Found problems: 85335
1976 Vietnam National Olympiad, 6
Show that $\frac{1}{x_1^n} + \frac{1}{x_2^n} +...+ \frac{1}{x_k^n} \ge k^{n+1}$ for positive real numbers $x_i $ with sum $1$.
2006 QEDMO 3rd, 7
Given a table with $2^n * n$ 1*1 squares ( $2^n$ rows and n column). In any square we put a number in {1, -1} such that no two rows are the same. Then we change numbers in some squares by 0. Prove that in new table we can choose some rows such that sum of all numbers in these rows equal to 0.
1964 AMC 12/AHSME, 7
Let $n$ be the number of real values of $p$ for which the roots of
\[ x^2-px+p=0 \]
are equal. Then $n$ equals:
${{ \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ \text{a finite number greater than 2} }\qquad\textbf{(E)}\ \text{an infinitely large number} } $
2011 N.N. Mihăileanu Individual, 3
Find $ \inf_{z\in\mathbb{C}} \left( |z^2+z+1|+|z^2-z+1| \right) . $
[i]Gheorghe Andrei[/i] and [i]Doru Constantin Caragea[/i]
2021 OMMock - Mexico National Olympiad Mock Exam, 1
Find all functions $f \colon \mathbb{R} \to \mathbb{R}$ that satisfy the following property for all real numbers $x$ and all polynomials $P$ with real coefficients:
If $P(f(x)) = 0$, then $f(P(x)) = 0$.
2020 Brazil Team Selection Test, 1
Consider an $n\times n$ unit-square board. The main diagonal of the board is the $n$ unit squares along the diagonal from the top left to the bottom right. We have an unlimited supply of tiles of this form:
[asy]
size(1.5cm);
draw((0,1)--(1,1)--(1,2)--(0,2)--(0,1)--(0,0)--(1,0)--(2,0)--(2,1)--(1,1)--(1,0));
[/asy]
The tiles may be rotated. We wish to place tiles on the board such that each tile covers exactly three unit squares, the tiles do not overlap, no unit square on the main diagonal is covered, and all other unit squares are covered exactly once. For which $n\geq 2$ is this possible?
[i]Proposed by Daniel Kohen[/i]
1956 AMC 12/AHSME, 22
Jones covered a distance of $ 50$ miles on his first trip. On a later trip he traveled $ 300$ miles while going three times as fast. His new time compared with the old time was:
$ \textbf{(A)}\ \text{three times as much} \qquad\textbf{(B)}\ \text{twice as much} \qquad\textbf{(C)}\ \text{the same}$
$ \textbf{(D)}\ \text{half as much} \qquad\textbf{(E)}\ \text{a third as much}$
2023 IFYM, Sozopol, 4
Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that
\[
f(2x + y + f(x + y)) + f(xy) = y f(x)
\]
for all real numbers $x$ and $y$.
2009 Junior Balkan Team Selection Tests - Romania, 4
To obtain a square $P$ of side length $2$ cm divided into $4$ unit squares it is sufficient to draw $3$ squares:
$P$ and another $2$ unit squares with a common vertex, as shown below:
[img]https://cdn.artofproblemsolving.com/attachments/1/d/827516518871ec8ff00a66424f06fda9812193.png[/img]
Find the minimum number of squares sufficient to obtain a square.of side length $n$ cm divided into $n^2$ unit squares ($n \ge 3$ is an integer).
2006 Estonia Math Open Senior Contests, 3
Let $ ABC$ be an acute triangle and choose points $ A_1, B_1$ and $ C_1$ on sides $ BC, CA$ and $ AB$, respectively. Prove that if the quadrilaterals $ ABA_1B_1, BCB_1C_1$ and $ CAC_1A_1$ are cyclic then their circumcentres lie on the sides of $ ABC$.
2007 Greece National Olympiad, 3
In a circular ring with radii $11r$ and $9r$, we put circles of radius $r$ which are tangent to the boundary circles and do not overlap. Determine the maximum number of circles that can be put this way. (You may use that $9.94<\sqrt{99}<9.95$)
2007 Romania Team Selection Test, 3
Find all subsets $A$ of $\left\{ 1, 2, 3, 4, \ldots \right\}$, with $|A| \geq 2$, such that for all $x,y \in A, \, x \neq y$, we have that $\frac{x+y}{\gcd (x,y)}\in A$.
[i]Dan Schwarz[/i]
2016 Korea USCM, 3
Given positive integers $m,n$ and a $m\times n$ matrix $A$ with real entries.
(1) Show that matrices $X = I_m + AA^T$ and $Y = I_n + A^T A$ are invertible. ($I_l$ is the $l\times l$ unit matrix.)
(2) Evaluate the value of $\text{tr}(X^{-1}) - \text{tr}(Y^{-1})$.
2014 239 Open Mathematical Olympiad, 5
Find all possible values of $k $ such that there exist a $k\times k$ table colored in $k$ colors such that there do not exist two cells in a column or a row with the same color or four cells made of intersecting two columns and two rows painted in exactly three colors.
2010 Contests, 2
The [i]rank[/i] of a rational number $q$ is the unique $k$ for which $q=\frac{1}{a_1}+\cdots+\frac{1}{a_k}$, where each $a_i$ is the smallest positive integer $q$ such that $q\geq \frac{1}{a_1}+\cdots+\frac{1}{a_i}$. Let $q$ be the largest rational number less than $\frac{1}{4}$ with rank $3$, and suppose the expression for $q$ is $\frac{1}{a_1}+\frac{1}{a_2}+\frac{1}{a_3}$. Find the ordered triple $(a_1,a_2,a_3)$.
2005 Swedish Mathematical Competition, 6
A regular tetrahedron of edge length $1$ is orthogonally projected onto a plane. Find the largest possible area of its image.
2017 Oral Moscow Geometry Olympiad, 1
One square is inscribed in a circle, and another square is circumscribed around the same circle so that its vertices lie on the extensions of the sides of the first (see figure). Find the angle between the sides of these squares.
[img]https://3.bp.blogspot.com/-8eLBgJF9CoA/XTodHmW87BI/AAAAAAAAKY0/xsHTx71XneIZ8JTn0iDMHupCanx-7u4vgCK4BGAYYCw/s400/sharygin%2Boral%2B2017%2B10-11%2Bp1.png[/img]
2012 Romania National Olympiad, 3
[color=darkred]Let $\mathcal{C}$ be the set of integrable functions $f\colon [0,1]\to\mathbb{R}$ such that $0\le f(x)\le x$ for any $x\in [0,1]$ . Define the function $V\colon\mathcal{C}\to\mathbb{R}$ by
\[V(f)=\int_0^1f^2(x)\ \text{d}x-\left(\int_0^1f(x)\ \text{d}x\right)^2\ ,\ f\in\mathcal{C}\ .\]
Determine the following two sets:
[list][b]a)[/b] $\{V(f_a)\, |\, 0\le a\le 1\}$ , where $f_a(x)=0$ , if $0\le x\le a$ and $f_a(x)=x$ , if $a<x\le 1\, ;$
[b]b)[/b] $\{V(f)\, |\, f\in\mathcal{C}\}\ .$[/list] [/color]
2024 USAMTS Problems, 3
A sequence of integers $x_1, x_2, \dots, x_k$ is called fibtastic if the difference between any two consecutive elements in the sequence is a Fibonacci number.
The integers from $1$ to $2024$ are split into two groups, each written in increasing order.
Group A is $a_1, a_2, \dots, a_m$ and Group B is $b_1, b_2, \dots, b_n.$
Find the largest integer $M$ such that we can guarantee that we can pick $M$ consecutive elements from either Group A or Group B which form a fibtastic sequence.
As an illustrative example, if a group of numbers is $2, 4, 11, 12, 13, 16, 18, 27, 29, 30,$ the
longest fibtastic sequence is $11, 12, 13, 16, 18,$ which has length $5.$
2013 Grand Duchy of Lithuania, 2
Let $ABC$ be an isosceles triangle with $AB = AC$. The points $D, E$ and $F$ are taken on the sides $BC, CA$ and $AB$, respectively, so that $\angle F DE = \angle ABC$ and $FE$ is not parallel to $BC$. Prove that the line $BC$ is tangent to the circumcircle of $\vartriangle DEF$ if and only if $D$ is the midpoint of the side $BC$.
1959 AMC 12/AHSME, 16
The expression $\frac{x^2-3x+2}{x^2-5x+6}\div \frac{x^2-5x+4}{x^2-7x+12}$, when simplified is:
$ \textbf{(A)}\ \frac{(x-1)(x-6)}{(x-3)(x-4)} \qquad\textbf{(B)}\ \frac{x+3}{x-3}\qquad\textbf{(C)}\ \frac{x+1}{x-1}\qquad\textbf{(D)}\ 1\qquad\textbf{(E)}\ 2$
2020 MBMT, 11
There are 8 distinct points on a plane, where no three are collinear. An ant starts at one of the points, then walks in a straight line to each one of the other points, visiting each point exactly once and stopping at the final point. This creates a trail of 7 line segments. What is the maximum number of times the ant can cross its own path as it walks?
[i]Proposed by Gabriel Wu[/i]
1999 Moldova Team Selection Test, 10
Let $n{}$ be a positive integer. Find the number of noncongruent triangles with integer sidelengths and a perimeter of $2n$.
2020 CMIMC Team, Estimation
Choose a point $(x,y)$ in the square bounded by $(0,0), (0,1), (1,0)$ and $(1,1)$. Your score is the minimal distance from your point to any other team's submitted point. Your answer must be in the form $(0.abcd, 0.efgh)$ where $a, b, c, d, e, f, g, h$ are decimal digits.
1970 Vietnam National Olympiad, 4
$AB$ and $CD$ are perpendicular diameters of a circle. $L$ is the tangent to the circle at $A$. $M$ is a variable point on the minor arc $AC$. The ray $BM, DM$ meet the line $L$ at $P$ and $Q$ respectively. Show that $AP\cdot AQ = AB\cdot PQ$.
Show how to construct the point $M$ which gives$ BQ$ parallel to $DP$.
If the lines $OP$ and $BQ$ meet at $N$ find the locus of $N$.
The lines $BP$ and $BQ$ meet the tangent at $D$ at $P'$ and $Q'$ respectively. Find the relation between $P'$ and $Q$'.
The lines $D$P and $DQ$ meet the line $BC$ at $P"$ and $Q"$ respectively. Find the relation between $P"$ and $Q"$.