Prove that if the polynomial $x^3 + ax^2 + bx + c$ has three distinct real roots, so does the polynomial $x^3 + ax^2 + \frac{1}{4}(a^2 + b)x + \frac{1}{8}(ab-c)$.

How many triples $(a,b,c)$ with $a,b,c \in \mathbb{R}$ satisfy the following system? $$\begin{cases} a^4-b^4=c \\ b^4-c^4=a \\ c^4-a^4=b \end{cases}$$.

Determine the triangle with sides $a,b,c$ and circumradius $R$ for which $R(b+c) = a\sqrt{bc}$. [i]Romania[/i]

[b]p1.[/b] What is the sum of the first $12$ positive integers? [b]p2.[/b] How many positive integers less than or equal to $100$ are multiples of both $2$ and $5$? [b]p3. [/b]Alex has a bag with $4$ white marbles and $4$ black marbles. She takes $2$ marbles from the bag without replacement. What is the probability that both marbles she took are black? Express your answer as a decimal or a fraction in lowest terms. [b]p4.[/b] How many $5$-digit numbers are there where each digit is either $1$ or $2$? [b]p5.[/b] An integer $a$ with $1\le a \le 10$ is randomly selected. What is the probability that $\frac{100}{a}$ is an integer? Express your answer as decimal or a fraction in lowest terms. [b]p6.[/b] Two distinct non-tangent circles are drawn so that they intersect each other. A third circle, distinct from the previous two, is drawn. Let $P$ be the number of points of intersection between any two circles. How many possible values of $P$ are there? [b]p7.[/b] Let $x, y, z$ be nonzero real numbers such that $x + y + z = xyz$. Compute $$\frac{1 + yz}{yz}+\frac{1 + xz}{xz}+\frac{1 + xy}{xy}.$$ [b]p8.[/b] How many positive integers less than $106$ are simultaneously perfect squares, cubes, and fourth powers? [b]p9.[/b] Let $C_1$ and $C_2$ be two circles centered at point $O$ of radii $1$ and $2$, respectively. Let $A$ be a point on $C_2$. We draw the two lines tangent to $C_1$ that pass through $A$, and label their other intersections with $C_2$ as $B$ and $C$. Let x be the length of minor arc $BC$, as shown. Compute $x$. [img]https://cdn.artofproblemsolving.com/attachments/7/5/915216d4b7eba0650d63b26715113e79daa176.png[/img] [b]p10.[/b] A circle of area $\pi$ is inscribed in an equilateral triangle. Find the area of the triangle. [b]p11.[/b] Julie runs a $2$ mile route every morning. She notices that if she jogs the route $2$ miles per hour faster than normal, then she will finish the route $5$ minutes faster. How fast (in miles per hour) does she normally jog? [b]p12.[/b] Let $ABCD$ be a square of side length $10$. Let $EFGH$ be a square of side length $15$ such that $E$ is the center of $ABCD$, $EF$ intersects $BC$ at $X$, and $EH$ intersects $CD$ at $Y$ (shown below). If $BX = 7$, what is the area of quadrilateral $EXCY$ ? [img]https://cdn.artofproblemsolving.com/attachments/d/b/2b2d6de789310036bc42d1e8bcf3931316c922.png[/img] [b]p13.[/b] How many solutions are there to the system of equations $$a^2 + b^2 = c^2$$ $$(a + 1)^2 + (b + 1)^2 = (c + 1)^2$$ if $a, b$, and $c$ are positive integers? [b]p14.[/b] A square of side length $ s$ is inscribed in a semicircle of radius $ r$ as shown. Compute $\frac{s}{r}$. [img]https://cdn.artofproblemsolving.com/attachments/5/f/22d7516efa240d00d6a9743a4dc204d23d190d.png[/img] [b]p15.[/b] $S$ is a collection of integers n with $1 \le n \le 50$ so that each integer in $S$ is composite and relatively prime to every other integer in $S$. What is the largest possible number of integers in $S$? [b]p16.[/b] Let $ABCD$ be a regular tetrahedron and let $W, X, Y, Z$ denote the centers of faces $ABC$, $BCD$, $CDA$, and $DAB$, respectively. What is the ratio of the volumes of tetrahedrons $WXYZ$ and $WAYZ$? Express your answer as a decimal or a fraction in lowest terms. [b]p17.[/b] Consider a random permutation $\{s_1, s_2, ... , s_8\}$ of $\{1, 1, 1, 1, -1, -1, -1, -1\}$. Let $S$ be the largest of the numbers $s_1$, $s_1 + s_2$, $s_1 + s_2 + s_3$, $...$ , $s_1 + s_2 + ... + s_8$. What is the probability that $S$ is exactly $3$? Express your answer as a decimal or a fraction in lowest terms. [b]p18.[/b] A positive integer is called [i]almost-kinda-semi-prime[/i] if it has a prime number of positive integer divisors. Given that there $are 168$ primes less than $1000$, how many almost-kinda-semi-prime numbers are there less than $1000$? [b]p19.[/b] Let $ABCD$ be a unit square and let $X, Y, Z$ be points on sides $AB$, $BC$, $CD$, respectively, such that $AX = BY = CZ$. If the area of triangle $XYZ$ is $\frac13$ , what is the maximum value of the ratio $XB/AX$? [img]https://cdn.artofproblemsolving.com/attachments/5/6/cf77e40f8e9bb03dea8e7e728b21e7fb899d3e.png[/img] [b]p20.[/b] Positive integers $a \le b \le c$ have the property that each of $a + b$, $b + c$, and $c + a$ are prime. If $a + b + c$ has exactly $4$ positive divisors, find the fourth smallest possible value of the product $c(c + b)(c + b + a)$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $\omega_1$ and $\omega_2$ be two circles that intersect at points $A$ and $B$. Let line $l$ be tangent to $\omega_1$ at $P$ and to $\omega_2$ at $Q$ such that $A$ is closer to $PQ$ than $B$. Let points $R$ and $S$ lie along rays $PA$ and $QA$, respectively, so that $PQ = AR = AS$ and $R$ and $S$ are on opposite sides of $A$ as $P$ and $Q$. Let $O$ be the circumcenter of triangle $ASR$, and $C$ and $D$ be the midpoints of major arcs $AP$ and $AQ$, respectively. If $\angle APQ$ is $45$ degrees and $\angle AQP$ is $30$ degrees, determine $\angle COD$ in degrees.

Let $ABC$ be a triangle for which the shortest side is $AC$. Its inscribed circle with center $I$ touches sides $AB$ and $BC$ in points $D$ and $E$ respectively. Point $M$ is the midpoint of $AC$. Points $F$ and $G$ lie on sides $BC$ and $AB$ respectively so that $FC=CA=AG$. The line through $I$ perpendicular to $MI$ intersects the line segments $AF$ and $CG$ in $P$ and $Q$ respectively. Prove that $AB=BC\Leftrightarrow PD=QE$.

Without lifting pen from paper, we draw a polygon in such away that from every two adjacent sides one of them is vertical. In addition, while drawing the polygon all vertical sides have been drawn from up to down. Prove that this polygon has cut itself.

One of the numbers $1$ or $-1$ is assigned to each vertex of a cube. To each face of the cube is assigned the integer which is the product of the four integers at the vertices of the face. Is it possible that the sum of the $14$ assigned integers is $0$? (G. Galperin)

Points $ A, B, C, D $ are vertices of an isosceles trapezoid, with $ \overline{AB} $ parallel to $ \overline{CD} $, $ AB = 1 $, $ CD = 2 $, and $ BC = 1 $. Point $ E $ is chosen uniformly and at random on $ \overline{CD} $, and let point $ F $ be the point on $ \overline{CD} $ such that $ EC = FD $. Let $ G $ denote the intersection of $ \overline{AE} $ and $ \overline{BF} $, not necessarily in the trapezoid. What is the probability that $ \angle AGB > 30^\circ $?

The trapezoid $ABCD$ has sides $AB$ and $CD$ that are parallel $\hat{DAB} = 6^{\circ}$ and $\hat{ABC} = 42^{\circ}$. Point $X$ lies on the side $AB$ , such that $\hat{AXD} = 78^{\circ}$ and $\hat{CXB} = 66^{\circ}$. The distance between $AB$ and $CD$ is $1$ unit . Prove that $AD + DX - (BC + CX) = 8$ units. (Heir of Ramanujan)

Let $\mathcal{P}$ be a parabola that passes through the points $(0, 0)$ and $(12, 5)$. Suppose that the directrix of $\mathcal{P}$ takes the form $y = b$. (Recall that a parabola is the set of points equidistant from a point called the focus and line called the directrix) Find the minimum possible value of $|b|$. [i]Proposed by Kevin You[/i]

[b]p1.[/b] Compute $2020 \cdot \left( 2^{(0\cdot1)} + 9 - \frac{(20^1)}{8}\right)$. [b]p2.[/b] Nathan has five distinct shirts, three distinct pairs of pants, and four distinct pairs of shoes. If an “outfit” has a shirt, pair of pants, and a pair of shoes, how many distinct outfits can Nathan make? [b]p3.[/b] Let $ABCD$ be a rhombus such that $\vartriangle ABD$ and $\vartriangle BCD$ are equilateral triangles. Find the angle measure of $\angle ACD$ in degrees. [b]p4.[/b] Find the units digit of $2019^{2019}$. [b]p5.[/b] Determine the number of ways to color the four vertices of a square red, white, or blue if two colorings that can be turned into each other by rotations and reflections are considered the same. [b]p6.[/b] Kathy rolls two fair dice numbered from $1$ to $6$. At least one of them comes up as a $4$ or $5$. Compute the probability that the sumof the numbers of the two dice is at least $10$. [b]p7.[/b] Find the number of ordered pairs of positive integers $(x, y)$ such that $20x +19y = 2019$. [b]p8.[/b] Let $p$ be a prime number such that both $2p -1$ and $10p -1$ are prime numbers. Find the sum of all possible values of $p$. [b]p9.[/b] In a square $ABCD$ with side length $10$, let $E$ be the intersection of $AC$ and $BD$. There is a circle inscribed in triangle $ABE$ with radius $r$ and a circle circumscribed around triangle $ABE$ with radius $R$. Compute $R -r$ . [b]p10.[/b] The fraction $\frac{13}{37 \cdot 77}$ can be written as a repeating decimal $0.a_1a_2...a_{n-1}a_n$ with $n$ digits in its shortest repeating decimal representation. Find $a_1 +a_2 +...+a_{n-1}+a_n$. [b]p11.[/b] Let point $E$ be the midpoint of segment $AB$ of length $12$. Linda the ant is sitting at $A$. If there is a circle $O$ of radius $3$ centered at $E$, compute the length of the shortest path Linda can take from $A$ to $B$ if she can’t cross the circumference of $O$. [b]p12.[/b] Euhan and Minjune are playing tennis. The first one to reach $25$ points wins. Every point ends with Euhan calling the ball in or out. If the ball is called in, Minjune receives a point. If the ball is called out, Euhan receives a point. Euhan always makes the right call when the ball is out. However, he has a $\frac34$ chance of making the right call when the ball is in, meaning that he has a $\frac14$ chance of calling a ball out when it is in. The probability that the ball is in is equal to the probability that the ball is out. If Euhan won, determine the expected number of wrong callsmade by Euhan. [b]p13.[/b] Find the number of subsets of $\{1, 2, 3, 4, 5, 6,7\}$ which contain four consecutive numbers. [b]p14.[/b] Ezra and Richard are playing a game which consists of a series of rounds. In each round, one of either Ezra or Richard receives a point. When one of either Ezra or Richard has three more points than the other, he is declared the winner. Find the number of games which last eleven rounds. Two games are considered distinct if there exists a round in which the two games had different outcomes. [b]p15.[/b] There are $10$ distinct subway lines in Boston, each of which consists of a path of stations. Using any $9$ lines, any pair of stations are connected. However, among any $8$ lines there exists a pair of stations that cannot be reached from one another. It happens that the number of stations is minimized so this property is satisfied. What is the average number of stations that each line passes through? [b]p16.[/b] There exist positive integers $k$ and $3\nmid m$ for which $$1 -\frac12 + \frac13 - \frac14 +...+ \frac{1}{53}-\frac{1}{54}+\frac{1}{55}=\frac{3^k \times m}{28\times 29\times ... \times 54\times 55}.$$ Find the value $k$. [b]p17.[/b] Geronimo the giraffe is removing pellets from a box without replacement. There are $5$ red pellets, $10$ blue pellets, and $15$ white pellets. Determine the probability that all of the red pellets are removed before all the blue pellets and before all of the white pellets are removed. [b]p18.[/b] Find the remainder when $$70! \left( \frac{1}{4 \times 67}+ \frac{1}{5 \times 66}+...+ \frac{1}{66\times 5}+ \frac{1}{67\times 4} \right)$$ is divided by $71$. [b]p19.[/b] Let $A_1A_2...A_{12}$ be the regular dodecagon. Let $X$ be the intersection of $A_1A_2$ and $A_5A_{11}$. Given that $X A_2 \cdot A_1A_2 = 10$, find the area of dodecagon. [b]p20.[/b] Evaluate the following infinite series: $$\sum^{\infty}_{n=1}\sum^{\infty}_{m=1} \frac{n \sec^2m -m \tan^2 n}{3^{m+n}(m+n)}$$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

If $n$ is a positive integer which can be expressed in the form $n=a^{2}+b^{2}+c^{2}$, where $a, b, c$ are positive integers, prove that for each positive integer $k$, $n^{2k}$ can be expressed in the form $A^2 +B^2 +C^2$, where $A, B, C$ are positive integers.

Let $A\in M_2(\mathbb Q)$ such that there is $n\in\mathbb N,n\ne0$, with $A^n=-I_2$. Prove that either $A^2=-I_2$ or $A^3=-I_2$.

If $ 8^x \equal{} 32$, then $ x$ equals: $ \textbf{(A)}\ 4 \qquad \textbf{(B)}\ \frac{5}{3} \qquad \textbf{(C)}\ \frac{3}{2} \qquad \textbf{(D)}\ \frac{3}{5} \qquad \textbf{(E)}\ \frac{1}{4}$

Determine all pairs $(a, b)$ of positive integers such that $ab^2+b+7$ divides $a^2 b+a+b$.

Calculate the integral of the Gaussian curvature of the surface \[z^4+(x^2+y^2-1)(2x^2+3y^2-1)=0\]

a) Are there rectangles $1\times \dfrac12$ rectangles lying strictly inside the interior of a unit square? b) Find the minimum number of equilateral triangles of unit side which can cover completely a unit square. [i]Laurențiu Panaitopol[/i]

An ant is at one corner of a unit cube. If the ant must travel on the box’s surface, the shortest distance the ant must crawl to reach the opposite corner of the cube can be written in the form $\sqrt{a}$, where $a$ is a positive integer. Compute $a$.

Show that any positive integer has at least as many positive divisors of the form $3k+1$ as of the form $3k-1$. [b](N. 7)[/b]

How many pairs of $2007$-digit numbers $\underline{a_1a_2}\cdots\underline{a_{2007}}$ and $\underline{b_1b_2}\cdots\underline{b_{2007}}$ are there such that $a_1b_1+a_2b_2+\cdots+a_{2007}b_{2007}$ is even? Express your answer as $a \** b^c + d \** e^f$ for integers $a$, $b$, $c$, $d$, $e$, and $f$ with $a \nmid b$ and $d \nmid e$.

Given are on a line three points $A, B, C$ such that $AB = 1$ and $BC = x$. Consider the circles $\Omega_a, \Omega_b$ and $\Omega_c$ which are tangent to the given line at the points $A, B, C$ respectively, and such that $\Omega_b$ is tangent externally with both $\Omega_a$ and $\Omega_c$ in points $M, N$ respectively. Find all values of the radius of the circle $\Omega_b$ for which the triangle $BMN$ is isosceles.

Find the number of positive integers $n$ with exactly $1974$ factors such that no prime greater than $40$ divides $n$, and $n$ ends in one of the digits $1$, $3$, $7$, $9$. (Note that $1974 = 2 \cdot 3 \cdot 7 \cdot 47$.) [i]Proposed by Yonah Borns-Weil[/i]

If $a, b, c, d$ are real numbers such that $a^2 + b^2 + c^2 + d^2 \leq 1$, find the maximum of the expression \[(a + b)^4 + (a + c)^4 + (a + d)^4 + (b + c)^4 + (b + d)^4 + (c + d)^4.\]

On the same set of axes are drawn the graph of $ y\equal{}ax^2\plus{}bx\plus{}c$ and the graph of the equation obtained by replacing $ x$ by $ \minus{}x$ in the given equation. If $ b \neq 0$ and $ c \neq 0$ these two graphs intersect: $ \textbf{(A)}\ \text{in two points, one on the x\minus{}axis and one on the y\minus{}axis}\\ \textbf{(B)}\ \text{in one point located on neither axis} \\ \textbf{(C)}\ \text{only at the origin} \\ \textbf{(D)}\ \text{in one point on the x\minus{}axis} \\ \textbf{(E)}\ \text{in one point on the y\minus{}axis}$