Found problems: 85335
2018 Purple Comet Problems, 21
Let $x$ be in the interval $\left(0, \frac{\pi}{2}\right)$ such that $\sin x - \cos x = \frac12$ . Then $\sin^3 x + \cos^3 x = \frac{m\sqrt{p}}{n}$ , where $m, n$, and $p$ are relatively prime positive integers, and $p$ is not divisible by the square of any prime. Find $m + n + p$.
MOAA Gunga Bowls, 2023.2
Harry wants to put $5$ identical blue books, $3$ identical red books, and $1$ white book on his bookshelf. If no two adjacent books may be the same color, how many distinct arrangements can Harry make?
[i]Proposed by Anthony Yang[/i]
2007 AIME Problems, 12
The increasing geometric sequence $x_{0},x_{1},x_{2},\ldots$ consists entirely of integral powers of $3.$ Given that \[\sum_{n=0}^{7}\log_{3}(x_{n}) = 308\qquad\text{and}\qquad 56 \leq \log_{3}\left ( \sum_{n=0}^{7}x_{n}\right ) \leq 57,\] find $\log_{3}(x_{14}).$
2022 CCA Math Bonanza, TB1
How many positive integer factors does the following expression have? \[ \sum_{n=1}^{999} \log_{10} \left(\frac{n+1}{n} \right) \]
[i]2022 CCA Math Bonanza Tiebreaker Round #1[/i]
2023 MOAA, 16
Compute the sum $$\frac{\varphi(50!)}{\varphi(49!)}+ \frac{\varphi(51!)}{\varphi(50!)} + \dots + \frac{\varphi(100!)}{\varphi(99!)}$$ where $\varphi(n)$ returns the number of positive integers less than $n$ that are relatively prime to $n$.
[i]Proposed by Andy Xu[/i]
2004 Purple Comet Problems, 25
In the addition problem
\[ \setlength{\tabcolsep}{1mm}\begin{tabular}{cccccc}& W & H & I & T & E\\ + & W & A & T & E & R \\\hline P & I & C & N & I & C\end{tabular} \] each distinct letter represents a different digit. Find the number represented by the answer PICNIC.
1993 Korea - Final Round, 2
Let be given a triangle $ABC$ with $BC = a, CA = b, AB = c$. Find point $P$ in the plane for which $aAP^{2}+bBP^{2}+cCP^{2}$ is minimum, and compute this minimum.
1941 Moscow Mathematical Olympiad, 073
Given a quadrilateral, the midpoints $A, B, C, D$ of its consecutive sides, and the midpoints of its diagonals, $P$ and $Q$. Prove that $\vartriangle BCP = \vartriangle ADQ$.
2010 ELMO Shortlist, 7
Find the smallest real number $M$ with the following property: Given nine nonnegative real numbers with sum $1$, it is possible to arrange them in the cells of a $3 \times 3$ square so that the product of each row or column is at most $M$.
[i]Evan O' Dorney.[/i]
V Soros Olympiad 1998 - 99 (Russia), 11.8
Side $BC$ of triangle $ABC$ is equal to $a$ and the opposite angle is equal to $\theta$. A straight line passing through the midpoint of $BC$ and the center of the inscribed circle intersects lines $AB$ and $AC$ at points $M$ and $P$, respectively. Find the area of the quadrilateral (non-convex) $BMPC$.
2011 ELMO Shortlist, 6
Do there exist positive integers $k$ and $n$ such that for any finite graph $G$ with diameter $k+1$ there exists a set $S$ of at most $n$ vertices such that for any $v\in V(G)\setminus S$, there exists a vertex $u\in S$ of distance at most $k$ from $v$?
[i]David Yang.[/i]
2002 USAMTS Problems, 2
Find four distinct positive integers, $a$, $b$, $c$, and $d$, such that each of the four sums $a+b+c$, $a+b+d$,$a+c+d$, and $b+c+d$ is the square of an integer. Show that infinitely many quadruples $(a,b,c,d)$ with this property can be created.
2022 Portugal MO, 5
In a badminton competition, $16$ players participate, of which $10$ are professionals and $6$ are amateurs. In the first phase, eight games are drawn. Among the eight winners of these games, four games are drawn. The four winners qualify for the semi-finals of the competition. Assuming that, whenever a professional player and an amateur play each other, the professional wins the game, what is the probability that an amateur player will reach the semi-finals of the competition?
LMT Team Rounds 2021+, 5
In regular hexagon $ABCDEF$ with side length $2$, let $P$, $Q$, $R$, and $S$ be the feet of the altitudes from $A$ to $BC$, $EF$, $CF$, and $BE$, respectively. Find the area of quadrilateral $PQRS$.
2022 Assara - South Russian Girl's MO, 2
There are $2022$ natural numbers written in a row. Product of any two adjacent numbers is a perfect cube. Prove that the product of the two extremes is also a perfect cube.
2018 Ramnicean Hope, 1
Let be two nonzero real numbers $ a,b $ such that $ |a|\neq |b| $ and let $ f:\mathbb{R}\longrightarrow\mathbb{R} $ be a function satisfying the functional relation
$$ af(x)+bf(-x)=(x^3+x)^5+\sin^5 x . $$
Calculate $ \int_{-2019}^{2019}f(x)dx . $
[i]Constantin Rusu[/i]
2015 AMC 12/AHSME, 14
What is the value of $a$ for which $\frac1{\log_2a}+\frac1{\log_3a}+\frac1{\log_4a}=1$?
$\textbf{(A) }9\qquad\textbf{(B) }12\qquad\textbf{(C) }18\qquad\textbf{(D) }24\qquad\textbf{(E) }36$
1998 Estonia National Olympiad, 5
The paper is marked with the finite number of blue and red dots and some these points are connected by lines. Let's name a point $P$ [i]special [/i] if more than half of the points connected with $P$ has a color other than point $P$. Juku selects one special point and reverses its color. Then Juku selects a new special point and changes its color, etc. Prove that by a finite number of integers Juku ends up in a situation where the paper has not made a special point.
2020 Kyiv Mathematical Festival, 1.1
(a) Find the numbers $a_0,. . . , a_{100}$, such that $a_0 = 0, a_{100} = 1$ and for all $k = 1,. . . , 99$ :
$$a_k = \frac12 a_{k- 1} + \frac12 a_{k+1 }$$
(b) Find the numbers $a_0,. . . , a_{100}$, such that $a_0 = 0, a_{100} = 1$ and for all $k = 1,. . . , 99$ :
$$a_k = 1+\frac12 a_{k- 1} + \frac12 a_{k+1 }$$.
2023 CMWMC, R3
[u]Set 3[/u]
[b]3.1[/b] Find the number of distinct values that can be made by inserting parentheses into the expression
$$1\,\,\,\,\, - \,\,\,\,\, 1 \,\,\,\,\, -\,\,\,\,\, 1 \,\,\,\,\, - \,\,\,\,\, 1 \,\,\,\,\, - \,\,\,\,\, 1\,\,\,\,\, - \,\,\,\,\, 1$$
such that you don’t introduce any multiplication. For example, $(1-1)-((1-1)-1-1)$ is a valid way to insert parentheses, but $1 - 1(-1 - 1) - 1 - 1$ is not.
[b]3.2[/b] Let $T$ be the answer from the previous problem. Katie rolls T fair 4-sided dice with faces labeled $0-3$. Considering all possible sums of these rolls, there are two sums that have the highest probability of occurring. Find the smaller of these two sums.
[b]3.3[/b] Let $T$ be the answer from the previous problem. Amy has a fair coin that she will repeatedly flip until her total number of heads is strictly greater than her total number of tails. Find the probability she will flip the coin exactly T times. (Hint: Finding a general formula in terms of T is hard, try solving some small cases while you wait for $T$.)
PS. You should use hide for answers.
1985 Tournament Of Towns, (081) T2
There are $68$ coins , each coin having a different weight than that of each other . Show how to find the heaviest and lightest coin in $100$ weighings on a balance beam.
(S. Fomin, Leningrad)
2025 AIME, 8
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 $i=\sqrt{-1}.$
2016 Polish MO Finals, 5
There are given two positive real number $a<b$. Show that there exist positive integers $p, \ q, \ r, \ s$ satisfying following conditions:
$1$. $a< \frac{p}{q} < \frac{r}{s} < b$.
$2.$ $p^2+q^2=r^2+s^2$.
2009 Purple Comet Problems, 4
John, Paul, George, and Ringo baked a circular pie. Each cut a piece that was a sector of the circle. John took one-third of the whole pie. Paul took one-fourth of the whole pie. George took one-fifth of the whole pie. Ringo took one-sixth of the whole pie. At the end the pie had one sector remaining. Find the measure in degrees of the angle formed by this remaining sector.
2023 Mexican Girls' Contest, 3
Find all triples $(a,b,c)$ of real numbers all different from zero that satisfies:
\begin{eqnarray} a^4+b^2c^2=16a\nonumber \\ b^4+c^2a^2=16b \nonumber\\ c^4+a^2b^2=16c \nonumber \end{eqnarray}