Found problems: 85335
Let $k$ be a real number such that the system \begin{align*} &|25+20i-z|=5\\ &|z-4-k|=|z-3i-k| \\ \end{align*} has exactly one complex solution $z.$ The sum of all possible values of $k$ can be written as $\dfrac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$ Here $i=\sqrt{-1}.$
Let $A$ be a set of positive integers containing the number $1$ and at least one more element. Given that for any two different elements $m, n$ of A the number $ \frac{m+1 }{(m+1,n+1) }$ is also an element of $A$, prove that $A$ coincides with the set of positive integers.
Sets A and B, shown in the venn diagram, have the same number of elements. Thier union has 2007 elements and their intersection has 1001 elements. Find the number of elements in A.
[asy]
defaultpen(linewidth(0.7));
draw(Circle(origin, 5));
draw(Circle((5,0), 5));
label("$A$", (0,5), N);
label("$B$", (5,5), N);
label("$1001$", (2.5, -0.5), N);[/asy]
$ \textbf{(A)}\: 503\qquad \textbf{(B)}\: 1006\qquad \textbf{(C)}\: 1504\qquad \textbf{(D)}\: 1507\qquad \textbf{(E)}\: 1510\qquad $
In the every cell of the board $5\times5$ there is one of the numbers: $-1$, $0$, $1$. It is true that in every $2 \times 2$ square there are three numbers summing up to $0$. Determine the maximal sum of all numbers in a board.
Find all positive integers $a,b$ such that $\frac{a^2+b}{b^2-a}$ and $\frac{b^2+a}{a^2-b}$ are also integers.
Find the smallest prime number $p$ that cannot be represented in the form $|3^{a} - 2^{b}|$, where $a$ and $b$ are non-negative integers.
Let $ABC$ be a non-isosceles triangle with circumcircle $k$, incenter $I$ and $C$-excenter $I_C$. Let $M$ be the midpoint of $AB$ and $N$ be the midpoint of arc $\widehat{ACB}$ on $k$. Prove that $\angle IMI_C + \angle INI_C = 180^{\circ}$.
Given infinite sequence $a_n$. It is known that the limit of $$b_n=a_{n+1}-a_n/2$$ equals zero. Prove that the limit of $a_n$ equals zero.
Let $\phi(k)$ denote the number of positive integers less than or equal to $k$ that are relatively prime to $k$. For example, $\phi(2)=1$ and $\phi(10)=4$. Compute the number of positive integers $n \leq 2020$ such that $\phi(n^2)=2\phi(n)^2$.
Joe and Kathryn are both on the school math team, which practices every Wednesday after school until $4$ PM for competitions. The team was preparing for the $ 2005$ iTest when Joe realized how crazy he was for not asking Kathryn out – the way she worked those iTest problems, solving question after question, almost made him go insane sitting there that day. He never felt the same way when she worked on preparing for other competitions – they just aren’t the same.
Kathryn always beat Joe at competitions, too. Joe admired her resolve and unwillingness to make herself look stupid, when so many other girls he knew at school tried to pretend they were stupid in order to attract guys.
So as time ticked away and that afternoon’s Wednesday practice neared an end, Joe was determined to strike up a conversation with Kathryn and ask her out. He really wanted to impress her, so he thought he’d ask her a really hard history of math question that she didn’t know. Naturally, she’d want the answer, and be so impressed with Joe’s brilliance that she’d go out with him on Friday night.
Great plan. Seriously.
When Joe asked Kathryn after class, “Who was the mathematician that died in approximately $200$ B.C. that developed a method for calculating all prime numbers?” Kathryn gave the correct response. What name did she say?
Determine the number of positive divisors of $2008^8$ that are less than $2008^4$.
Prove that there exists just one function $ f:\mathbb{N}^2\longrightarrow\mathbb{N} $ which simultaneously satisfies:
$ \text{(1)}\quad f(m,n)=f(n,m),\quad\forall m,n\in\mathbb{N} $
$ \text{(2)}\quad f(n,n)=n,\quad\forall n\in\mathbb{N} $
$ \text{(3)}\quad n>m\implies (n-m)f(m,n)=nf(m,n-m), \quad\forall m,n\in\mathbb{N} $
Let three lines forming a triangle $ABC$ be given. Using a two-sided ruler and drawing at most eight lines construct a point $D$ on the side $AB$ such that $\frac{AD}{BD}=\frac{BC}{AC}.$
Find all pairs $ (a, b)$ of real numbers such that whenever $ \alpha$ is a root of $ x^{2} \plus{} ax \plus{} b \equal{} 0$, $ \alpha^{2} \minus{} 2$ is also a root of the equation.
[b][Weightage 17/100][/b]
In the following equation, each of the letters represents uniquely a different digit in base ten:
\[ (YE) \cdot (ME) \equal{} TTT\]
The sum $ E\plus{}M\plus{}T\plus{}Y$ equals
$ \textbf{(A)}\ 19 \qquad
\textbf{(B)}\ 20 \qquad
\textbf{(C)}\ 21 \qquad
\textbf{(D)}\ 22 \qquad
\textbf{(E)}\ 24$
A natural number $n$ is given. Determine all $(n - 1)$-tuples of nonnegative integers $a_1, a_2, ..., a_{n - 1}$ such that
$$\lfloor \frac{m}{2^n - 1}\rfloor + \lfloor \frac{2m + a_1}{2^n - 1}\rfloor + \lfloor \frac{2^2m + a_2}{2^n - 1}\rfloor + \lfloor \frac{2^3m + a_3}{2^n - 1}\rfloor + ... + \lfloor \frac{2^{n - 1}m + a_{n - 1}}{2^n - 1}\rfloor = m$$
holds for all $m \in \mathbb{Z}$.
8. You have been kidnapped by a witch and are stuck in the [i]Terrifying Tower[/i], which has an infinite number of floors, starting with floor 1, each initially having 0 boxes. The witch allows you to do the following two things:[list]
[*] For a floor $i$, put 2 boxes on floor $i+5$, 6 on floor $i+4$, 13 on floor $i+3$, 12 on floor $i+2$, 8 on floor $i+1$, and 1 on floor $i$, or remove the corresponding number of boxes from each floor if possible.
[*] For a floor $i$, put 1 box on floor $i+4$, put 3 boxes on floor $i+3$, 6 on floor $i+2$, 5 on floor $i+1$, and 3 on floor $i$, or remove the corresponding number of boxes from each floor if possible. [/list]
At the end, suppose the witch wishes to have exactly $n$ boxes in the tower. Specifically, she wants them to be on the first 10 floors. Let $T(n)$ be the number of distinct distributions of these $n$ boxes that you can make. Find $\displaystyle\sum_{n=1}^{15} T(n)$.
[i]Proposed by Monkey_king1[/i]
A coloring of all plane points with coordinates belonging to the set $S=\{0,1,\ldots,99\}$ into red and white colors is said to be [i]critical[/i] if for each $i,j\in S$ at least one of the four points $(i,j),(i + 1,j),(i,j + 1)$ and $(i + 1, j + 1)$ $(99 + 1\equiv0)$ is colored red. Find the maximal possible number of red points in a critical coloring which loses its property after recoloring of any red point into white.
Let $\mathcal{C}_1$ and $\mathcal{C}_2$ be circles whose centers are $10$ units apart, and whose radii are $1$ and $3$. Find, with proof, the locus of all points $M$ for which there exists points $X\in \mathcal{C}_1,Y\in \mathcal{C}_2$ such that $M$ is the midpoint of $XY$.
Let $\triangle ABC$ be an acute triangle and $AD$ the bisector of the angle $\angle BAC$ with $D\in(BC)$. Let $E$ and $F$ denote feet of perpendiculars from $D$ to $AB$ and $AC$ respectively. If $BF\cap CE=K$ and $\odot AKE\cap BF=L$ prove that $DL\perp BF$.
Let $ n$ be a nonzero positive integer. Find $ n$ such that there exists a permutation $ \sigma \in S_{n}$ such that
\[ \left| \{ |\sigma(k) \minus{} k| \ : \ k \in \overline{1, n} \}\right | = n.\]
Prove that:
[list=a][*]$\displaystyle\left( \frac{1}{2}\right) ^{3}+\left( \frac{2}{3}\right)^{3}+\left( \frac{5}{6}\right) ^{3}=1$
[*]$3^{33}+4^{33}+5^{33}<6^{33}$[/list]
For a natural number $k>1$, define $S_k$ to be the set of all triplets $(n,a,b)$ of natural numbers, with $n$ odd and $\gcd (a,b)=1$, such that $a+b=k$ and $n$ divides $a^n+b^n$. Find all values of $k$ for which $S_k$ is finite.
Let $M$ be the midpoint of side $BC$ of triangle $ABC$ ($AB>AC$), and let $AL$ be the bisector of the angle $A$. The line passing through $M$ perpendicular to $AL$ intersects the side $AB$ at the point $D$. Prove that $AD+MC$ is equal to half the perimeter of triangle $ABC$.
Let $ABC$ be an acute triangle. Let $M$ be the midpoint of side $BC$, and let $E$ and $F$ be the feet of the altitudes from $B$ and $C$, respectively. Suppose that the common external tangents to the circumcircles of triangles $BME$ and $CMF$ intersect at a point $K$, and that $K$ lies on the circumcircle of $ABC$. Prove that line $AK$ is perpendicular to line $BC$.
[i]Kevin Cong[/i]