Found problems: 85335
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.
2005 Belarusian National Olympiad, 1
Prove for positive numbers:
$$(a^2+b+\frac{3}{4})(b^2+a+\frac{3}{4}) \geq (2a+\frac{1}{2})(2b+\frac{1}{2})$$
2022 Chile Junior Math Olympiad, 3
By dividing $2023$ by a natural number $m$, the remainder is $23$. How many numbers $m$ are there with this property?
1996 National High School Mathematics League, 7
The number of proper subsets of the set $\left\{x|-1\leq\log_{\frac{1}{x}}10<-\frac{1}{2},x\in\mathbb{Z}_{\geq0}\right\}$ is________.
2005 AMC 10, 12
Twelve fair dice are rolled. What is the probability that the product of the numbers on the top faces is prime?
$ \textbf{(A)}\ \left(\frac{1}{12}\right)^{12}\qquad
\textbf{(B)}\ \left(\frac{1}{6}\right)^{12}\qquad
\textbf{(C)}\ 2\left(\frac{1}{6}\right)^{11}\qquad
\textbf{(D)}\ \frac{5}{2}\left(\frac{1}{6}\right)^{11}\qquad
\textbf{(E)}\ \left(\frac{1}{6}\right)^{10}$
2021 Sharygin Geometry Olympiad, 10-11.4
Can a triangle be a development of a quadrangular pyramid?
1971 AMC 12/AHSME, 29
Given the progression $10^{\dfrac{1}{11}}, 10^{\dfrac{2}{11}}, 10^{\dfrac{3}{11}}, 10^{\dfrac{4}{11}},\dots , 10^{\dfrac{n}{11}}$. The least positive integer $n$ such that the product of the first $n$ terms of the progression exceeds $100,000$ is
$\textbf{(A) }7\qquad\textbf{(B) }8\qquad\textbf{(C) }9\qquad\textbf{(D) }10\qquad \textbf{(E) }11$
2013 China Second Round Olympiad, 3
$n$ students take a test with $m$ questions, where $m,n\ge 2$ are integers. The score given to every question is as such: for a certain question, if $x$ students fails to answer it correctly, then those who answer it correctly scores $x$ points, while those who answer it wrongly scores $0$. The score of a student is the sum of his scores for the $m$ questions. Arrange the scores in descending order $p_1\ge p_2\ge \ldots \ge p_n$. Find the maximum value of $p_1+p_n$.
PEN P Problems, 16
Prove that there exist infinitely many integers $n$ such that $n, n+1, n+2$ are each the sum of the squares of two integers.
2009 Middle European Mathematical Olympiad, 2
Suppose that we have $ n \ge 3$ distinct colours. Let $ f(n)$ be the greatest integer with the property that every side and every diagonal of a convex polygon with $ f(n)$ vertices can be coloured with one of $ n$ colours in the following way:
(i) At least two colours are used,
(ii) any three vertices of the polygon determine either three segments of the same colour or of three different colours.
Show that $ f(n) \le (n\minus{}1)^2$ with equality for infintely many values of $ n$.
2022 Balkan MO, 1
Let $ABC$ be an acute triangle such that $CA \neq CB$ with circumcircle $\omega$ and circumcentre $O$. Let $t_A$ and $t_B$ be the tangents to $\omega$ at $A$ and $B$ respectively, which meet at $X$. Let $Y$ be the foot of the perpendicular from $O$ onto the line segment $CX$. The line through $C$ parallel to line $AB$ meets $t_A$ at $Z$. Prove that the line $YZ$ passes through the midpoint of the line segment $AC$.
[i]Proposed by Dominic Yeo, United Kingdom[/i]
1999 National Olympiad First Round, 12
\[ \begin{array}{c} {x^{2} \plus{} y^{2} \plus{} z^{2} \equal{} 21} \\
{x \plus{} y \plus{} z \plus{} xyz \equal{} \minus{} 3} \\
{x^{2} yz \plus{} y^{2} xz \plus{} z^{2} xy \equal{} \minus{} 40} \end{array}
\]
The number of real triples $ \left(x,y,z\right)$ satisfying above system is
$\textbf{(A)}\ 0 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 6 \qquad\textbf{(D)}\ 12 \qquad\textbf{(E)}\ \text{None}$
OIFMAT II 2012, 1
A circle is divided into $ n $ equal parts. Marceline sets out to assign whole numbers from $ 1 $ to $ n $ to each of these pieces so that the distance between two consecutive numbers is always the same. The numbers $ 887 $, $ 217 $ and $ 1556 $ occupy consecutive positions. How many parts was the circumference divided into?
LMT Team Rounds 2021+, A9
Find the sum of all positive integers $n$ such that $7<n < 100$ and $1573_{n}$ has $6$ factors when written in base $10$.
[i]Proposed by Aidan Duncan[/i]
PEN L Problems, 6
Prove that no Fibonacci number can be factored into a product of two smaller Fibonacci numbers, each greater than 1.
2024 Korea Junior Math Olympiad (First Round), 4.
There is a shape like this (Attachment down below)
Find the number of triangles made by choosing 3 vertex from the 8 vertex in the attachment.
2006 Stanford Mathematics Tournament, 12
What is the largest prime factor of 8091?
1996 AMC 12/AHSME, 18
A circle of radius 2 has center at (2,0). A circle of radius 1 has center at (5,0). A line is tangent to the two circles at points in the first quadrant. Which of the following is closest to the $y$-intercept of the line?
$\text{(A)} \ \sqrt{2}/4 \qquad \text{(B)} \ 8/3 \qquad \text{(C)} \ 1 + \sqrt 3 \qquad \text{(D)} \ 2 \sqrt 2 \qquad \text{(E)} \ 3$
2015 AMC 10, 4
Four siblings ordered an extra large pizza. Alex ate $\frac15$, Beth $\frac13$, and Cyril $\frac14$ of the pizza. Dan got the leftovers. What is the sequence of the siblings in decreasing order of the part of pizza they consumed?
$\textbf{(A) } \text{Alex, Beth, Cyril, Dan}$
$\textbf{(B) } \text{Beth, Cyril, Alex, Dan}$
$\textbf{(C) } \text{Beth, Cyril, Dan, Alex}$
$\textbf{(D) } \text{Beth, Dan, Cyril, Alex}$
$\textbf{(E) } \text{Dan, Beth, Cyril, Alex}$
2015 Purple Comet Problems, 5
The two diagonals of a quadrilateral have lengths $12$ and $9$, and the two diagonals are perpendicular to each other. Find the area of the quadrilateral.
2020 AMC 12/AHSME, 1
What is the value in simplest form of the following expression? \[\sqrt{1} + \sqrt{1+3} + \sqrt{1+3+5} + \sqrt{1+3+5+7}\]
$\textbf{(A) }5 \qquad \textbf{(B) }4 + \sqrt{7} + \sqrt{10} \qquad \textbf{(C) } 10 \qquad \textbf{(D) } 15 \qquad \textbf{(E) } 4 + 3\sqrt{3} + 2\sqrt{5} + \sqrt{7}$
2024 Belarusian National Olympiad, 11.6
Let $2=p_1<p_2<\ldots<p_n<\ldots$ be all prime numbers.
Prove that for any positive integer $n \geq 3$ there exist at least $p_n+n-1$ prime numbers, that do not exceed $p_1p_2\ldots p_n$
[i]I. Voronovich[/i]
2021 JHMT HS, 6
Suppose $JHMT$ is a convex quadrilateral with perimeter $68$ and satisfies $\angle HJT = 120^\circ,$ $HM = 20,$ and $JH + JT = JM > HM.$ Furthermore, $\overrightarrow{JM}$ bisects $\angle HJT.$ Compute $JM.$
III Soros Olympiad 1996 - 97 (Russia), 11.3
A chord $AB$ is drawn in a certain circle. The smaller of the two arcs $AB$ corresponds to a central angle of $120^o$. A tangent $p$ to this arc is drawn. Two circles with radii $R$ and $r$ are constructed, touching this smaller arc $AB$ and straight lines $AB$ and $p$. Find the radius of the original circle.
2010 Balkan MO Shortlist, C1
In a soccer tournament each team plays exactly one game with all others. The winner gets $3$ points, the loser gets $0$ and each team gets $1$ point in case of a draw. It is known that $n$ teams ($n \geq 3$) participated in the tournament and the final classification is given by the arithmetical progression of the points, the last team having only 1 point.
[list=a]
[*] Prove that this configuration is unattainable when $n=12$
[*] Find all values of $n$ and all configurations when this is possible
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