Found problems: 85335
2007 AIME Problems, 9
Rectangle $ABCD$ is given with $AB=63$ and $BC=448.$ Points $E$ and $F$ lie on $AD$ and $BC$ respectively, such that $AE=CF=84.$ The inscribed circle of triangle $BEF$ is tangent to $EF$ at point $P,$ and the inscribed circle of triangle $DEF$ is tangent to $EF$ at point $Q.$ Find $PQ.$
2012 China Second Round Olympiad, 11
In the Cartesian plane $XOY$, there is a rhombus $ABCD$ whose side lengths are all $4$ and $|OB|=|OD|=6$, where $O$ is the origin.
[b](1)[/b] Prove that $|OA|\cdot |OB|$ is a constant.
[b](2)[/b] Find the locus of $C$ if $A$ is a point on the semicircle
\[(x-2)^2+y^2=4 \quad (2\le x\le 4).\]
2019 Harvard-MIT Mathematics Tournament, 5
Contessa is taking a random lattice walk in the plane, starting at $(1,1)$. (In a random lattice walk, one moves up, down, left, or right $1$ unit with equal probability at each step.) If she lands on a point of the form $(6m,6n)$ for $m,n \in \mathbb{Z}$, she ascends to heaven, but if she lands on a point of the form $(6m+3,6n+3)$ for $m,n \in \mathbb{Z}$, she descends to hell. What is the probability she ascends to heaven?
2023 Malaysia IMONST 2, 1
Prove that for all positive integers $n$, $1^3 + 2^3 + 3^3 +\dots+n^3$ is a perfect square.
2017 BmMT, Ind. Tie
[b]p1.[/b] Consider a $4 \times 4$ lattice on the coordinate plane. At $(0,0)$ is Mori’s house, and at $(4,4)$ is Mori’s workplace. Every morning, Mori goes to work by choosing a path going up and right along the roads on the lattice. Recently, the intersection at $(2, 2)$ was closed. How many ways are there now for Mori to go to work?
[b]p2.[/b] Given two integers, define an operation $*$ such that if a and b are integers, then a $*$ b is an integer. The operation $*$ has the following properties:
1. $a * a$ = 0 for all integers $a$.
2. $(ka + b) * a = b * a$ for integers $a, b, k$.
3. $0 \le b * a < a$.
4. If $0 \le b < a$, then $b * a = b$.
Find $2017 * 16$.
[b]p3.[/b] Let $ABC$ be a triangle with side lengths $AB = 13$, $BC = 14$, $CA = 15$. Let $A'$, $B'$, $C'$, be the midpoints of $BC$, $CA$, and $AB$, respectively. What is the ratio of the area of triangle $ABC$ to the area of triangle $A'B'C'$?
[b]p4.[/b] In a strange world, each orange has a label, a number from $0$ to $10$ inclusive, and there are an infinite number of oranges of each label. Oranges with the same label are considered indistinguishable. Sally has 3 boxes, and randomly puts oranges in her boxes such that
(a) If she puts an orange labelled a in a box (where a is any number from 0 to 10), she cannot put any other oranges labelled a in that box.
(b) If any two boxes contain an orange that have the same labelling, the third box must also contain an orange with that labelling.
(c) The three boxes collectively contain all types of oranges (oranges of any label).
The number of possible ways Sally can put oranges in her $3$ boxes is $N$, which can be written as the product of primes: $$p_1^{e_1} p_2^{e_2}... p_k^{e_k}$$ where $p_1 \ne p_2 \ne p_3 ... \ne p_k$ and $p_i$ are all primes and $e_i$ are all positive integers. What is the sum $e_1 + e_2 + e_3 +...+ e_k$?
[b]p5.[/b] Suppose I want to stack $2017$ identical boxes. After placing the first box, every subsequent box must either be placed on top of another one or begin a new stack to the right of the rightmost pile. How many different ways can I stack the boxes, if the order I stack them doesn’t matter? Express your answer as $$p_1^{e_1} p_2^{e_2}... p_n^{e_n}$$ where $p_1, p_2, p_3, ... , p_n$ are distinct primes and $e_i$ are all positive integers.
PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
2000 AIME Problems, 14
In triangle $ABC,$ it is given that angles $B$ and $C$ are congruent. Points $P$ and $Q$ lie on $\overline{AC}$ and $\overline{AB},$ respectively, so that $AP=PQ=QB=BC.$ Angle $ACB$ is $r$ times as large as angle $APQ,$ where $r$ is a positive real number. Find the greatest integer that does not exceed $1000r.$
2014 PUMaC Number Theory B, 6
Let $S = \{2,5,8,11,14,17,20,\dots\}$. Given that one can choose $n$ different numbers from $S$, $\{A_1,A_2,\dots,A_n\}$ s.t. $\sum_{i=1}^n \frac{1}{A_i} = 1$, find the minimum possible value of $n$.
2003 Germany Team Selection Test, 2
Given a triangle $ABC$ and a point $M$ such that the lines $MA,MB,MC$ intersect the lines $BC,CA,AB$ in this order in points $D,E$ and $F,$ respectively. Prove that there are numbers $\epsilon_1, \epsilon_2, \epsilon_3 \in \{-1, 1\}$ such that:
\[\epsilon_1 \cdot \frac{MD}{AD} + \epsilon_2 \cdot \frac{ME}{BE} + \epsilon_3 \cdot \frac{MF}{CF} = 1.\]
2024 CMIMC Combinatorics and Computer Science, 7
If $S=\{s_1,s_2,\dots,s_n\}$ is a set of integers with $s_1<s_2<\dots<s_n$, define
$$f(S)=\sum_{k=1}^n (-1)^k k^2 s_k.$$
(If $S$ is empty, $f(S)=0$.) Compute the average value of $f(S)$ as $S$ ranges over all subsets of $\{1^2,2^2,\dots,100^2\}$.
[i]Proposed by Connor Gordon and Nairit Sarkar[/i]
Mathematical Minds 2023, P2
Let $a,b,c$ be real numbers with sum equal to zero. Prove that \[ab^3+bc^3+ca^3\leqslant 0.\]
2017 Peru Iberoamerican Team Selection Test, P2
Determine if there exists a positive integer $n$ such that $n^2+11$ is a prime number and $n+4$ is a perfect cube.
2011 APMO, 1
Let $a,b,c$ be positive integers. Prove that it is impossible to have all of the three numbers $a^2+b+c,b^2+c+a,c^2+a+b$ to be perfect squares.
2011 Today's Calculation Of Integral, 719
Compute $\int_0^x \sin t\cos t\sin (2\pi\cos t)\ dt$.
2024 Czech-Polish-Slovak Junior Match, 1
Let $G$ be the barycenter of triangle $ABC$. Let $D$ be a point such that $AGDB$ is a parallelogram. Show that $BG \parallel CD$.
2005 China Team Selection Test, 2
Let $n$ be a positive integer, and $x$ be a positive real number. Prove that $$\sum_{k=1}^{n} \left( x \left[\frac{k}{x}\right] - (x+1)\left[\frac{k}{x+1}\right]\right) \leq n,$$ where $[x]$ denotes the largest integer not exceeding $x$.
2019 IMO Shortlist, N7
Prove that there is a constant $c>0$ and infinitely many positive integers $n$ with the following property: there are infinitely many positive integers that cannot be expressed as the sum of fewer than $cn\log(n)$ pairwise coprime $n$th powers.
[i]Canada[/i]
1966 Miklós Schweitzer, 7
Does there exist a function $ f(x,y)$ of two real variables that takes natural numbers as its values and for which $ f(x,y)\equal{}f(y,z)$ implies $ x\equal{}y\equal{}z?$
[i]A. Hajnal[/i]
ICMC 7, 5
Is it possible to dissect an equilateral triangle into three congruent polygonal pieces (not necessarily convex), one of which contains the triangle’s centre in its interior?
[i]Note:[/i] The interior of a polygon is the polygon without its boundary.
[i]Proposed by Dylan Toh[/i]
2018 Purple Comet Problems, 18
Rectangle $ABCD$ has side lengths $AB = 6\sqrt3$ and $BC = 8\sqrt3$. The probability that a randomly chosen point inside the rectangle is closer to the diagonal $\overline{AC}$ than to the outside of the rectangle is $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
2016 Harvard-MIT Mathematics Tournament, 34
$\textbf{(Caos)}$ A cao [sic] has 6 legs, 3 on each side. A walking pattern for the cao is defined as an ordered sequence of raising and lowering each of the legs exactly once (altogether 12 actions), starting and ending with all legs on the ground. The pattern is safe if at any point, he has at least 3 legs on the ground and not all three legs are on the same side. Estimate $N$, the number of safe patterns.
An estimate of $E > 0$ earns $\left\lfloor 20\min(N/E, E/N)^4 \right\rfloor$ points.
2002 HKIMO Preliminary Selection Contest, 2
A clock has an hour hand of length 3 and a minute hand of length 4. From 1:00 am to 1:00 pm of the same day, find the number of occurrences when the distance between the tips of the two hands is an integer.
2019 Putnam, B5
Let $F_m$ be the $m$'th Fibonacci number, defined by $F_1=F_2=1$ and $F_m = F_{m-1}+F_{m-2}$ for all $m \geq 3$. Let $p(x)$ be the polynomial of degree 1008 such that $p(2n+1)=F_{2n+1}$ for $n=0,1,2,\ldots,1008$. Find integers $j$ and $k$ such that $p(2019) = F_j - F_k$.
2010 Indonesia TST, 2
Consider a polynomial with coefficients of real numbers $ \phi(x)\equal{}ax^3\plus{}bx^2\plus{}cx\plus{}d$ with three positive real roots. Assume that $ \phi(0)<0$, prove that \[ 2b^3\plus{}9a^2d\minus{}7abc \le 0.\]
[i]Hery Susanto, Malang[/i]
2000 AMC 12/AHSME, 18
In year $ N$, the $ 300^\text{th}$ day of the year is a Tuesday. In year $ N \plus{} 1$, the $ 200^{\text{th}}$ day of the year is also a Tuesday. On what day of the week did the $ 100^\text{th}$ day of year $ N \minus{} 1$ occur?
$ \textbf{(A)}\ \text{Thursday} \qquad \textbf{(B)}\ \text{Friday} \qquad \textbf{(C)}\ \text{Saturday} \qquad \textbf{(D)}\ \text{Sunday} \qquad \textbf{(E)}\ \text{Monday}$
2016 IFYM, Sozopol, 3
Find the least natural number $n\geq 5$, for which $x^n\equiv 16\, (mod\, p)$ has a solution for any prime number $p$.