Found problems: 85335
Let $a$ and $b$ be real numbers such that $\frac{1}{a^2} +\frac{3}{b^2} = 2018a$ and $\frac{3}{a^2} +\frac{1}{b^2} = 290b$. Then $\frac{ab}{b-a }= \frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
Call a positive integer "useful but not optimized " (!), if it can be written as a sum of distinct powers of $3$ and powers of $5$.
Prove that there exist infinitely many positive integers which they are not "useful but not optimized".
(e.g. $37=(3^0+3^1+3^3)+(5^0+5^1)$ is a " useful but not optimized" number)
[i]Proposed by Mohsen Jamali[/i]
A triangle with perimeter $1$ has side lengths $a, b, c$. Show that $a^2 + b^2 + c^2 + 4abc <\frac 12$.
Prove the inequalities $ 0<n\left( \sqrt[n]{2} -1 \right) -\left( \frac{1}{n+1} +\frac{1}{n+2} +\cdots +\frac{1}{n+n}\right) <\frac{1}{2n} , $ where $ n\ge 2. $
[i]Marius Cavachi[/i]
Mary mixes $2$ gallons of a solution that is $40$ percent alcohol with $3$ gallons of a solution that is $60$ percent alcohol. Sandra mixes $4$ gallons of a solution that is $30$ percent alcohol with $\frac{m}{n}$ gallons of a solution that is $80$ percent alcohol, where $m$ and $n$ are relatively prime positive integers. Mary and Sandra end up with solutions that are the same percent alcohol. Find $m + n$.
The sum of the reciprocals of the roots of the equation $ ax^2 \plus{} bx \plus{} c \equal{} 0$ is:
$ \textbf{(A)}\ \frac {1}{a} \plus{} \frac {1}{b} \qquad \textbf{(B)}\ \minus{} \frac {c}{b} \qquad \textbf{(C)}\ \frac {b}{c} \qquad \textbf{(D)}\ \minus{} \frac {a}{b} \qquad \textbf{(E)}\ \minus{} \frac {b}{c}$
Given $n\geq2$, let $\mathcal{A}$ be a family of subsets of the set $\{1,2,\dots,n\}$ such that, for any $A_1,A_2,A_3,A_4 \in \mathcal{A}$, it holds that $|A_1 \cup A_2 \cup A_3 \cup A_4| \leq n -2$.
Prove that $|\mathcal{A}| \leq 2^{n-2}.$
You enter an elevator on floor $0$ of a building with some other people, and request to go to floor $10$. In order to be efficient, it doesn’t stop at adjacent floors (so, if it’s at floor $0$, its next stop cannot be floor $ 1$). Given that the elevator will stop at floor $10$, no matter what other floors it stops at, how many combinations of stops are there for the elevator?
Let $x_1,x_2,\ldots, x_{2007}$ be real numbers such that $-1\leq x_i\leq 1$ for $1\leq i\leq 2007$, and \[\sum_{i=1}^{2007}x_i^3 = 0.\] Find the maximum possible value of $\Big\lfloor\sum_{i=1}^{2007}x_i\Big\rfloor$.
Find the smallest positive integer $N$ such that there are no different sets $A, B$ that satisfy the following conditions. (Here, $N$ is not a power of $2$. That is, $N \neq 1, 2^1, 2^2, \dots$.)
[list]
[*] $A, B \subseteq \{1, 2^1, 2^2, 2^3, \dots, 2^{2023}\} \cup \{ N \}$
[*] $|A| = |B| \geq 1$
[*] Sum of elements in $A$ and sum of elements in $B$ are equal.
[/list]
Prove that every finite triangle-free graph can be embedded as an induced subgraph in a finite triangle-free graph of diameter 2.
Find the number of $n$ that follow the following:
$ \bigstar $ The number of integers $ (x,y,z) $ following this equation is not a multiple of 4.
$ 2n=x^2+2y^2+2x^2+2xy+2yz $
In the five-sided star shown, the letters $A,B,C,D,$ and $E$ are replaced by the numbers $3,5,6,7,$ and $9$, although not necessarily in this order. The sums of the numbers at the ends of the line segments $\overline{AB}$,$\overline{BC}$,$\overline{CD}$,$\overline{DE}$, and $\overline{EA}$ form an arithmetic sequence, although not necessarily in this order. What is the middle term of the arithmetic sequence?
[asy]
size(150);
defaultpen(linewidth(0.8));
string[] strng = {'A','D','B','E','C'};
pair A=dir(90),B=dir(306),C=dir(162),D=dir(18),E=dir(234);
draw(A--B--C--D--E--cycle);
for(int i=0;i<=4;i=i+1)
{
path circ=circle(dir(90-72*i),0.125);
unfill(circ);
draw(circ);
label("$"+strng[i]+"$",dir(90-72*i));
}
[/asy]
$ \textbf{(A)}\ 9\qquad
\textbf{(B)}\ 10\qquad
\textbf{(C)}\ 11\qquad
\textbf{(D)}\ 12\qquad
\textbf{(E)}\ 13$
In a ten-mile race First beats Second by $2$ miles and First beats Third by $4$ miles. If the runners maintain constant speeds throughout the race, by how many miles does Second beat Third?
$ \textbf{(A)}\ 2\qquad\textbf{(B)}\ 2\frac{1}{4}\qquad\textbf{(C)}\ 2\frac{1}{2}\qquad\textbf{(D)}\ 2\frac{3}{4}\qquad\textbf{(E)}\ 3 $
The value of $ 21!$ is $ 51{,}090{,}942{,}171{,}abc{,}440{,}000$, where $ a$, $ b$, and $ c$ are digits. What is the value of $ 100a \plus{} 10b \plus{} c$?
In rectangle $ABCD$, points $E$ and $F$ lie on sides $AB$ and $CD$ respectively such that both $AF$ and $CE$ are perpendicular to diagonal $BD$. Given that $BF$ and $DE$ separate $ABCD$ into three polygons with equal area, and that $EF = 1$, find the length of $BD$.
On sides $BC$ and $AC$ of $\triangle ABC$ given are $D$ and $E$, respectively. Let $F$ ($F \neq C$) be a point of intersection of circumcircle of $\triangle CED$ and line that is parallel to $AB$ and passing through C. Let $G$ be a point of intersection of line $FD$ and side $AB$, and let $H$ be on line $AB$ such that $\angle HDA = \angle GEB$ and $H-A-B$. If $DG=EH$, prove that point of intersection of $AD$ and $BE$ lie on angle bisector of $\angle ACB$.
[i]Proposed by Milos Milosavljevic[/i]
For $i = 1, 2$ let $\vartriangle A_iB_iC_i$ be a triangle with side lengths $a_i, b_i, c_i$ and altitude lengths $p_i, q_i, r_i$.
Define $a_3 =\sqrt{a_1^2 + a_2^2}, b_3 =\sqrt{b_1^2 + b_2^2}$ , and $c_3 =\sqrt{c_1^2 + c_2^2}$.
Prove that $a_3, b_3, c_3$ are side lengths of a triangle, and if $p_3, q_3, r_3$ are the lengths of altitudes of this triangle,
then $p_3^2 \ge p_1^2 +p_2^2$, $q_3^2 \ge q_1^2 +q_2^2$ , and $r_3^2 \ge r_1^2 +r_2^2$
Find the positive integer $n$ such that a convex polygon with $3n + 2$ sides has $61.5$ percent fewer diagonals than a convex polygon with $5n - 2$ sides.
Suppose $n$ lines in plane are such that no two are parallel and no three are concurrent. For each two lines their angle is a real number in $[0,\frac{\pi}2]$. Find the largest value of the sum of the $\binom n2$ angles between line.
[i]By Aliakbar Daemi[/i]
Let $a_{1}, a_{2}, a_{3}, \cdots$ be an increasing sequence of nonnegative integers such that every nonnegative integer can be expressed uniquely in the form $a_{i}+2a_{j}+4a_{k}$, where $i, j, $ and $k$ are not necessarily distinct. Determine $a_{1998}$.
Solve the equation $5^x7^y+4=3^z$ in nonnegative integers.
Screws are sold in packs of $10$ and $12$. Harry and Sam independently go to the hardware store, and by coincidence each of them buys exactly $k$ screws. However, the number of packs of screws Harry buys is different than the number of packs Sam buys. What is the smallest possible value of $k$?
Find all functions $f: \mathbb{R} \to \mathbb{R}$ that have a continuous second derivative and for which the equality $f(7x+1)=49f(x)$ holds for all $x \in \mathbb{R}$.
An unlimited supply of 8-cent and 15-cent stamps is available. Some amounts of postage cannot be made up exactly, e.g., 7 cents, 29 cents. What is the largest unattainable amount, i.e., the amount, say $n$, of postage which is unattainable while all amounts larger than $n$ are attainable? (Justify your answer.)