Found problems: 85335
2014 Chile National Olympiad, 1
Let $a, b,c$ real numbers that are greater than $ 0$ and less than $1$. Show that there is at least one of these three values $ab(1-c)^2$, $bc(1-a)^2$ , $ca(1- b)^2$ which is less than or equal to $\frac{1}{16}$ .
2000 Moldova National Olympiad, Problem 1
What is the greatest possible number of Fridays by the date $13$ in a year?
1989 IMO Longlists, 4
Ali Barber, the carpet merchant, has a rectangular piece of carpet whose dimensions are unknown. Unfortunately, his tape measure is broken and he has no other measuring instruments. However, he finds that if he lays it flat on the floor of either of his storerooms, then each corner of the carpet touches a different wall of that room. He knows that the sides of the carpet are integral numbers of feet and that his two storerooms have the same (unknown) length, but widths of 38 feet and 50 feet respectively. What are the carpet dimensions?
2022 Thailand Mathematical Olympiad, 5
Determine all functions $f:\mathbb{R}\times\mathbb{R}\to\mathbb{R}$ that satisfies the equation
$$f\left(\frac{x+y+z}{3},\frac{a+b+c}{3}\right)=f(x,a)f(y,b)f(z,c)$$
for any real numbers $x,y,z,a,b,c$ such that $az+bx+cy\neq ay+bz+cx$.
2000 France Team Selection Test, 2
A function from the positive integers to the positive integers satisfies these properties
1. $f(ab)=f(a)f(b)$ for any two coprime positive integers $a,b$.
2. $f(p+q)=f(p)+f(q)$ for any two primes $p,q$.
Prove that $f(2)=2, f(3)=3, f(1999)=1999$.
MathLinks Contest 5th, 6.1
Let $ABC$ be a triangle and let $C$ be a circle that intersects the sides $BC, CA$ and $AB$ in the points $A_1, A_2, B_1, B_2$ and $C_1, C_2$ respectively. Prove that if $AA_1, BB_1$ and $CC_1$ are concurrent lines then $AA_2, BB_2$ and $CC_2$ are also concurrent lines.
2020 Denmark MO - Mohr Contest, 1
The figure shows $9$ circles connected by $12$ lines. Georg must colour each circle either red or blue. He gets one point for each line connecting circles with different colours. How many points can he at most achieve?
[img]https://cdn.artofproblemsolving.com/attachments/3/9/983d3c5755547246899891db141fe2383f3dc1.png[/img]
2007 Nicolae Coculescu, 4
Prove that there exists a nonconstant function $ f:\mathbb{R}^2\longrightarrow\mathbb{R} $ verifying the following system of relations:
$$ \left\{ \begin{matrix} f(x,x+y)=f(x,y) ,& \quad \forall x,y\in\mathbb{R} \\f(x,y+z)=f(x,y) +f(x,z) ,& \quad \forall x,y\in\mathbb{R} \end{matrix} \right. $$
2024 Putnam, A4
Find all primes $p>5$ for which there exists an integer $a$ and an integer $r$ satisfying $1\leq r\leq p-1$ with the following property: the sequence $1,\,a,\,a^2,\,\ldots,\,a^{p-5}$ can be rearranged to form a sequence $b_0,\,b_1,\,b_2,\,\ldots,\,b_{p-5}$ such that $b_n-b_{n-1}-r$ is divisible by $p$ for $1\leq n\leq p-5$.
1973 Dutch Mathematical Olympiad, 1
Given is a triangle $ABC$, $\angle C = 60^o$, $R$ the midpoint of side $AB$. There exist a point $P$ on the line $BC$ and a point $Q$ on the line $AC$ such that the perimeter of the triangle $PQR$ is minimal.
a) Prove that and also indicate how the points $P$ and $Q$ can be constructed.
b) If $AB = c$, $AC = b$, $BC = a$, then prove that the perimeter of the triangle $PQR$ equals $\frac12\sqrt{3c^2+6ab}$ .
1998 Switzerland Team Selection Test, 8
Let $\vartriangle ABC$ be an equilateral triangle and let $P$ be a point in its interior. Let the lines $AP,BP,CP$ meet the sides $BC,CA,AB$ in the points $X,Y,Z$ respectively. Prove that $XY \cdot YZ\cdot ZX \ge XB\cdot YC\cdot ZA$.
2007 Irish Math Olympiad, 1
Let $ r,s,$ and $ t$ be the roots of the cubic polynomial: $ p(x)\equal{}x^3\minus{}2007x\plus{}2002.$
Determine the value of: $ \frac{r\minus{}1}{r\plus{}1}\plus{}\frac{s\minus{}1}{s\plus{}1}\plus{}\frac{t\minus{}1}{t\plus{}1}$.
2007 Switzerland - Final Round, 10
The plane is divided into equilateral triangles of side length $1$. Consider a equilateral triangle of side length $n$ whose sides lie on the grid lines. On every grid point on the edge and inside of this triangle lies a stone. In a move, a unit triangle is selected, which has exactly $2$ corners with is covered with a stone. The two stones are removed, and the third corner is turned a new stone was laid. For which $n$ is it possible that after finitely many moves only one stone left?
1970 Putnam, A1
Show that the power series for the function
$$e^{ax} \cos bx,$$
where $a,b >0$, has either no zero coefficients or infinitely many zero coefficients.
2013 China Northern MO, 1
Find the largest positive integer $n$ ($n \ge 3$), so that there is a convex $n$-gon, the tangent of each interior angle is an integer.
2008 Princeton University Math Competition, B2
Let $P$ be a convex polygon, and let $n \ge 3$ be a positive integer. On each side of $P$, erect a regular $n$-gon that shares that side of $P$, and is outside $P$. If none of the interiors of these regular n-gons overlap, we call P $n$-[i]good[/i].
(a) Find the largest value of $n$ such that every convex polygon is $n$-[i]good[/i].
(b) Find the smallest value of $n$ such that no convex polygon is $n$-[i]good[/i].
2003 Poland - Second Round, 6
Each pair $(x, y)$ of nonnegative integers is assigned number $f(x, y)$ according the conditions:
$f(0, 0) = 0$;
$f(2x, 2y) = f(2x + 1, 2y + 1) = f(x, y)$,
$f(2x + 1, 2y) = f(2x, 2y + 1) = f(x ,y) + 1$ for $x, y \ge 0$.
Let $n$ be a fixed nonnegative integer and let $a$, $b$ be nonnegative integers such that $f(a, b) = n$. Decide how many numbers satisfy the equation $f(a, x) + f(b, x) = n$.
1967 IMO Longlists, 12
Given a segment $AB$ of the length 1, define the set $M$ of points in the
following way: it contains two points $A,B,$ and also all points obtained from $A,B$ by iterating the following rule: With every pair of points $X,Y$ the set $M$ contains also the point $Z$ of the segment $XY$ for which $YZ = 3XZ.$
Swiss NMO - geometry, 2017.5
Let $ABC$ be a triangle with $AC> AB$. Let $P$ be the intersection of $BC$ and the tangent through $A$ around the triangle $ABC$. Let $Q$ be the point on the straight line $AC$, so that $AQ = AB$ and $A$ is between $C$ and $Q$. Let $X$ and $Y$ be the center of $BQ$ and $AP$. Let $R$ be the point on $AP$ so that $AR = BP$ and $R$ is between $A$ and $P$. Show that $BR = 2XY$.
1996 Hungary-Israel Binational, 2
$ n>2$ is an integer such that $ n^2$ can be represented as a difference of cubes of 2 consecutive positive integers. Prove that $ n$ is a sum of 2 squares of positive integers, and that such $ n$ does exist.
2017 Princeton University Math Competition, A5/B7
Let $f_{0}(x)=x$, and for each $n\geq 0$, let $f_{n+1}(x)=f_{n}(x^{2}(3-2x))$. Find the smallest real number that is at least as large as
\[ \sum_{n=0}^{2017} f_{n}(a) + \sum_{n=0}^{2017} f_{n}(1-a)\]
for all $a \in [0,1]$.
2003 AMC 10, 25
How many distinct four-digit numbers are divisible by $ 3$ and have $ 23$ as their last two digits?
$ \textbf{(A)}\ 27 \qquad
\textbf{(B)}\ 30 \qquad
\textbf{(C)}\ 33 \qquad
\textbf{(D)}\ 81 \qquad
\textbf{(E)}\ 90$
2010 Federal Competition For Advanced Students, Part 1, 1
Let $f(n)=\sum_{k=0}^{2010}n^k$. Show that for any integer $m$ satisfying $2\leqslant m\leqslant 2010$, there exists no natural number $n$ such that $f(n)$ is divisible by $m$.
[i](41st Austrian Mathematical Olympiad, National Competition, part 1, Problem 1)[/i]
2016 CMIMC, 10
Let $\mathcal{P}$ be the unique parabola in the $xy$-plane which is tangent to the $x$-axis at $(5,0)$ and to the $y$-axis at $(0,12)$. We say a line $\ell$ is $\mathcal{P}$-friendly if the $x$-axis, $y$-axis, and $\mathcal{P}$ divide $\ell$ into three segments, each of which has equal length. If the sum of the slopes of all $\mathcal{P}$-friendly lines can be written in the form $-\tfrac mn$ for $m$ and $n$ positive relatively prime integers, find $m+n$.
2010 Turkey Junior National Olympiad, 3
In an exam every question is solved by exactly four students, every pair of questions is solved by exactly one student, and none of the students solved all of the questions. Find the maximum possible number of questions in this exam.