Found problems: 85335
MBMT Team Rounds, 2020.2
Daniel, Clarence, and Matthew split a \$20.20 dinner bill so that Daniel pays half of what Clarence pays. If Daniel pays \$6.06, what is the ratio of Clarence's pay to Matthew's pay?
[i]Proposed by Henry Ren[/i]
Dumbest FE I ever created, 1.
Determine all functions $f\colon\mathbb{Z}_{>0}\to\mathbb{Z}_{>0}$ such that, for all positive integers $m$ and $n$,
$$ m^{\phi(n)}+n^{\phi(m)} \mid f(m)^n + f(n)^m$$
1972 Spain Mathematical Olympiad, 1
Let $K$ be a ring with unit and $M$ the set of $2 \times 2$ matrices constituted with elements of $K$. An addition and a multiplication are defined in $M$ in the usual way between arrays. It is requested to:
a) Check that $M$ is a ring with unit and not commutative with respect to the laws of defined composition.
b) Check that if $K$ is a commutative field, the elements of$ M$ that have inverse they are characterized by the condition $ad - bc \ne 0$.
c) Prove that the subset of $M$ formed by the elements that have inverse is a multiplicative group.
2011 Purple Comet Problems, 12
Find the area of the region in the coordinate plane satisfying the three conditions
$\star$ $x \le 2y$
$\star$ $y \le 2x$
$\star$ $x + y \le 60.$
2014 International Zhautykov Olympiad, 2
Let $U=\{1, 2,\ldots, 2014\}$. For positive integers $a$, $b$, $c$ we denote by $f(a, b, c)$ the number of ordered 6-tuples of sets $(X_1,X_2,X_3,Y_1,Y_2,Y_3)$ satisfying the following conditions:
(i) $Y_1 \subseteq X_1 \subseteq U$ and $|X_1|=a$;
(ii) $Y_2 \subseteq X_2 \subseteq U\setminus Y_1$ and $|X_2|=b$;
(iii) $Y_3 \subseteq X_3 \subseteq U\setminus (Y_1\cup Y_2)$ and $|X_3|=c$.
Prove that $f(a,b,c)$ does not change when $a$, $b$, $c$ are rearranged.
[i]Proposed by Damir A. Yeliussizov, Kazakhstan[/i]
2004 Canada National Olympiad, 1
Find all ordered triples $ (x,y,z)$ of real numbers which satisfy the following system of equations:
\[ \left\{\begin{array}{rcl} xy & \equal{} & z \minus{} x \minus{} y \\
xz & \equal{} & y \minus{} x \minus{} z \\
yz & \equal{} & x \minus{} y \minus{} z \end{array} \right.
\]
2010 Romania Team Selection Test, 3
Given a positive integer $a$, prove that $\sigma(am) < \sigma(am + 1)$ for infinitely many positive integers $m$. (Here $\sigma(n)$ is the sum of all positive divisors of the positive integer number $n$.)
[i]Vlad Matei[/i]
2021 JBMO Shortlist, G3
Let $ABC$ be an acute triangle with circumcircle $\omega$ and circumcenter $O$. The perpendicular from $A$ to $BC$ intersects $BC$ and $\omega$ at $D$ and $E$, respectively. Let $F$ be a point on the segment $AE$, such that $2 \cdot FD = AE$. Let $l$ be the perpendicular to $OF$ through $F$. Prove that $l$, the tangent to $\omega$ at $E$, and the line $BC$ are concurrent.
Proposed by [i] Stefan Lozanovski, Macedonia[/i]
2016 Germany National Olympiad (4th Round), 3
Let $I_a$ be the $A$-excenter of a scalene triangle $ABC$. And let $M$ be the point symmetric to $I_a$ about line $BC$.
Prove that line $AM$ is parallel to the line through the circumcenter and the orthocenter of triangle $I_aCB$.
2015 Switzerland Team Selection Test, 6
Find all polynomial function $P$ of real coefficients such that for all $x \in \mathbb{R}$ $$P(x)P(x+1)=P(x^2+2)$$
2015 Math Prize for Girls Problems, 6
In baseball, a player's [i]batting average[/i] is the number of hits divided by the number of at bats, rounded to three decimal places. Danielle's batting average is $.399$. What is the fewest number of at bats that Danielle could have?
2017 Czech-Polish-Slovak Junior Match, 5
In each square of the $100\times 100$ square table, type $1, 2$, or $3$. Consider all subtables $m \times n$, where $m = 2$ and $n = 2$. A subtable will be called [i]balanced [/i] if it has in its corner boxes of four identical numbers boxes . For as large a number $k$ prove, that we can always find $k$ balanced subtables, of which no two overlap, i.e. do not have a common box.
1966 IMO Shortlist, 30
Let $n$ be a positive integer, prove that :
[b](a)[/b] $\log_{10}(n + 1) > \frac{3}{10n} +\log_{10}n ;$
[b](b)[/b] $ \log n! > \frac{3n}{10}\left( \frac 12+\frac 13 +\cdots +\frac 1n -1\right).$
2019 Macedonia National Olympiad, 4
Determine all functions $f: \mathbb {N} \to \mathbb {N}$ such that
$n!\hspace{1mm} +\hspace{1mm} f(m)!\hspace{1mm} |\hspace{1mm} f(n)!\hspace{1mm} +\hspace{1mm} f(m!)$,
for all $m$, $n$ $\in$ $\mathbb{N}$.
1918 Eotvos Mathematical Competition, 3
If $a, b,c,p,q, r $are real numbers such that, for every real number $x,$
$$ax^2 - 2bx + c \ge 0 \ \ and \ \ px^2 + 2qx + r \ge 0;$$
prove that then
$$apx^2 + bqx + cr \ge 0$$
for all real $x$.
1964 Poland - Second Round, 5
Given is a trihedral angle with edges $ SA $, $ SB $, $ SC $, all plane angles of which are acute, and the dihedral angle at edge $ SA $ is right. Prove that the section of this triangle with any plane perpendicular to any edge, at a point different from the vertex $ S $, is a right triangle.
2021 China Team Selection Test, 5
Find the smallest real $\alpha$, such that for any convex polygon $P$ with area $1$, there exist a point $M$ in the plane, such that the area of convex hull of $P\cup Q$ is at most $\alpha$, where $Q$ denotes the image of $P$ under central symmetry with respect to $M$.
2012 India IMO Training Camp, 2
Find the least positive integer that cannot be represented as $\frac{2^a-2^b}{2^c-2^d}$ for some positive integers $a, b, c, d$.
2016 BMT Spring, 8
Simplify $\frac{1}{\sqrt[3]{81} + \sqrt[3]{72} + \sqrt[3]{64}}$
2013 Serbia National Math Olympiad, 6
Find the largest constant $K\in \mathbb{R}$ with the following property:
if $a_1,a_2,a_3,a_4>0$ are numbers satisfying $a_i^2 + a_j^2 + a_k^2 \geq 2
(a_ia_j + a_ja_k + a_ka_i)$, for every $1\leq i<j<k\leq 4$, then \[a_1^2+a_2^2+a_3^2+a_4^2 \geq K
(a_1a_2+a_1a_3+a_1a_4+a_2a_3+a_2a_4+a_3a_4).\]
1952 Miklós Schweitzer, 9
Let $ C$ denote the set of functions $ f(x)$, integrable (according to either Riemann or Lebesgue) on $ (a,b)$, with $ 0\le f(x)\le1$. An element $ \phi(x)\in C$ is said to be an "extreme point" of $ C$ if it can not be represented as the arithmetical mean of two different elements of $ C$. Find the extreme points of $ C$ and the functions $ f(x)\in C$ which can be obtained as "weak limits" of extreme points $ \phi_n(x)$ of $ C$.
(The latter means that
$ \lim_{n\to \infty}\int_a^b \phi_n(x)h(x)\,dx\equal{}\int_a^bf(x)h(x)\,dx$
holds for every integrable function $ h(x)$.)
2001 Tuymaada Olympiad, 3
Do there exist quadratic trinomials $P, \ \ Q, \ \ R$ such that for every integers $x$ and $y$ an integer $z$ exists satisfying $P(x)+Q(y)=R(z)?$
[i]Proposed by A. Golovanov[/i]
2010 Flanders Math Olympiad, 4
In snack bar Pita Goras, the owner checks his accounts. He writes on every line either a positive amount in case of an income or a negative amount in case of an expense. He says to his accountant, “If I change the amounts of random $5$ adding consecutive lines, I always get a strictly positive result.” "Indeed," the accountant answers him, “but if you put the sums of $7$ consecutive lines add up, you always get a strictly negative result.” How many lines are there maximum
on his sheet?
2009 AMC 12/AHSME, 14
A triangle has vertices $ (0,0)$, $ (1,1)$, and $ (6m,0)$, and the line $ y \equal{} mx$ divides the triangle into two triangles of equal area. What is the sum of all possible values of $ m$?
$ \textbf{(A)}\minus{} \!\frac {1}{3} \qquad \textbf{(B)} \minus{} \!\frac {1}{6} \qquad \textbf{(C)}\ \frac {1}{6} \qquad \textbf{(D)}\ \frac {1}{3} \qquad \textbf{(E)}\ \frac {1}{2}$
2005 ITAMO, 1
Determine all $n \geq 3$ for which there are $n$ positive integers $a_1, \cdots , a_n$ any two of which have a common divisor greater than $1$, but any three of which are coprime. Assuming that, moreover, the numbers $a_i$ are less than $5000$, find the greatest possible $n$.