Found problems: 85335
1996 Romania National Olympiad, 4
Let $f:[0,1) \to \mathbb{R}$ be a monotonic function. Prove that the limits [center]$\lim_{x \nearrow 1} \int_0^x f(t) \mathrm{d}t$ and $\lim_{n \to \infty} \frac{1}{n} \left[ f(0) + f \left(\frac{1}{n}\right) + \ldots + f \left( \frac{n-1}{n} \right) \right]$[/center] exist and are equal.
2013 Stanford Mathematics Tournament, 8
Rational Man and Irrational Man both buy new cars, and they decide to drive around two racetracks from time $t=0$ to time $t=\infty$. Rational Man drives along the path parametrized by \begin{align*}x&=\cos(t)\\y&=\sin(t)\end{align*} and Irrational Man drives along the path parametrized by \begin{align*}x&=1+4\cos\frac{t}{\sqrt{2}}\\ y&=2\sin\frac{t}{\sqrt{2}}.\end{align*} Find the largest real number $d$ such that at any time $t$, the distance between Rational Man and Irrational Man is not less than $d$.
1975 Putnam, A3
Let $0<\alpha<\beta <\gamma\in \mathbb{R}$. Let $K=\{(x,y,z)\in \mathbb{R}^{3}\;|\; x,y,z\geq 0\; \text{and}\; x^{\beta}+y^{\beta}+z^{\beta}=1\}$. Define $f:K\rightarrow \mathbb{R},\; (x,y,z)\mapsto x^{\alpha}+y^{\beta}+z^{\gamma}$. At what points of $K$ does $f$ assume its minimal and maximal values?
2005 Iran MO (3rd Round), 2
$n$ vectors are on the plane. We can move each vector forward and backeard on the line that the vector is on it. If there are 2 vectors that their endpoints concide we can omit them and replace them with their sum (If their sum is nonzero). Suppose with these operations with 2 different method we reach to a vector. Prove that these vectors are on a common line
2002 Tournament Of Towns, 5
Let $AA_1,BB_1,CC_1$ be the altitudes of acute $\Delta ABC$. Let $O_a,O_b,O_c$ be the incentres of $\Delta AB_1C_1,\Delta BC_1A_1,\Delta CA_1B_1$ respectively. Also let $T_a,T_b,T_c$ be the points of tangency of the incircle of $\Delta ABC$ with $BC,CA,AB$ respectively. Prove that $T_aO_cT_bO_aT_cO_b$ is an equilateral hexagon.
2018 Iranian Geometry Olympiad, 4
We have a polyhedron all faces of which are triangle. Let $P$ be an arbitrary point on one of the edges of this polyhedron such that $P$ is not the midpoint or endpoint of this edge. Assume that $P_0 = P$. In each step, connect $P_i$ to the centroid of one of the faces containing it. This line meets the perimeter of this face again at point $P_{i+1}$. Continue this process with $P_{i+1}$ and the other face containing $P_{i+1}$. Prove that by continuing this process, we cannot pass through all the faces. (The centroid of a triangle is the point of intersection of its medians.)
Proposed by Mahdi Etesamifard - Morteza Saghafian
2017 Online Math Open Problems, 14
Let $S$ be the set of all points $(x_1, x_2, x_3, \dots, x_{2017})$ in $\mathbb{R}^{2017}$ satisfying $|x_i|+|x_j|\leq 1$ for any $1\leq i< j\leq 2017$. The volume of $S$ can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $100m+n$.
[i]Proposed by Yannick Yao[/i]
2012 Middle European Mathematical Olympiad, 6
Let $ ABCD $ be a convex quadrilateral with no pair of parallel sides, such that $ \angle ABC = \angle CDA $. Assume that the intersections of the pairs of neighbouring angle bisectors of $ ABCD $ form a convex quadrilateral $ EFGH $. Let $ K $ be the intersection of the diagonals of $ EFGH$. Prove that the lines $ AB $ and $ CD $ intersect on the circumcircle of the triangle $ BKD $.
2020 USMCA, 8
Let $n, m$ be positive integers, and let $\alpha$ be an irrational number satisfying $1 < \alpha < n$. Define the set
\[ X = \{a + b\alpha : 0 \le a \le n \text{ and } 0 \le b \le m \}. \]
Let $x_0\le x_1\le \cdots \le x_{(n+1)(m+1)-1}$ be the elements of $X$. Show that for all $i+j\le (n+1)(m+1)-1$, we have that $x_{i+j} \le x_i + x_j$ .
1993 Bundeswettbewerb Mathematik, 1
In a regular nonagon, each vertex is colored either red or green. Three corners of the nonagon determine a triangle. Such a triangle is called [i]red [/i] or [i]green [/i] if all its vertices are red or green if all are green. Prove that for each such coloring of the nonagon there are at least two different ones , that are congruent triangles of the same color.
1994 USAMO, 1
Let $\, k_1 < k_2 < k_3 < \cdots \,$ be positive integers, no two consecutive, and let $\, s_m = k_1 + k_2 + \cdots + k_m \,$ for $\, m = 1,2,3, \ldots \; \;$. Prove that, for each positive integer $\, n, \,$ the interval $\, [s_n, s_{n+1}) \,$ contains at least one perfect square.
2002 Croatia National Olympiad, Problem 1
For each $x$ with $|x|<1$, compute the sum of the series
$$1+4x+9x^2+\ldots+n^2x^{n-1}+\ldots.$$
2014 District Olympiad, 2
[list=a]
[*]Let $f\colon\mathbb{R}\rightarrow\mathbb{R}$ be a function such that
$g\colon\mathbb{R}\rightarrow\mathbb{R}$, $g(x)=f(x)+f(2x)$, and
$h\colon\mathbb{R}\rightarrow\mathbb{R}$, $h(x)=f(x)+f(4x)$, are continuous
functions. Prove that $f$ is also continuous.
[*]Give an example of a discontinuous function $f\colon\mathbb{R}
\rightarrow\mathbb{R}$, with the following property: there exists an interval
$I\subset\mathbb{R}$, such that, for any $a$ in $I$, the function $g_{a}
\colon\mathbb{R}\rightarrow\mathbb{R}$, $g_{a}(x)=f(x)+f(ax)$, is continuous.[/list]
1965 IMO, 6
In a plane a set of $n\geq 3$ points is given. Each pair of points is connected by a segment. Let $d$ be the length of the longest of these segments. We define a diameter of the set to be any connecting segment of length $d$. Prove that the number of diameters of the given set is at most $n$.
2015 Azerbaijan National Olympiad, 3
Find all polynomials $P(x)$ with real coefficents such that \[P(P(x))=(x^2+x+1)\cdot P(x)\] where $x \in \mathbb{R}$
2022 USAMTS Problems, 2
Given a sphere, a great circle of the sphere is a circle on the sphere whose diameter is also a diameter of the sphere. For a given positive integer $n,$ the surface of a sphere is divided into several regions by $n$ great circles, and each region is colored black or white. We say that a coloring is good if any two adjacent regions (that share an arc as boundary, not just a finite number of points) have different colors. Find, with proof, all positive integers $n$ such that in every good coloring with $n$ great circles, the sum of the areas of the black regions is equal to the sum of the areas of the white regions.
2012 VJIMC, Problem 1
Let $f:[0,1]\to[0,1]$ be a differentiable function such that $|f'(x)|\ne1$ for all $x\in[0,1]$. Prove that there exist unique $\alpha,\beta\in[0,1]$ such that $f(\alpha)=\alpha$ and $f(\beta)=1-\beta$.
2020/2021 Tournament of Towns, P3
For which $n{}$ is it possible that a product of $n{}$ consecutive positive integers is equal to a sum of $n{}$ consecutive (not necessarily the same) positive integers?
[i]Boris Frenkin[/i]
2024 Princeton University Math Competition, A5 / B7
Real numbers $a,b,c$ satisfy $\tfrac{1}{ab} = b+2c, \tfrac{1}{bc} = 2c+3a, \tfrac{1}{ca}=3a+b.$ Then, $(a+b+c)^3$ can be written as $\tfrac{m}{n}$ for relatively prime positive integers $m$ and $n.$ Find $m+n.$
2011 Purple Comet Problems, 13
The diagram shows two equilateral triangles with side length $4$ mounted on two adjacent sides of a square also with side length $4$. The distance between the two vertices marked $A$ and $B$ can be written as $\sqrt{m}+\sqrt{n}$ for two positive integers $m$ and $n$. Find $m + n$.
[asy]
size(120);
defaultpen(linewidth(0.7)+fontsize(11pt));
draw(unitsquare);
draw((0,1)--(1/2,1+sqrt(3)/2)--(1,1)--(1+sqrt(3)/2,1/2)--(1,0));
label("$A$",(1/2,1+sqrt(3)/2),N);
label("$B$",(1+sqrt(3)/2,1/2),E);
[/asy]
2014 Korea National Olympiad, 2
How many one-to-one functions $f : \{1, 2, \cdots, 9\} \rightarrow \{1, 2, \cdots, 9\}$ satisfy (i) and (ii)?
(i) $f(1)>f(2)$, $f(9)<9$.
(ii) For each $i=3, 4, \cdots, 8$, if $f(1), \cdots, f(i-1)$ are smaller than $f(i)$, then $f(i+1)$ is also smaller than $f(i)$.
1973 Bulgaria National Olympiad, Problem 6
In the tetrahedron $ABCD$, $E$ and $F$ are the midpoints of $BC$ and $AD$, $G$ is the midpoint of the segment $EF$. Construct a plane through $G$ intersecting the segments $AB$, $AC$, $AD$ in the points $M,N,P$ respectively in such a way that the sum of the volumes of the tetrahedrons $BMNP$, $CMNP$ and $DMNP$ to be minimal.
[i]H. Lesov[/i]
2016 Chile National Olympiad, 6
Let $P_1$ and $P_2$ be two non-parallel planes in space, and $A$ a point that does not It is in none of them. For each point $X$, let $X_1$ denote its reflection with respect to $P_1$, and $X_2$ its reflection with respect to $P_2$. Determine the locus of points $X$ for the which $X_1, X_2$ and $A$ are collinear.
2006 Junior Balkan Team Selection Tests - Moldova, 2
Prove that there infinitely many numbers of the form $18^{m}+45^{m}+50^{m}+125^{m}$, divisible by 2006. $m\in N$
2023 CMIMC Team, 6
A positive integer $n$ is said to be base-able if there exists positive integers $a$ and $b,$ with $b>1,$ such that $n=a^b.$ How many positive integer divisors of $729000000$ are base-able?
[i]Proposed by Kyle Lee[/i]