Found problems: 85335
2022 MMATHS, 10
Suppose that $A_1A_2A_3$ is a triangle with $A_1A_2 = 16$ and $A_1A_3 = A_2A_3 = 10$. For each integer $n \ge 4$, set An to be the circumcenter of triangle $A_{n-1}A_{n-2}A_{n-3}$. There exists a unique point $Z$ lying in the interiors of the circumcircles of triangles $A_kA_{k+1}A_{k+2}$ for all integers $k \ge 1$. If $ZA^2_1+ ZA^2_2+ ZA^2_3+ ZA^2_4$ can be expressed as $\frac{a}{b}$ for positive integers $a, b$ with $gcd(a, b) = 1$, find $a + b$.
2003 Cuba MO, 2
Let $A$ be a point outside the circle $\omega$ . The tangents from $A$ touch the circle at $B$ and $C$. Let $P$ be an arbitrary point on extension of $AC$ towards $C$, $Q$ the projection of $C$ onto $PB$ and $E$ the second intersection point of the circumcircle of $ABP$ with the circle $\omega$ . Prove that $\angle PEQ = 2\angle APB$
2022 CMIMC, 1.8
Find the largest $c > 0$ such that for all $n \ge 1$ and $a_1,\dots,a_n, b_1,\dots, b_n > 0$ we have
$$\sum_{j=1}^n a_j^4 \ge c\sum_{k = 1}^n \frac{\left(\sum_{j=1}^k a_jb_{k+1-j}\right)^4}{\left(\sum_{j=1}^k b_j^2j!\right)^2}$$
[i]Proposed by Grant Yu[/i]
2023 CMIMC Algebra/NT, 7
Let $\phi(n)$ denote the number of positive integers less than or equal to $n$ which are relatively prime to $n$. Compute $\displaystyle \sum_{i=1}^{\phi(2023)} \dfrac{\gcd(i,\phi(2023))}{\phi(2023)}$.
[i]Proposed by Giacomo Rizzo[/i]
2007 VJIMC, Problem 2
Alice has got a circular key ring with $n$ keys, $n\ge3$. When she takes it out of her pocket, she does not know whether it got rotated and/or flipped. The only way she can distinguish the keys is by coloring them (a color is assigned to each key). What is the minimum number of colors needed?
1999 National Olympiad First Round, 14
Find the sum of squares of the digits of the least positive integer having $72$ positive divisors.
$\textbf{(A)}\ 41 \qquad\textbf{(B)}\ 65 \qquad\textbf{(C)}\ 110 \qquad\textbf{(D)}\ 123 \qquad\textbf{(E)}\ \text{None}$
2018 Mathematical Talent Reward Programme, SAQ: P 5
[list=1]
[*] Prove that, the sequence of remainders obtained when the Fibonacci numbers are divided by $n$ is periodic, where $n$ is a natural number.
[*] There exists no such non-constant polynomial with integer coefficients such that for every Fibonacci number $n,$ $ P(n)$ is a prime.
[/list]
2008 Iran MO (3rd Round), 5
Prove that the following polynomial is irreducible in $ \mathbb Z[x,y]$:
\[ x^{200}y^5\plus{}x^{51}y^{100}\plus{}x^{106}\minus{}4x^{100}y^5\plus{}x^{100}\minus{}2y^{100}\minus{}2x^6\plus{}4y^5\minus{}2\]
2018 CCA Math Bonanza, L4.2
A subset of $\left\{1,2,3,\ldots,2017,2018\right\}$ has the property that none of its members are $5$ times another. What is the maximum number of elements that such a subset could have?
[i]2018 CCA Math Bonanza Lightning Round #4.2[/i]
1996 Tournament Of Towns, (504) 1
Do there exist $10$ consecutive positive integers such that the sum of their squares is equal to the sum of squares of the next $9$ integers?
(Inspired by a diagram in an old text book)
2022 Belarusian National Olympiad, 10.8
A sequence $a_1,\ldots,a_n$ of positive integers is given. For each $l$ from $1$ to $n-1$ the array $(gcd(a_1,a_{1+l}),\ldots,gcd(a_n,a_{n+l}))$ is considered, where indices are taken modulo $n$. It turned out that all this arrays consist of the same $n$ pairwise distinct numbers and differ only,possibly, by their order.
Can $n$ be a) $21$ b) $2021$
2008 China Northern MO, 4
As shown in figure , it is known that $ABCD$ is parallelogram, $A,B,C$ lie on circle $\odot O_1$, $AD$ and $BD$ intersect $\odot O$ at points $E$ and $F$ respectively, $C,D,F$ lie on circle $\odot O_2$, $AD$ intersects $\odot O_2$ at point $G$. If the radii of circles $\odot O_1$, $\odot O_2$ are $R_1, R_2$ respectively, prove that $\frac{EG}{AD}=\frac{R_2^2}{R_1^2}$.
[img]https://cdn.artofproblemsolving.com/attachments/d/f/1d9925a77d4f3fe068bd24364fb396eaa9a27a.png[/img]
2012 Indonesia MO, 3
Let $n$ be a positive integer. Show that the equation \[\sqrt{x}+\sqrt{y}=\sqrt{n}\] have solution of pairs of positive integers $(x,y)$ if and only if $n$ is divisible by some perfect square greater than $1$.
[i]Proposer: Nanang Susyanto[/i]
2005 AIME Problems, 1
A game uses a deck of $n$ different cards, where $n$ is an integer and $n \geq 6$. The number of possible sets of $6$ cards that can be drawn from the deck is $6$ times the number of possible sets of $3$ cards that can be drawn. Find $n$.
2004 Harvard-MIT Mathematics Tournament, 3
Compute \[ \left\lfloor \dfrac {2005^3}{2003 \cdot 2004} - \dfrac {2003^3}{2004 \cdot 2005} \right\rfloor \]
2009 Thailand Mathematical Olympiad, 4
Let $k$ be a positive integer. Show that there are infinitely many positive integer solutions $(m, n)$ to
$(m - n)^2 = kmn + m + n$.
1994 Putnam, 6
For $a\in \mathbb{Z}$ define \[ n_a=101a-100\cdot 2^a \]
Show that, for $0\le a,b,c,d\le 99$
\[ n_a+n_b\equiv n_c+n_d\pmod{10100}\implies \{a,b\}=\{c,d\} \]
2004 AMC 10, 25
Three mutually tangent spheres of radius 1 rest on a horizontal plane. A sphere of radius 2 rests on them. What is the distance from the plane to the top of the larger sphere?
$ \textbf{(A)}\; 3+\frac{\sqrt{30}}2\qquad
\textbf{(B)}\; 3+\frac{\sqrt{69}}3\qquad
\textbf{(C)}\; 3+\frac{\sqrt{123}}4\qquad
\textbf{(D)}\; \frac{52}9\qquad
\textbf{(E)}\; 3+2\sqrt{2} $
2022/2023 Tournament of Towns, P2
А positive integer $n{}$ is given. For every $x{}$ consider the sum \[Q(x)=\sum_{k=1}^{10^n}\left\lfloor\frac{x}{k}\right\rfloor.\]Find the difference $Q(10^n)-Q(10^n-1)$.
[i]Alexey Tolpygo[/i]
2005 AMC 12/AHSME, 9
There are two values of $ a$ for which the equation $ 4x^2 \plus{} ax \plus{} 8x \plus{} 9 \equal{} 0$ has only one solution for $ x$. What is the sum of these values of $ a$?
$ \textbf{(A)}\ \minus{}16\qquad
\textbf{(B)}\ \minus{}8\qquad
\textbf{(C)}\ 0\qquad
\textbf{(D)}\ 8\qquad
\textbf{(E)}\ 20$
2003 AMC 12-AHSME, 22
Objects $A$ and $B$ move simultaneously in the coordinate plane via a sequence of steps, each of length one. Object $A$ starts at $(0,0)$ and each of its steps is either right or up, both equally likely. Object $B$ starts at $(5,7)$ and each of its steps is either left or down, both equally likely. Which of the following is closest to the probability that the objects meet?
$ \textbf{(A)}\ 0.10 \qquad
\textbf{(B)}\ 0.15 \qquad
\textbf{(C)}\ 0.20 \qquad
\textbf{(D)}\ 0.25 \qquad
\textbf{(E)}\ 0.30$
2008 Romania National Olympiad, 2
a) Prove that
\[ \dfrac{1}{2} \plus{} \dfrac{1}{3} \plus{} ... \plus{} \dfrac{1}{2^{2n}} > n,
\]
for all positive integers $ n$.
b) Prove that for every positive integer $ n$ we have $ \min\left\{ k \in \mathbb{Z}, k\geq 2 \mid \dfrac{1}{2} \plus{} \dfrac{1}{3} \plus{} \cdots \plus{} \dfrac{1}{k}>n \right\} > 2^n$.
2009 Today's Calculation Of Integral, 442
Evaluate $ \int_0^{\frac{\pi}{2}} \frac{\cos \theta \minus{}\sin \theta}{(1\plus{}\cos \theta)(1\plus{}\sin \theta)}\ d\theta$
2012 Portugal MO, 1
A five-digit positive integer $abcde_{10}$ ($a\neq 0$) is said to be a [i]range[/i] if its digits satisfy the inequalities $a<b>c<d>e$. For example, $37452$ is a range. How many ranges are there?
2010 IMO Shortlist, 1
Find the least positive integer $n$ for which there exists a set $\{s_1, s_2, \ldots , s_n\}$ consisting of $n$ distinct positive integers such that
\[ \left( 1 - \frac{1}{s_1} \right) \left( 1 - \frac{1}{s_2} \right) \cdots \left( 1 - \frac{1}{s_n} \right) = \frac{51}{2010}.\]
[i]Proposed by Daniel Brown, Canada[/i]