Found problems: 85335
Let $a, b, c$ and $d$ be positive real numbers. Prove that
$$\frac{a - b}{b + c}+\frac{b - c}{c + d}+\frac{c - d}{d + a} +\frac{d - a}{a + b } \ge 0 $$
In triangle $ABC$, points $P$, $Q$, $R$ lie on sides $BC$, $CA$, $AB$ respectively. Let $\omega_A$, $\omega_B$, $\omega_C$ denote the circumcircles of triangles $AQR$, $BRP$, $CPQ$, respectively. Given the fact that segment $AP$ intersects $\omega_A$, $\omega_B$, $\omega_C$ again at $X$, $Y$, $Z$, respectively, prove that $YX/XZ=BP/PC$.
Let $ABC$ be a triangle with $AB = 12$, $BC = 5$, $AC = 13$. Let$ D$ and $E$ be the feet of the internal and external angle bisectors from $B$, respectively. (The external angle bisector from $B$ bisects the angle between $BC$ and the extension of $AB$.) Let $\omega$ be the circumcircle of $\vartriangle BDE$, extend $AB$ so that it intersects $\omega$ again at $F$. Extend $F C$ to meet $\omega$ again at $X$, and extend $AX$ to meet $\omega$ again at $G$. Find $F G$.
Find all integers $x,y$ which satisfy the equation $xy=20-3x+y$.
Two regular tetrahedrons $A$ and $B$ are made with the 8 vertices of a unit cube. (this way is unique)
What's the volume of $A\cup B$?
The sequences $ a_0, a_1, \ldots$ and $ b_0, b_1, \ldots$ are defined for $ n \equal{} 0, 1, 2, \ldots$ by the equalities
\[ a_0 \equal{} \frac {\sqrt {2}}{2}, \quad a_{n \plus{} 1} \equal{} \frac {\sqrt {2}}{2} \cdot \sqrt {1 \minus{} \sqrt {1 \minus{} a^2_n}}
\]
and
\[ b_0 \equal{} 1, \quad b_{n \plus{} 1} \equal{} \frac {\sqrt {1 \plus{} b^2_n} \minus{} 1}{b_n}
\]
Prove the inequalities for every $ n \equal{} 0, 1, 2, \ldots$
\[ 2^{n \plus{} 2} a_n < \pi < 2^{n \plus{} 2} b_n.
\]
On a circumference of a unit radius, take points $A$ and $B$ such that section $AB$ has length one. $C$ can be any point on the longer arc of the circle between $A$ and $B$. How do we take $C$ to make the perimeter of the triangle $ABC$ as large as possible?
Suppose \(40\) objects are placed along a circle at equal distances. In how many ways can \(3\) objects be chosen from among them so that no two of the three chosen objects are adjacent nor diametrically opposite?
A street on Stanford can be modeled by a number line. Four Stanford students, located at positions $1$, $9$, $25$ and $49$ along the line, want to take an UberXL to Berkeley, but are not sure where to meet the driver. Find the smallest possible total distance walked by the students to a single position on the street. (For example, if they were to meet at position $46$, then the total distance walked by the students would be $45 + 37 + 21 + 3 = 106$, where the distances walked by the students at positions $1$, $9$, $25$ and $49$ are summed in that order.)
Let $ABC$ be a triangle such that $\angle BAC = 90^{\circ}$ and $AB = AC$. Let $M$ be the midpoint of $BC$. A point $D \neq A$ is chosen on the semicircle with diameter $BC$ that contains $A$. The circumcircle of triangle $DAM$ cuts lines $DB$ and $DC$ at $E$ and $F$ respectively. Show that $BE = CF$.
Let $S$ be the sum of all positive real numbers $x$ for which $$x^{2^{\sqrt2}}=\sqrt2^{2^x}.$$ Which of the following statements is true?
$\textbf{(A) }S<\sqrt2 \qquad \textbf{(B) }S=\sqrt2 \qquad \textbf{(C) }\sqrt2<S<2\qquad \textbf{(D) }2\le S<6 \qquad \textbf{(E) }S\ge 6$
Zachary tries to simplify the fraction $\frac{2020}{5050}$ by dividing the numerator and denominator by the same integer to get the fraction $\frac{m}{n}$ , where $m$ and $n$ are both positive integers. Find the sum of the (not necessarily distinct) prime factors of the sum of all the possible values of $m +n$
Positive intenger $n\geq3$. $a_1,a_2,\cdots,a_n$ are $n$ positive intengers that are pairwise coprime, satisfying that there exists $k_1,k_2,\cdots,k_n\in\{-1,1\}, \sum_{i=1}^{n}k_ia_i=0$. Are there positive intengers $b_1,b_2,\cdots,b_n$, for any $k\in\mathbb{Z}_+$, $b_1+ka_1,b_2+ka_2,\cdots,b_n+ka_n$ are pairwise coprime?
All the natural numbers from $1$ to $1982$ are gathered in an array in an arbitrary order in computer's memory. The program looks through all the sequent pairs (first and second, second and third,...) and exchanges numbers in the pair, if the number on the lower place is greater than another. Then the program repeats the process, but moves from another end of the array. The number, that stand initially on the $100$-th place reserved its place. Find that number.
Evaluate $\sum_{n=0}^{\infty}\frac{1}{(2n+1)2^{2n+1}}.$
A convex quadrilateral $ABCD$ has no parallel sides. The angles between the diagonal $AC$ and the four sides are $55^o, 55^o, 19^o$ and $16^o$ in some order. Determine all possible values of the acute angle between $AC$ and $BD$.
With the conditions $a,b,c\in\mathbb{R^+}$ and $a+b+c=1$, prove that \[\frac{7+2b}{1+a}+\frac{7+2c}{1+b}+\frac{7+2a}{1+c}\geq\frac{69}{4}\]
Consider the function $ f(x) \equal{} \frac {2x^3 \minus{} 3}{3x^2 \minus{} 1}$.
$ 1.$ Prove that there is a continuous function $ g(x)$ on $ \mathbb{R}$ satisfying $ f(g(x)) \equal{} x$ and $ g(x) > x$ for all real $ x$.
$ 2.$ Show that there exists a real number $ a > 1$ such that the sequence $ \{a_n\}$, $ n \equal{} 1, 2, \ldots$, defined as follows $ a_0 \equal{} a$, $ a_{n \plus{} 1} \equal{} f(a_n)$, $ \forall n\in\mathbb{N}$ is periodic with the smallest period $ 1995$.
Find the smallest prime $p$ such that the digits of $p$ (in base 10) add up to a prime number greater than $10$.
Show that system of equations
$2ab=6(a+b)-13$
$a^2+b^2=4$
has not solutions in set of real numbers.
Let $ x$, $ y$ be two integers with $ 2\le x, y\le 100$. Prove that $ x^{2^n} \plus{} y^{2^n}$ is not a prime for some positive integer $ n$.
Let $ABC$ be a triangle with incentre $I$. A line $r$ that passes through $I$ intersects the circumcircles of triangles $AIB$ and $AIC$ at points $P$ and $Q$, respectively. Prove that the circumcentre of triangle $APQ$ is on the circumcircle of $ABC$.
Samuel Tsui and Jason Yang each chose a different integer between $1$ and $60$, inclusive. They don’t know each others’ numbers, but they both know that the other person’s number is between $1$ and $60$ and distinct from their own. They have the following conversation:
Samuel Tsui: Do our numbers have any common factors greater than $1$?
Jason Yang: Definitely not. However their least common multiple must be less than$ 2023$.
Samuel Tsui: Ok, thismeans that the sumof the factors of our two numbers are equal.
What is the sumof Samuel Tsui’s and Jason Yang’s numbers?
[i]Proposed by Samuel Tsui[/i]
In how many ways rooks can be placed on a $8$ by $8$ chess board such that every row and every column has at least one rook?
(Any number of rooks are available,each square can have at most one rook and there is no relation of attacking between them)
Let $t\in (1,2)$. Show that there exists a polynomial $P(x)=a_nx^n+a_{n-1}x^{n-1}+...+a_1x+a_0$ with the coefficients in $\{1,-1\}$ such that $\left|P(t)-2019\right| \leqslant 1.$
[i]Proposed by N. Safaei (Iran)[/i]