Found problems: 85335
Denote the three real roots of the cubic $ x^3 \minus{} 3x \minus{} 1 \equal{} 0$ by $ x_1$, $ x_2$, $ x_3$ in order of increasing magnitude. (You may assume that the equation in fact has three distinct real roots.) Prove that $ x_3^2 \minus{} x_2^2 \equal{} x_3 \minus{} x_1$.
If a parabola is given in the plane, find a geometric construction (ruler and compass) for the focus.
What is the smallest positive integer with exactly $7$ distinct proper divisors?
A semicircle of radius $5$ and a quarter of a circle of radius $8$ touch each other and are located inside the square as shown in the figure. Find the length of the part of the common tangent, enclosed in the same square.
[img]https://cdn.artofproblemsolving.com/attachments/f/2/010f501a7bc1d34561f2fe585773816f168e93.png[/img]
For $a,b,c$ positive real numbers satisfying $a^2+b^2+c^2 \geq 3$,show that:
$\sqrt[3]{\frac{a^3+b^3+c^3}{3}}+\frac{a+b+c}{9} \geq \frac{4}{3}$.
Let $ABC$ be a triangle with circumcircle $\omega$, incenter $I$, and $A$-excenter $I_A$. Let the incircle and the $A$-excircle hit $BC$ at $D$ and $E$, respectively, and let $M$ be the midpoint of arc $BC$ without $A$. Consider the circle tangent to $BC$ at $D$ and arc $BAC$ at $T$. If $TI$ intersects $\omega$ again at $S$, prove that $SI_A$ and $ME$ meet on $\omega$.
[i]Amol Aggarwal.[/i]
Real nonzero numbers $x, y, z$ are such that $x+y +z = 0$. Moreover, it is known that $$A =\frac{x}{y}+\frac{y}{z}+\frac{z}{x}=\frac{x}{z}+\frac{z}{y}+\frac{y}{x}+ 1$$Determine $A$.
Let $a,b,A,B$ be given reals. We consider the function defined by \[ f(x) = 1 - a \cdot \cos(x) - b \cdot \sin(x) - A \cdot \cos(2x) - B \cdot \sin(2x). \] Prove that if for any real number $x$ we have $f(x) \geq 0$ then $a^2 + b^2 \leq 2$ and $A^2 + B^2 \leq 1.$
Find all functions $f : \mathbb Z \to \mathbb Z$ such that $$(f(x + y))^2 = f(x^2) + f(y^2)$$ for all $x, y \in \mathbb Z$.
Given that $a$ and $b$ are real numbers such that for infinitely many positive integers $m$ and $n$,
\[ \lfloor an + b \rfloor \ge \lfloor a + bn \rfloor \]
\[ \lfloor a + bm \rfloor \ge \lfloor am + b \rfloor \]
Prove that $a = b$.
Assuming that the roots of $x^3 + p \cdot x^2 + q \cdot x + r = 0$ are real and positive, find a relation between $p,q$ and $r$ which gives a necessary condition for the roots to be exactly the cosines of the three angles of a triangle.
Test whether equation $$\frac{1}{x - a} + \frac{1}{x - b} + \frac{1}{x - c} = 0,$$ where $ a $, $ b $, $ c $ denote the given real numbers, has real roots.
We call a tetrahedron a "trirectangular " if it has a vertex (we call this is called a "right-angled" vertex) in which the planes of the three sides of the tetrahedron intersect at right angles.
Prove the "three-dimensional Pythagorean theorem":
The square of the area of the opposite face of the "right-angled" vertex of the ""trirectangular " tetrahedron is equal to the sum of the squares of the areas of three other sides of the tetrahedron .
Given a prime number $p$. A positive integer $x$ is divided by $p$ with a remainder, and the number $p^2$ is divided by $x$ with a remainder. The remainders turned out to be equal. Find them
[i]Proposed by Sergey Berlov[/i]
For each positive integer $n$, let $\tau (n)$ be the number of positive divisors of $n$.
a) Find all positive integers $n$ such that $\tau(n)+2023=n$.
b) Prove that there exist infinitely many positive integers $k$ such that there are exactly two positive integers $n$ satisfying $\tau(kn)+2023=n$.
[b]p1.[/b] What is the value of the sum $2 + 20 + 202 + 2022$?
[b]p2.[/b] Find the smallest integer greater than $10000$ that is divisible by $12$.
[b]p3.[/b] Valencia chooses a positive integer factor of $6^{10}$ at random. The probability that it is odd can be expressed in the form $\frac{m}{n}$ where $m$ and $n$ are relatively prime integers. Find $m + n$.
[b]p4.[/b] How many three digit positive integers are multiples of $4$ but not $8$?
[b]p5.[/b] At the Jane Street store, Andy accidentally buys $5$ dollars more worth of shirts than he had planned. Originally, including the tip to the cashier, he planned to spend all of the remaining $90$ dollars on his giftcard. To compensate for his gluttony, Andy instead gives the cashier a smaller, $12.5\%$ tip so that he still spends $90$ dollars total. How much percent tip was Andy originally planning on giving?
[b]p6.[/b] Let $A,B,C,D$ be four coplanar points satisfying the conditions $AB = 16$, $AC = BC =10$, and $AD = BD = 17$. What is the minimum possible area of quadrilateral $ADBC$?
[b]p7.[/b] How many ways are there to select a set of three distinct points from the vertices of a regular hexagon so that the triangle they form has its smallest angle(s) equal to $30^o$?
[b]p8.[/b] Jaeyong rolls five fair $6$-sided die. The probability that the sum of some three rolls is exactly $8$ times the sum of the other two rolls can be expressed as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
[b]p9.[/b] Find the least positive integer n for there exists some positive integer $k > 1$ for which $k$ and $k + 2$ both divide $\underbrace{11...1}_{n\,\,\,1's}$.
[b]p10.[/b] For some real constant $k$, line $y = k$ intersects the curve $y = |x^4-1|$ four times: points $A$,$B$,$C$ and $D$, labeled from left to right. If $BC = 2AB = 2CD$, then the value of $k$ can be expressed as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
[b]p11.[/b] Let a be a positive real number and $P(x) = x^2 -8x+a$ and $Q(x) = x^2 -8x+a+1$ be quadratics with real roots such that the positive difference of the roots of $P(x)$ is exactly one more than the positive difference of the roots of $Q(x)$. The value of a can be written as a common fraction $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
[b]p12.[/b] Let $ABCD$ be a trapezoid satisfying $AB \parallel CD$, $AB = 3$, $CD = 4$, with area $35$. Given $AC$ and $BD$ intersect at $E$, and $M$, $N$, $P$, $Q$ are the midpoints of segments $AE$,$BE$,$CE$,$DE$, respectively, the area of the intersection of quadrilaterals $ABPQ$ and $CDMN$ can be expressed as $\frac{m}{n}$ where $m, n$ are relatively prime positive integers. Find $m + n$.
[b]p13.[/b] There are $8$ distinct points $P_1, P_2, ... , P_8$ on a circle. How many ways are there to choose a set of three distinct chords such that every chord has to touch at least one other chord, and if any two chosen chords touch, they must touch at a shared endpoint?
[b]p14.[/b] For every positive integer $k$, let $f(k) > 1$ be defined as the smallest positive integer for which $f(k)$ and $f(k)^2$ leave the same remainder when divided by $k$. The minimum possible value of $\frac{1}{x}f(x)$ across all positive integers $x \le 1000$ can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m, n$. Find $m + n$.
[b]p15.[/b] In triangle $ABC$, let $I$ be the incenter and $O$ be the circumcenter. If $AO$ bisects $\angle IAC$, $AB + AC = 21$, and $BC = 7$, then the length of segment $AI$ can be expressed as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
Given $a;b;c$ satisfying $a^{2}+b^{2}+c^{2}=2$ . Prove that:
a) $\left | a+b+c-abc \right |\leqslant 2$ .
b) $\left | a^{3}+b^{3}+c^{3}-3abc \right |\leqslant 2\sqrt{2}$
In quadrilateral $ABCD$, $AB=2$, $AD=3$, $BC=CD=\sqrt7$, and $\angle DAB=60^\circ$. Semicircles $\gamma_1$ and $\gamma_2$ are erected on the exterior of the quadrilateral with diameters $\overline{AB}$ and $\overline{AD}$; points $E\neq B$ and $F\neq D$ are selected on $\gamma_1$ and $\gamma_2$ respectively such that $\triangle CEF$ is equilateral. What is the area of $\triangle CEF$?
Let $n$ be a positive integer.
Consider sequences $a_0, a_1, ..., a_k$ and $b_0, b_1,,..,b_k$ such that $a_0 = b_0 = 1$ and $a_k = b_k = n$ and such that for all $i$ such that $1 \le i \le k $, we have that $(a_i, b_i)$ is either equal to $(1 + a_{i-1}, b_{i-1})$ or $(a_{i-1}; 1 + b_{i-1})$.
Consider for $1 \le i \le k$ the number $c_i = \begin{cases} a_i \,\,\, if \,\,\, a_i = a_{i-1} \\
b_i \,\,\, if \,\,\, b_i = b_{i-1}\end{cases}$
Show that $c_1 + c_2 + ... + c_k = n^2 - 1$.
The vertices and the circumcenter of an isosceles triangle lie on four different sides of a square. Find the angles of this triangle.
(I. Bogdanov, B. Frenkin)
Let $ABC$ be a triangle with circumcircle $c$ and circumcenter $O$, and let $D$ be a point on the side $BC$ different from the vertices and the midpoint of $BC$. Let $K$ be the point where the circumcircle $c_1$ of the triangle $BOD$ intersects $c$ for the second time and let $Z$ be the point where $c_1$ meets the line $AB$. Let $M$ be the point where the circumcircle $c_2$ of the triangle $COD$ intersects $c$ for the second time and let $E$ be the point where $c_2$ meets the line $AC$. Finally let $N$ be the point where the circumcircle $c_3$ of the triangle $AEZ$ meets $c$ again. Prove that the triangles $ABC$ and $NKM$ are congruent.
Given $m\in\mathbb{N}$. Find all functions $f:\mathbb{R^{+}}\rightarrow\mathbb{R^{+}}$ such that $$f(f(x)+y)-f(x)=\left( \frac{f(y)}{y}-1\right)x+f^m(y)$$
holds for all $x,y\in\mathbb{R^{+}}.$
($f^m(x) =$ $f$ applies $m$ times.)
Let $ABCD$ be a quadrilateral with an inscribed circle centered at $I$. Let $CI$ intersect $AB$ at $E$. If $\angle IDE=35^\circ$, $\angle ABC=70^\circ$, and $\angle BCD=60^\circ$, then what are all possible measures of $\angle CDA$?
Prove the following inequality.
\[ \frac{2}{2\plus{}e^{\frac 12}}<\int_0^1 \frac{dx}{1\plus{}xe^{x}}<\frac{2\plus{}e}{2(1\plus{}e)}\]
The Professor chooses to assign homework problems from a set of problems labeled $1$ to $100$, inclusive. He will not assign two problems whose numbers share a common factor greater than $1$. If the Professor chooses to assign the maximum number of homework problems possible, how many different combinations of problems can he assign?