Found problems: 85335
For each pair of real numbers $ a\not\equal{} b$, define the operation $ \star$ as \[(a \star b) \equal{} \frac{a \plus{} b}{a \minus{} b}.\] What is the value of $ ((1 \star 2) \star 3)$?
$ \textbf{(A)}\ \minus{}\frac{2}{3}\qquad
\textbf{(B)}\ \minus{}\frac{1}{5}\qquad
\textbf{(C)}\ 0\qquad
\textbf{(D)}\ \frac{1}{2}\qquad
\textbf{(E)}\ \text{This value is not defined.}$
Fix a point $O$ in the plane and an integer $n\geq 3$. Consider a finite family $\mathcal{D}$ of closed unit discs in the plane such that:
(a) No disc in $\mathcal{D}$ contains the point $O$; and
(b) For each positive integer $k < n$, the closed disc of radius $k + 1$ centred at $O$ contains the centres of at least $k$ discs in $\mathcal{D}$.
Show that some line through $O$ stabs at least $\frac{2}{\pi} \log \frac{n+1}{2}$ discs in $\mathcal{D}$.
Show that there are infinitely many positive real numbers a which are not integers such that a(a-3{a}) is an integer.
In triangle $ ABC$, $ CD$ is the altitude to $ AB$ and $ AE$ is the altitude to $ BC.$ If the lengths of $ AB, CD,$ and $ AE$ are known, the length of $ DB$ is:
$ \textbf{(A)}\ \text{not determined by the information given} \qquad$
$ \textbf{(B)}\ \text{determined only if A is an acute angle} \qquad$
$ \textbf{(C)}\ \text{determined only if B is an acute angle} \qquad$
$ \textbf{(D)}\ \text{determined only in ABC is an acute triangle} \qquad$
$ \textbf{(E)}\ \text{none of these is correct}$
A positive integer $n$ is called an untouchable number if there is no positive integer $m$ for which the sum of the factors of $m$ (including $m$ itself) is $n + m$. Find the sum of all of the untouchable numbers between $1$ and $10$ (inclusive)
Let $n=2k-1$, where $k\ge 6$ is an integer. Let $T$ be the set of all $n$-tuples $$\textbf{x}=(x_1,x_2,\dots ,x_n), \text{ where, for } i=1,2,\dots ,n, \text{ } x_i \text{ is } 0 \text{ or } 1.$$ For $\textbf{x}=(x_1,x_2,\dots ,x_n)$ and $\textbf{y}=(y_1,y_2,\dots ,y_n)$ in $T$, let $d(\textbf{x},\textbf{y})$ denote the number of integers $j$ with $1\le j\le n$ such that $x_j\neq x_y$. $($In particular, $d(\textbf{x},\textbf{x})=0)$.
Suppose that there exists a subset $S$ of $T$ with $2^k$ elements which has the following property: given any element $\textbf{x}$ in $T$, there is a unique $\textbf{y}$ in $S$ with $d(\textbf{x},\textbf{y})\le 3$.
Prove that $n=23$.
With a compass and a ruler, split a triangle into two smaller triangles with the same sum of squares of sides.
Two distinct real numbers are written on each vertex of a convex $2012-$gon. Show that we can remove a number from each vertex such that the remaining numbers on any two adjacent vertices are different.
Let $n^2-6n+1=0$. Find $n^6+\frac1{n^6}$.
Let $ABC$ be an acute scalene triangle with circumcircle $\omega$. Let $P$ and $Q$ be interior points of the sides $AB$ and $AC$, respectively, such that $PQ$ is parallel to $BC$. Let $L$ be a point on $\omega$ such that $AL$ is parallel to $BC$. The segments $BQ$ and $CP$ intersect at $S$. The line $LS$ intersects $\omega$ at $K$. Prove that $\angle BKP = \angle CKQ$.
Proposed by [i]Ervin Macić, Bosnia and Herzegovina[/i]
Are there integers $m$ and $n$ such that $5m^2 -6mn+7n^2 =1985$?
In triangle $ABC$, $\angle ABC = 120^{\circ}$, $AB = 3$ and $BC = 4$. If perpendiculars constructed to $\overline{AB}$ at $A$ and to $\overline{BC}$ at $C$ meet at $D$, then $CD = $
$ \textbf{(A)}\ 3\qquad\textbf{(B)}\ \frac{8}{\sqrt{3}}\qquad\textbf{(C)}\ 5\qquad\textbf{(D)}\ \frac{11}{2}\qquad\textbf{(E)}\ \frac{10}{\sqrt{3}} $
Alice is counting up by fives, starting with the number $3$. Meanwhile, Bob is counting down by fours, starting with the number $2021$. How many numbers between $3$ and $2021$, inclusive, are counted by both Alice and Bob?
Let $ABCD$ be a convex quadrilateral with the line $CD$ being tangent to the circle on diameter $AB$. Prove that the line $AB$ is tangent to the circle on diameter $CD$ if and only if the lines $BC$ and $AD$ are parallel.
Let $f(x)$ be a non-constant polynomial with integer coefficients such that $f(1) \neq 1$. For a positive integer $n$, define $\text{divs}(n)$ to be the set of positive divisors of $n$.
A positive integer $m$ is $f$-cool if there exists a positive integer $n$ for which $$f[\text{divs}(m)]=\text{divs}(n).$$
Prove that for any such $f$, there are finitely many $f$-cool integers.
(The notation $f[S]$ for some set $S$ denotes the set $\{f(s):s \in S\}$.)
In the acute-angled triangle $ABC$, the point $F$ is the foot of the altitude from $A$, and $P$ is a point on the segment $AF$. The lines through $P$ parallel to $AC$ and $AB$ meet $BC$ at $D$ and $E$, respectively. Points $X \ne A$ and $Y \ne A$ lie on the circles $ABD$ and $ACE$, respectively, such that $DA = DX$ and $EA = EY$.
Prove that $B, C, X,$ and $Y$ are concyclic.
It's pouring down rain, and the amount of rain hitting point $(x,y)$ is given by
$$f(x,y)=|x^3+2x^2y-5xy^2-6y^3|.$$
If you start at the origin $(0,0)$, find all the possibilities for $m$ such that $y=mx$ is a straight line along which you could walk without any rain falling on you.
How many roots does equation $\sin x = \frac{x}{100}$ have?
Find the sum of all integers from $1$ to $1000$ inclusive which contain at least one $7$ in their digits, i.e. find $$7 + 17 +... + 979 + 987 + 997.$$
$(a_{1}, a_{2}, ... , a_{2n})$ is a permutation of $\{1, 2, ... , 2n\}$ such that $|a_{i}-a_{i+1}| \neq |a_{j}-a_{j+1}|$ for $i \neq j$. Show that $a_{1}= a_{2n}+n$ iff $1 \leq a_{2i}\leq n$ for $i = 1, 2, ... n.$
Determine all functions $f:\mathbb{R} \rightarrow \mathbb{R}$ such that for every $x,y \in \mathbb{R}$
$$f(x^{4}-y^{4})+4f(xy)^{2}=f(x^{4}+y^{4})$$
Find largest possible constant $M$ such that, for any sequence $a_n$, $n=0,1,2,...$ of real numbers, that satisfies the conditions :
i) $a_0=1$, $a_1=3$
ii) $a_0+a_1+...+a_{n-1} \ge 3 a_n - a_{n+1}$ for any integer $n\ge 1$
to be true that
$$\frac{a_{n+1}}{a_n} >M$$ for any integer $n\ge 0$.
In one criminal kingdom, an underdeveloped state, the King decided to start a fight against corruption and, as an example, punish one of his $199$ ministers. The ministers were summoned to the palace and seated at a large round table. At first they wanted to find the one who had the most money in his bank account and declare him the main corrupt official. It takes $20$ minutes to determine the amount of money in the bank account of one minister. But the King ordered that the accused be found within four hours while he underwent medical procedures. According to the Noble Court Administrator, any minister can be accused, you just need to find a legal justification.The Chief Lawyer proposed that the first minister discovered, who has more money in his bank account than each of his two neighbors (one on the right and one on the left), be declared corrupt. How can one be sure to find a minister who meets this condition within the allotted $4$ hours? (During this time, it is possible to consistently determine the size of the bank accounts of no more than $12$ ministers. It is assumed that the amount of money in bank accounts is different.)
Let $N$ be a positive integer, and consider an $N \times N$ grid. A [i]right-down path[/i] is a sequence of grid cells such that each cell is either one cell to the right of or one cell below the previous cell in the sequence. A [i]right-up path[/i] is a sequence of grid cells such that each cell is either one cell to the right of or one cell above the previous cell in the sequence.
Prove that the cells of the $N \times N$ grid cannot be partitioned into less than $N$ right-down or right-up paths. For example, the following partition of the $5 \times 5$ grid uses $5$ paths.
[asy]
size(4cm);
draw((5,-1)--(0,-1)--(0,-2)--(5,-2)--(5,-3)--(0,-3)--(0,-4)--(5,-4),gray+linewidth(0.5)+miterjoin);
draw((1,-5)--(1,0)--(2,0)--(2,-5)--(3,-5)--(3,0)--(4,0)--(4,-5),gray+linewidth(0.5)+miterjoin);
draw((0,0)--(5,0)--(5,-5)--(0,-5)--cycle,black+linewidth(2.5)+miterjoin);
draw((0,-1)--(3,-1)--(3,-2)--(1,-2)--(1,-4)--(4,-4)--(4,-3)--(2,-3)--(2,-2),black+linewidth(2.5)+miterjoin);
draw((3,0)--(3,-1),black+linewidth(2.5)+miterjoin);
draw((1,-4)--(1,-5),black+linewidth(2.5)+miterjoin);
draw((4,-3)--(4,-1)--(5,-1),black+linewidth(2.5)+miterjoin);
[/asy]
[i]Proposed by Zixiang Zhou, Canada[/i]
Let $a, b, c, p, q, r > 0$ such that $(a,b,c)$ is a geometric progression and $(p, q, r)$ is an arithmetic progression. If \[a^p b^q c^r = 6 \quad \text{and} \quad a^q b^r c^p = 29\] then compute $\lfloor a^r b^p c^q \rfloor$.
[i]Proposed by Michael Tang[/i]