Found problems: 85335
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
[/list]
1991 Chile National Olympiad, 1
Determine all nonnegative integer solutions of the equation $2^x-2^y = 1$
2022 Dutch BxMO TST, 1
Find all functions $f : Z_{>0} \to Z_{>0}$ for which $f(n) | f(m) - n$ if and only if $n | m$ for all natural numbers $m$ and $n$.
2016 Junior Regional Olympiad - FBH, 3
In trapezoid $ABCD$ holds $AD \mid \mid BC$, $\angle ABC = 30^{\circ}$, $\angle BCD = 60^{\circ}$ and $BC=7$. Let $E$, $M$, $F$ and $N$ be midpoints of sides $AB$, $BC$, $CD$ and $DA$, respectively. If $MN=3$, find $EF$
2014 China Western Mathematical Olympiad, 7
In the plane, Point $ O$ is the center of the equilateral triangle $ABC$ , Points $P,Q$ such that $\overrightarrow{OQ}=2\overrightarrow{PO}$.
Prove that\[|PA|+|PB|+|PC|\le |QA|+|QB|+|QC|.\]
STEMS 2021 Math Cat B, Q4
Let $n$ be a fixed positive integer.
- Show that there exist real polynomials $p_1, p_2, p_3, \cdots, p_k \in \mathbb{R}[x_1, \cdots, x_n]$ such that
\[(x_1 + x_2 + \cdots + x_n)^2 + p_1(x_1, \cdots, x_n)^2 + p_2(x_1, \cdots, x_n)^2 + \cdots + p_k(x_1, \cdots, x_n)^2 = n(x_1^2 + x_2^2 + \cdots + x_n^2)\]
- Find the least natural number $k$, depending on $n$, such that the above polynomials $p_1, p_2, \cdots, p_k$ exist.