Found problems: 85335
2007 Princeton University Math Competition, 8
Find the minimum number $n$ such that for any coloring of the integers from $1$ to $n$ into two colors, one can find monochromatic $a$, $b$, $c$, and $d$ (not necessarily distinct) such that $a+b+c=d$.
1999 Moldova Team Selection Test, 15
Distinct integers $x,y,z{}$ verify the relation $(x-y)(y-z)(z-x)=x+y+z$. Find the smallest possibile value of $|x+y+z|$.
1986 Flanders Math Olympiad, 1
A circle with radius $R$ is divided into twelve equal parts. The twelve dividing points are connected with the centre of the circle, producing twelve rays. Starting from one of the dividing points a segment is drawn perpendicular to the next ray in the clockwise sense; from the foot of this perpendicular another perpendicular segment is drawn to the next ray, and the process is continued [i]ad infinitum[/i]. What is the limit of the sum of these segments (in terms of $R$)?
[img]https://cdn.artofproblemsolving.com/attachments/2/6/83705b54ecc817b7d913468cd8467d7b8d9f8f.png[/img]
1998 Croatia National Olympiad, Problem 3
Let $AA_1,BB_1,CC_1$ be the altitudes of a triangle $ABC$. If $\overrightarrow{AA_1}+\overrightarrow{BB_1}+\overrightarrow{CC_1}=0$ prove that the triangle $ABC$ is equilateral.
2014 Costa Rica - Final Round, 4
Consider the isosceles triangle $ABC$ inscribed in the semicircle of radius $ r$. If the $\vartriangle BCD$ and $\vartriangle CAE$ are equilateral, determine the altitude of $\vartriangle DEC$ on the side $DE$ in terms of $ r$.
[img]https://cdn.artofproblemsolving.com/attachments/6/3/772ff9a1fd91e9fa7a7e45ef788eec7a1ba48e.png[/img]
1992 AMC 8, 18
On a trip, a car traveled $80$ miles in an hour and a half, then was stopped in traffic for $30$ minutes, then traveled $100$ miles during the next $2$ hours. What was the car's average speed in miles per hour for the $4$-hour trip?
$\text{(A)}\ 45 \qquad \text{(B)}\ 50 \qquad \text{(C)}\ 60 \qquad \text{(D)}\ 75 \qquad \text{(E)}\ 90$
III Soros Olympiad 1996 - 97 (Russia), 11.5
Prove that this triangle cut out of paper can be folded so that the surface of a regular unit tetradragon (i.e., a triangular pyramid, all edges of which are equal to $1$) is obtained if:
a) this triangle is isosceles, the lateral sides are equal to $2$ , the angle between them is $120^o$,
b) two sides of this triangle are equal to $2$ and $2\sqrt3$, the angle between them is $150^o$.
2019 Brazil Team Selection Test, 2
We say that a distribution of students lined upen in collumns is $\textit{bacana}$ when there are no two friends in the same column. We know that all contestants in a math olympiad can be arranged in a $\textit{bacana}$ configuration with $n$ columns, and that this is impossible with $n-1$ columns. Show that we can choose competitors $M_1, M_2, \cdots, M_n$ in such a way that $M_i$ is on the $i$-th column, for each $i = 1, 2, 3, \ldots, n$ and $M_i$ is a friend of $M_{i+1}$ for each $i = 1, 2, \ldots, n - 1$.
2014 HMNT, 8
Consider the parabola consisting of the points $(x, y)$ in the real plane satisfying
$$(y + x) = (y - x)^2 + 3(y - x) + 3.$$
Find the minimum possible value of $y$.
2021 Flanders Math Olympiad, 2
Catherine lowers five matching wooden discs over bars placed on the vertices of a regular pentagon. Then she leaves five smaller congruent checkers these rods drop. Then she stretches a ribbon around the large discs and a second ribbon around the small discs. The first ribbon has a length of $56$ centimeters and the second one of $50$ centimeters. Catherine looks at her construction from above and sees an area demarcated by the two ribbons. What is the area of that area?
[img]https://cdn.artofproblemsolving.com/attachments/1/0/68e80530742f1f0775aff5a265e0c9928fa66c.png[/img]
2018 Oral Moscow Geometry Olympiad, 4
On the side $AB$ of the triangle $ABC$, point $M$ is selected. In triangle $ACM$ point $I_1$ is the center of the inscribed circle, $J_1$ is the center of excircle wrt side $CM$. In the triangle $BCM$ point $I_2$ is the center of the inscribed circle, $J_2$ is the center of excircle wrt side $CM$. Prove that the line passing through the midpoints of the segments $I_1I_2$ and $J_1J_2$ is perpendicular to $AB$.
2013 IMO Shortlist, A6
Let $m \neq 0 $ be an integer. Find all polynomials $P(x) $ with real coefficients such that
\[ (x^3 - mx^2 +1 ) P(x+1) + (x^3+mx^2+1) P(x-1) =2(x^3 - mx +1 ) P(x) \]
for all real number $x$.
2012 Gheorghe Vranceanu, 2
$ G $ is the centroid of $ ABC. $ The incircle of $ ABC $ touches $ BC,CA,AB $ at $ D,E,F, $ respectively. Show that $ ABC $ is equilateral if and only if $ BC\cdot\overrightarrow{GD}+ AC\cdot\overrightarrow{GE} +AB\cdot\overrightarrow{GF} =0. $
[i]Marian Ursărescu[/i]
2007 Harvard-MIT Mathematics Tournament, 10
$ABCD$ is a convex quadrilateral such that $AB=2$, $BC=3$, $CD=7$, and $AD=6$. It also has an incircle. Given that $\angle ABC$ is right, determine the radius of this incircle.
1998 IberoAmerican Olympiad For University Students, 3
The positive divisors of a positive integer $n$ are written in increasing order starting with 1.
\[1=d_1<d_2<d_3<\cdots<n\]
Find $n$ if it is known that:
[b]i[/b]. $\, n=d_{13}+d_{14}+d_{15}$
[b]ii[/b]. $\,(d_5+1)^3=d_{15}+1$
2003 Kazakhstan National Olympiad, 2
For positive real numbers $ x, y, z $, prove the inequality: $$ \displaylines {\frac {x ^ 3} {x + y} + \frac {y ^ 3} {y + z} + \frac {z ^ 3} {z + x} \geq \frac {xy + yz + zx} {2}.} $$
2022 Canadian Junior Mathematical Olympiad, 4
I think we are allowed to discuss since its after 24 hours
How do you do this
Prove that $d(1)+d(3)+..+d(2n-1)\leq d(2)+d(4)+...d(2n)$ which $d(x)$ is the divisor function
2019 Brazil Undergrad MO, 6
In a hidden friend, suppose no one takes oneself. We say that the hidden friend has "marmalade" if
there are two people $A$ and $ B$ such that A took $B$ and $ B $ took $A$. For each positive integer n, let $f (n)$ be the number of hidden friends with n people where there is no “marmalade”, i.e. $f (n)$ is equal to the number of permutations $\sigma$ of {$1, 2,. . . , n$} such that:
*$\sigma (i) \neq i$ for all $i=1,2,...,n$
* there are no $ 1 \leq i <j \leq n $ such that $ \sigma (i) = j$ and $\sigma (j) = i. $
Determine the limit
$\lim_{n \to + \infty} \frac{f(n)}{n!}$
1936 Eotvos Mathematical Competition, 3
Let $a$ be any positive integer. Prove that there exists a unique pair of positive integers $x$ and $y$ such that
$$x +\frac12 (x + y - 1)(x + y- 2) = a.$$
2001 National Olympiad First Round, 7
How many ordered triples of positive integers $(a,b,c)$ are there such that $(2a+b)(2b+a)=2^c$?
$
\textbf{(A)}\ 0
\qquad\textbf{(B)}\ 1
\qquad\textbf{(C)}\ 2
\qquad\textbf{(D)}\ 3
\qquad\textbf{(E)}\ \text{None of the preceding}
$
1938 Eotvos Mathematical Competition, 1
Prove that an integer $n$ can be expressed as the sum of two squares if and only if $2n$ can be expressed as the sum of two squares.
2017 Harvard-MIT Mathematics Tournament, 9
Let $n$ be an odd positive integer greater than $2$, and consider a regular $n$-gon $\mathcal{G}$ in the plane centered at the origin. Let a [i]subpolygon[/i] $\mathcal{G}'$ be a polygon with at least $3$ vertices whose vertex set is a subset of that of $\mathcal{G}$. Say $\mathcal{G}'$ is [i]well-centered[/i] if its centroid is the origin. Also, say $\mathcal{G}'$ is [i]decomposable[/i] if its vertex set can be written as the disjoint union of regular polygons with at least $3$ vertices. Show that all well-centered subpolygons are decomposable if and only if $n$ has at most two distinct prime divisors.
2012 Tournament of Towns, 2
The number $4$ has an odd number of odd positive divisors, namely $1$, and an even number of even positive divisors, namely $2$ and $4$. Is there a number with an odd number of even positive divisors and an even number of odd positive divisors?
2010 CHMMC Winter, 4
Compute the number of integer solutions $(x, y)$ to $xy - 18x - 35y = 1890$.
2020 Tournament Of Towns, 4
We say that a nonconstant polynomial $p(x)$ with real coefficients is split into two squares if it is represented as $a(x) +b(x)$ where $a(x)$ and $b(x)$ are squares of polynomials with real coefficients. Is there such a polynomial $p(x)$ that it may be split into two squares:
a) in exactly one way;
b) in exactly two ways?
Note: two splittings that differ only in the order of summands are considered to be the same.
Sergey Markelov