_1,\ldots,A_n$ be points of an ellipsoid with center $O$ in $\mathbb R^n$ such that $OA_i$, for $i=1,\ldots,n$, are mutually orthogonal. Prove that the distance of the point $O$ from the hyperplane $A_1A_2\ldots A_n$ does not depend on the choice of the points $A_1,\ldots,A_n$.

How many zeroes does the function $f(x)=2^x -1 -x^2 $ have on the real line?

Let $A$, $B$, $C$, $D$ be four points on a circle in that order. Also, $AB=3$, $BC=5$, $CD=6$, and $DA=4$. Let diagonals $AC$ and $BD$ intersect at $P$. Compute $\frac{AP}{CP}$.

Determine all $n \in \mathbb{Z}^+$ such that a regular hexagon (i.e. all sides equal length, all interior angles same size) can be partitioned in finitely many $n-$gons such that they can be composed into $n$ congruent regular hexagons in a non-overlapping way upon certain rotations and translations.

Let $ n\ge 5$ be a positive integer. Prove that the set $ \{1,2,\ldots,n\}$ can be partitioned into two non-zero subsets $ S_n$ and $ P_n$ such that the sum of elements in $ S_n$ is equal to the product of elements in $ P_n$.

Diagonals of a cyclic quadrilateral $ABCD$ intersect at point $P$. The circumscribed circles of triangles $APD$ and $BPC$ intersect the line $AB$ at points $E, F$ correspondingly. $Q$ and $R$ are the projections of $P$ onto the lines $FC, DE$ correspondingly. Show that $AB \parallel QR$. [i](Proposed by Mykhailo Shtandenko)[/i]

Find all functions $f:\mathbb{N} \rightarrow \mathbb{N}$ such that for all positive integers $n$, there exists an unique positive integer $k$, satisfying $f^k(n)\leq n+k+1$.

Three points $A,B,C$ are such that $B \in ]AC[$. On the side of $AC$ we draw the three semicircles with diameters $[AB], [BC]$ and $[AC]$. The common interior tangent at $B$ to the first two semi-circles meets the third circle in $E$. Let $U$ and $V$ be the points of contact of the common exterior tangent to the first two semi-circles. Denote the area of the triangle $ABC$ as $S(ABC)$. Evaluate the ratio $R=\frac{S(EUV)}{S(EAC)}$ as a function of $r_1 = \frac{AB}{2}$ and $r_2 = \frac{BC}{2}$.

Find the continuous function such that $ f(x)\equal{}\frac{e^{2x}}{2(e\minus{}1)}\int_0^1 e^{\minus{}y}f(y)dy\plus{}\int_0^{\frac 12} f(y)dy\plus{}\int_0^{\frac 12}\sin ^ 2(\pi y)dy$.

A strip of width $w$ is the set of all points which lie on, or between, two parallel lines distance $w$ apart. Let $S$ be a set of $n$ ($n \ge 3$) points on the plane such that any three different points of $S$ can be covered by a strip of width $1$. Prove that $S$ can be covered by a strip of width $2$.

Show that $3x^5 +5x^3 -8x$ is divisible by $120$ for any integer $x$

There are $5$ people who start with $1, 2, 3, 4,$ and $5$ cookies, respectively. Every minute, two different people are chosen uniformly at random. If they have $a$ and $b$ cookies and $a\neq b,$ the person with more cookies eats $|a-b|$ of their own cookies. If $a = b,$ the minute still passes with nothing happening. Compute the expected number of minutes until all $5$ people have an equal number of cookies.

Given six points $ A, B, C, D, E, F $ such that $ \triangle BCD \stackrel{+}{\sim} \triangle ECA \stackrel{+}{\sim} \triangle BFA $ and let $ I $ be the incenter of $ \triangle ABC. $ Prove that the circumcenter of $ \triangle AID, \triangle BIE, \triangle CIF $ are collinear. [i]Proposed by Telv Cohl[/i]

Let \begin{align*}X&=\begin{pmatrix}7&8&9\\8&-9&-7\\-7&-7&9\end{pmatrix}\\Y&=\begin{pmatrix}9&8&-9\\8&-7&7\\7&9&8\end{pmatrix}.\end{align*}Let $A=Y^{-1}X$ and let $B$ be the inverse of $X^{-1}+A^{-1}$. Find a matrix $M$ such that $M^2=XY-BY$ (you may assume that $A$ and $X^{-1}+A^{-1}$ are invertible).

The complete set of $x$-values satisfying the inequality $\dfrac{x^2-4}{x^2-1}>0$ is the set of all $x$ such that: $\textbf{(A) }x>2\text{ or }x<-2\text{ or }-1<x<1\qquad\, \textbf{(B) }x>2\text{ or }x<-2$ $\textbf{(C) }x>1\text{ or }x<-2\qquad\qquad\qquad\qquad\,\,\,\,\,\,\, \textbf{(D) }x>1\text{ or }x<-2\qquad$ $\textbf{(E) }x\text{ is any real number except }1\text{ or }-1$

[b]p1.[/b] Let $p > 5$ be a prime. It is known that the average of all of the prime numbers that are at least $5$ and at most $p$ is $12$. Find $p$. [b]p2.[/b] The numbers $1, 2,..., n$ are written down in random order. What is the probability that $n-1$ and $n$ are written next to each other? (Give your answer in term of $n$.) [b]p3.[/b] The Duke Blue Devils are playing a basketball game at home against the UNC Tar Heels. The Tar Heels score $N$ points and the Blue Devils score $M$ points, where $1 < M,N < 100$. The first digit of $N$ is $a$ and the second digit of $N$ is $b$. It is known that $N = a+b^2$. The first digit of $M$ is $b$ and the second digit of $M$ is $a$. By how many points do the Blue Devils win? [b]p4.[/b] Let $P(x)$ be a polynomial with integer coefficients. It is known that $P(x)$ gives a remainder of $1$ upon polynomial division by $x + 1$ and a remainder of $2$ upon polynomial division by $x + 2$. Find the remainder when $P(x)$ is divided by $(x + 1)(x + 2)$. [b]p5.[/b] Dracula starts at the point $(0,9)$ in the plane. Dracula has to pick up buckets of blood from three rivers, in the following order: the Red River, which is the line $y = 10$; the Maroon River, which is the line $y = 0$; and the Slightly Crimson River, which is the line $x = 10$. After visiting all three rivers, Dracula must then bring the buckets of blood to a castle located at $(8,5)$. What is the shortest distance that Dracula can walk to accomplish this goal? [b]p6.[/b] Thirteen hungry zombies are sitting at a circular table at a restaurant. They have five identical plates of zombie food. Each plate is either in front of a zombie or between two zombies. If a plate is in front of a zombie, that zombie and both of its neighbors can reach the plate. If a plate is between two zombies, only those two zombies may reach it. In how many ways can we arrange the plates of food around the circle so that each zombie can reach exactly one plate of food? (All zombies are distinct.) [b]p7.[/b] Let $R_I$ , $R_{II}$ ,$R_{III}$ ,$R_{IV}$ be areas of the elliptical region $$\frac{(x - 10)^2}{10}+ \frac{(y-31)^2}{31} \le 2009$$ that lie in the first, second, third, and fourth quadrants, respectively. Find $R_I -R_{II} +R_{III} -R_{IV}$ . [b]p8.[/b] Let $r_1, r_2, r_3$ be the three (not necessarily distinct) solutions to the equation $x^3+4x^2-ax+1 = 0$. If $a$ can be any real number, find the minimum possible value of $$\left(r_1 +\frac{1}{r_1} \right)^2+ \left(r_2 +\frac{1}{r_2} \right)^2+ \left(r_3 +\frac{1}{r_3} \right)^2$$ [b]p9.[/b] Let $n$ be a positive integer. There exist positive integers $1 = a_1 < a_2 <... < a_n = 2009$ such that the average of any $n - 1$ of elements of $\{a_1, a_2,..., a_n\}$ is a positive integer. Find the maximum possible value of $n$. [b]p10.[/b] Let $A(0) = (2, 7, 8)$ be an ordered triple. For each $n$, construct $A(n)$ from $A(n - 1)$ by replacing the $k$th position in $A(n - 1)$ by the average (arithmetic mean) of all entries in $A(n - 1)$, where $k \equiv n$ (mod $3$) and $1 \le k \le 3$. For example, $A(1) = \left( \frac{17}{3} , 7, 8 \right)$ and $A(2) = \left( \frac{17}{3} , \frac{62}{9}, 8\right)$. It is known that all entries converge to the same number $N$. Find the value of $N$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].