$(CZS 3)$ Let $a$ and $b$ be two positive real numbers. If $x$ is a real solution of the equation $x^2 + px + q = 0$ with real coefficients $p$ and $q$ such that $|p| \le a, |q| \le b,$ prove that $|x| \le \frac{1}{2}(a +\sqrt{a^2 + 4b})$ Conversely, if $x$ satisfies the above inequality, prove that there exist real numbers $p$ and $q$ with $|p|\le a, |q|\le b$ such that $x$ is one of the roots of the equation $x^2+px+ q = 0.$

We say that a triangle $ABC$ is great if the following holds: for any point $D$ on the side $BC$, if $P$ and $Q$ are the feet of the perpendiculars from $D$ to the lines $AB$ and $AC$, respectively, then the reflection of $D$ in the line $PQ$ lies on the circumcircle of the triangle $ABC$. Prove that triangle $ABC$ is great if and only if $\angle A = 90^{\circ}$ and $AB = AC$. [i]Senior Problems Committee of the Australian Mathematical Olympiad Committee[/i]

Juan chooses a five-digit positive integer. Maria erases the ones digit and gets a four-digit number. The sum of this four-digit number and the original five-digit number is $52,713$. What can the sum of the five digits of the original number be?

Find all positive integers $ n$ for which the numbers in the set $ S \equal{} \{1,2, \ldots,n \}$ can be colored red and blue, with the following condition being satisfied: The set $ S \times S \times S$ contains exactly $ 2007$ ordered triples $ \left(x, y, z\right)$ such that: [b](i)[/b] the numbers $ x$, $ y$, $ z$ are of the same color, and [b](ii)[/b] the number $ x \plus{} y \plus{} z$ is divisible by $ n$. [i]Author: Gerhard Wöginger, Netherlands[/i]

Let $x_1, x_2, \dots, x_n$ be different real numbers. Prove that \[\sum_{1 \leqslant i \leqslant n} \prod_{j \neq i} \frac{1-x_{i} x_{j}}{x_{i}-x_{j}}=\left\{\begin{array}{ll} 0, & \text { if } n \text { is even; } \\ 1, & \text { if } n \text { is odd. } \end{array}\right.\]

Let $ABCD$ be a cyclic quadrilateral $\angle{BAD}< 90^\circ$ and $\angle BCA = \angle DCA$. Point $E$ is taken on segment $DA$ such that $BD=2DE$. The line through $E$ parallel to $CD$ intersects the diagonal $AC$ at $F$. Prove that \[ \frac{AC\cdot BD}{AB\cdot FC}=2.\]

What is the smallest $n$ for which there is a solution to $$\begin{cases} \sin x_1 + \sin x_2 + ... + \sin x_n = 0 \\ \sin x_1 + 2 \sin x_2 + ... + n \sin x_n = 100 \end{cases}$$ ?

Give positive numbers $p,q,a,b,c$, if $p,a,q$ is a geometric series, $p,b,c,q$ is an arithmetic sequence. Then, wich is true about the equation $bx^2-ax+c=0$? $\text{(A)}$ It has no real roots. $\text{(B)}$ It has two equal real roots. $\text{(C)}$ It has two different real roots, and their product is positive. $\text{(D)}$ It has two different real roots, and their product is negative.

Let $ABCD$ be an inscribed quadrilateral of some circle $\omega$ with $AC\ \bot \ BD$. Define $E$ to be the intersection of segments $AC$ and $BD$. Let $F$ be some point on segment $AD$ and define $P$ to be the intersection point of half-line $FE$ and $\omega$. Let $Q$ be a point on segment $PE$ such that $PQ\cdot PF = PE^2$. Let $R$ be a point on $BC$ such that $QR\ \bot \ AD$. Prove that $PR=QR$.

Moe uses a mower to cut his rectangular $ 90$-foot by $ 150$-foot lawn. The swath he cuts is $ 28$ inches wide, but he overlaps each cut by $ 4$ inches to make sure that no grass is missed. He walks at the rate of $ 5000$ feet per hour while pushing the mower. Which of the following is closest to the number of hours it will take Moe to mow his lawn? $ \textbf{(A)}\ 0.75 \qquad \textbf{(B)}\ 0.8 \qquad \textbf{(C)}\ 1.35 \qquad \textbf{(D)}\ 1.5 \qquad \textbf{(E)}\ 3$

Let $n$ be an arbitrary positive integer. (a) For every positive integers $a$ and $b$, show that $gcd(n^a + 1, n^b + 1) \le n^{gcd(a,b)} + 1$. (b) Show that there exist infinitely many composite pairs ($a, b)$, such that each of them is not a multiply of the other number and equality holds in (a).

Let $ P (x) $ be a polynomial with real coefficients such that $ P (x)> 0 $ for all $ x \geq 0 $. Prove that there is a positive integer $ n $ such that $ (1 + x) ^ n P (x) $ polynomial with nonnegative coefficients.

Determine the smallest positive integer $m$ with the property that $m^3-3m^2+2m$ is divisible by both $79$ and $83$.

[b]p1.[/b] Farmer John wants to bring some cows to a pasture with grass that grows at a constant rate. Initially, the pasture has some nonzero amount of grass and it will stop growing if there is no grass left. The pasture sustains $100$ cows for ten days. The pasture can also sustain $100$ cows for five days, and then $120$ cows for three more days. If cows eat at a constant rate, fund the maximum number of cows Farmer John can bring to the pasture so that they can be sustained indefinitely. [b]p2.[/b] Sam is learning basic arithmetic. He may place either the operation $+$ or $-$ in each of the blank spots between the numbers below: $$5\,\, \_ \,\, 8\,\, \_ \,\,9\,\, \_ \,\,7\,\,\_ \,\,2\,\,\_ \,\,3$$ In how many ways can he place the operations so the result is divisible by $3$? [b]p3.[/b] Will loves the color blue, but he despises the color red. In the $5\times 6$ rectangular grid below, how many rectangles are there containing at most one red square and with sides contained in the gridlines? [img]https://cdn.artofproblemsolving.com/attachments/1/7/7ce55bdc9e05c7c514dddc7f8194f3031b93c4.png[/img] [b]p4.[/b] Let $r_1, r_2, r_3$ be the three roots of a cubic polynomial $P(x)$. Suppose that $$\frac{P(2) + P(-2)}{P(0)}= 200.$$ If $\frac{1}{r_1r_2}+ \frac{1}{r_2r_3}+\frac{1}{r_3r_1}= \frac{m}{n}$ for relatively prime positive integers $m$ and $n$, compute $m + n$. [b]p5.[/b] Consider a rectangle $ABCD$ with $AB = 3$ and $BC = 1$. Let $O$ be the intersection of diagonals $AC$ and $BD$. Suppose that the circumcircle of $ \vartriangle ADO$ intersects line $AB$ again at $E \ne A$. Then, the length $BE$ can be written as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Find $m + n$. [b]p6.[/b] Let $ABCD$ be a square with side length $100$ and $M$ be the midpoint of side $AB$. The circle with center $M$ and radius $50$ intersects the circle with center $D$ and radius $100$ at point $E$. $CE$ intersects $AB$ at $F$. If $AF = \frac{m}{n}$ for relatively prime positive integers $m$ and $n$, find $m + n$. [b]p7.[/b] How many pairs of real numbers $(x, y)$, with $0 < x, y < 1$ satisfy the property that both $3x + 5y$ and $5x + 2y$ are integers? [b]p8.[/b] Sebastian is coloring a circular spinner with $4$ congruent sections. He randomly chooses one of four colors for each of the sections. If two or more adjacent sections have the same color, he fuses them and considers them as one section. (Sections meeting at only one point are not adjacent.) Suppose that the expected number of sections in the final colored spinner is equal to $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m + n$. [b]p9.[/b] Let $ABC$ be a triangle and $D$ be a point on the extension of segment $BC$ past $C$. Let the line through $A$ perpendicular to $BC$ be $\ell$. The line through $B$ perpendicular to $AD$ and the line through $C$ perpendicular to $AD$ intersect $\ell$ at $H_1$ and $H_2$, respectively. If $AB = 13$, $BC = 14$, $CA = 15$, and $H_1H_2 = 1001$, find $CD$. [b]p10.[/b] Find the sum of all positive integers $k$ such that $$\frac21 -\frac{3}{2 \times 1}+\frac{4}{3\times 2\times 1} + ...+ (-1)^{k+1} \frac{k+1}{k\times (k - 1)\times ... \times 2\times 1} \ge 1 + \frac{1}{700^3}$$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Compute the number of ordered triples $(x,y,z)$ of integers satisfying \[x^2+y^2+z^2=9.\] [i]Proposed by Nathan Xiong[/i]

Tuesday’s high temperature was 4°C warmer than that of Monday’s. Wednesday’s high temperature was 6°C cooler than that of Monday’s. If Tuesday’s high temperature was 22°C, what was Wednesday’s high temperature? $\textbf{(A)}\ 20 \qquad \textbf{(B)}\ 24 \qquad \textbf{(C)}\ 12 \qquad \textbf{(D)}\ 32 \qquad \textbf{(E)}\ 16$

[b]p1.[/b] Find the largest possible number of consecutive $9$’s in which an integer between $10,000,000$ and $13,371,337$ can end. For example, $199$ ends in two $9$’s, while $92,999$ ends in three $9$’s. [b]p2.[/b] Let $ABCD$ be a square of side length $2$. Equilateral triangles $ABP$, $BCQ$, $CDR$, and $DAS$ are constructed inside the square. Compute the area of quadrilateral $PQRS$. [b]p3.[/b] Evaluate the expression $7 \cdot 11 \cdot 13 \cdot 1003 - 3 \cdot 17 \cdot 59 \cdot 331$. [b]p4.[/b] Compute the number of positive integers $c$ such that there is a non-degenerate obtuse triangle with side lengths $21$, $29$, and $c$. [b]p5.[/b] Consider a $5\times 5$ board, colored like a chessboard, such that the four corners are black. Determine the number of ways to place $5$ rooks on black squares such that no two of the rooks attack one another, given that the rooks are indistinguishable and the board cannot be rotated. (Two rooks attack each other if they are in the same row or column.) [b]p6.[/b] Let $ABCD$ be a trapezoid of height $6$ with bases $AB$ and $CD$. Suppose that $AB = 2$ and $CD = 3$, and let $F$ and $G$ be the midpoints of segments $AD$ and $BC$, respectively. If diagonals $AC$ and $BD$ intersect at point $E$, compute the area of triangle $FGE$. [b]p7.[/b] A regular octahedron is a solid with eight faces that are congruent equilateral triangles. Suppose that an ant is at the center of one face of a regular octahedron of edge length $10$. The ant wants to walk along the surface of the octahedron to reach the center of the opposite face. (Two faces of an octahedron are said to be opposite if they do not share a vertex.) Determine the minimum possible distance that the ant must walk. [b]p8.[/b] Let $A_1A_2A_3$, $B_1B_2B_3$, $C_1C_2C_3$, and $D_1D_2D_3$ be triangles in the plane. All the sides of the four triangles are extended into lines. Determine the maximum number of pairs of these lines that can meet at $60^o$ angles. [b]p9.[/b] For an integer $n$, let $f_n(x)$ denote the function $f_n(x) =\sqrt{x^2 - 2012x + n}+1006$. Determine all positive integers $a$ such that $f_a(f_{2012}(x)) = x$ for all $x \ge 2012$. [b]p10.[/b] Determine the number of ordered triples of integers $(a, b, c)$ such that $(a + b)(b + c)(c + a) = 1800$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

What is the area of the smallest triangle with all side lengths rational and all vertices lattice points?

Talya went for a 6 kilometer run. She ran 2 kilometers at 12 kilometers per hour followed by 2 kilometers at 10 kilometers per hour followed by 2 kilometers at 8 kilometers per hour. Talya’s average speed for the 6 kilometer run was $\frac{m}{n}$ kilometers per hour, where m and n are relatively prime positive integers. Find m + n.

An integer sequence is defined by \[{ a_n = 2 a_{n-1} + a_{n-2}}, \quad (n > 1), \quad a_0 = 0, a_1 = 1.\] Prove that $2^k$ divides $a_n$ if and only if $2^k$ divides $n$.

$ABCDEF$ is a convex hexagon, such that $|\angle FAB| = |\angle BCD| =|\angle DEF|$ and $|AB| =|BC|,$ $|CD| = |DE|$, $|EF| = |FA|$. Prove that the lines $AD$, $BE$ and $CF$ are concurrent.

Derek is bored in math class and is drawing a flower. He first draws $8$ points $A_1, A_2, \ldots, A_8$ equally spaced around an enormous circle. He then draws $8$ arcs outside the circle where the $i$th arc for $i = 1, 2, \ldots, 8$ has endpoints $A_i, A_{i+1}$ with $A_9 = A_1,$ such that all of the arcs have radius $1$ and any two consecutive arcs are tangent. Compute the perimeter of Derek’s $8$-petaled flower (not including the central circle). [center] [img] https://cdn.artofproblemsolving.com/attachments/8/4/e8b23c587762c089adb77b29cae155209f5db5.png [/img] [/center]

The table size $8 \times 8$ contains the numbers $1,2,...,8$ in each amount as much as you want provided that two numbers that are adjacent vertically, horizontally, diagonally are relative primes. Prove that some number appears in the table at least $12$ times. [i](PP-nine)[/i]

Determine the maximal length $L$ of a sequence $a_1,\dots,a_L$ of positive integers satisfying both the following properties: [list=disc] [*]every term in the sequence is less than or equal to $2^{2023}$, and [*]there does not exist a consecutive subsequence $a_i,a_{i+1},\dots,a_j$ (where $1\le i\le j\le L$) with a choice of signs $s_i,s_{i+1},\dots,s_j\in\{1,-1\}$ for which \[s_ia_i+s_{i+1}a_{i+1}+\dots+s_ja_j=0.\] [/list]

Show that there exists a set $S$ of $15$ distinct circles on the surface of a sphere, all having the same radius and such that $5$ touch exactly $5$ others, $5$ touch exactly $4$ others, and $5$ touch exactly $3$ others. [i][General Problem: http://www.artofproblemsolving.com/Forum/viewtopic.php?f=46&t=384764][/i]