Found problems: 85335
2016 APMC, 6
Let $a$ be a natural number, $a>3$. Prove there is an infinity of numbers n, for which $a+n|a^{n}+1$
1978 All Soviet Union Mathematical Olympiad, 264
Given $0 < a \le x_1\le x_2\le ... \le x_n \le b$. Prove that $$(x_1+x_2+...+x_n)\left ( \frac{1}{x_1}+ \frac{1}{x_2}+...+ \frac{1}{x_n}\right)\le \frac{(a+b)^2}{4ab}n^2$$
PEN H Problems, 68
Consider the system \[x+y=z+u,\] \[2xy=zu.\] Find the greatest value of the real constant $m$ such that $m \le \frac{x}{y}$ for any positive integer solution $(x, y, z, u)$ of the system, with $x \ge y$.
1896 Eotvos Mathematical Competition, 1
If $k$ is the number of distinct prime divisors of a natural number $n$, prove that log $n \geq k$ log $2$.
2014 AMC 8, 17
George walks $1$ mile to school. He leaves home at the same time each day, walks at a steady speed of $3$ miles per hour, and arrives just as school begins. Today he was distracted by the pleasant weather and walked the first $\frac{1}{2}$ mile at a speed of only $2$ miles per hour. At how many miles per hour must George run the last $\frac{1}{2}$ mile in order to arrive just as school begins today?
$\textbf{(A) }4\qquad\textbf{(B) }6\qquad\textbf{(C) }8\qquad\textbf{(D) }10\qquad \textbf{(E) }12$
2017 Korea USCM, 4
For a real coefficient cubic polynomial $f(x)=ax^3+bx^2+cx+d$, denote three roots of the equation $f(x)=0$ by $\alpha,\beta,\gamma$. Prove that the three roots $\alpha,\beta,\gamma$ are distinct real numbers iff the real symmetric matrix
$$\begin{pmatrix} 3 & p_1 & p_2 \\ p_1 & p_2 & p_3 \\ p_2 & p_3 & p_4 \end{pmatrix},\quad p_i = \alpha^i + \beta^i + \gamma^i$$
is positive definite.
2021 USAMTS Problems, 2
Let $n$ be a fixed positive integer. Which is greater?[list=1]
[*]The number of $n$-tuples of integers whose largest value is $7$ and whose smallest value
is $0$; or
[*]The number of ordered triples $(A, B, C)$ that satisfy the following property: $A$, $B$, $C$
are subsets of $\{1, 2, 3, \dots , n\}$, and neither $C\subseteq A\cup B$, nor $B\subseteq A\cup C$.
[/list]
Your answer can be: $(1)$, $(2)$, the two counts are equal, or it depends on $n$.
2016 Sharygin Geometry Olympiad, 6
The sidelines $AB$ and $CD$ of a trapezoid meet at point $P$, and the diagonals of this trapezoid meet at point $Q$. Point $M$ on the smallest base $BC$ is such that $AM=MD$. Prove that $\angle PMB=\angle QMB$.
2006 Estonia Team Selection Test, 4
The side $AC$ of an acute triangle $ABC$ is the diameter of the circle $c_1$ and side $BC$ is the diameter of the circle $c_2$. Let $E$ be the foot of the altitude drawn from the vertex $B$ of the triangle and $F$ the foot of the altitude drawn from the vertex $A$. In addition, let $L$ and $N$ be the points of intersection of the line $BE$ with the circle $c_1$ (the point $L$ lies on the segment $BE$) and the points of intersection of $K$ and $M$ of line $AF$ with circle $c_2$ (point $K$ is in section $AF$). Prove that $K LM N$ is a cyclic quadrilateral.
1951 AMC 12/AHSME, 35
If $ a^x \equal{} c^q \equal{} b$ and $ c^y \equal{} a^z \equal{} d$, then
$ \textbf{(A)}\ xy \equal{} qz \qquad\textbf{(B)}\ \frac {x}{y} \equal{} \frac {q}{z} \qquad\textbf{(C)}\ x \plus{} y \equal{} q \plus{} z \qquad\textbf{(D)}\ x \minus{} y \equal{} q \minus{} z$
$ \textbf{(E)}\ x^y \equal{} q^z$
2011 Postal Coaching, 1
Prove that, for any positive integer $n$, there exists a polynomial $p(x)$ of degree at most $n$ whose coefficients are all integers such that, $p(k)$ is divisible by $2^n$ for every even integer $k$, and $p(k) -1$ is divisible by $2^n$ for every odd integer $k$.
2010 Balkan MO, 2
Let $ABC$ be an acute triangle with orthocentre $H$, and let $M$ be the midpoint of $AC$. The point $C_1$ on $AB$ is such that $CC_1$ is an altitude of the triangle $ABC$. Let $H_1$ be the reflection of $H$ in $AB$. The orthogonal projections of $C_1$ onto the lines $AH_1$, $AC$ and $BC$ are $P$, $Q$ and $R$, respectively. Let $M_1$ be the point such that the circumcentre of triangle $PQR$ is the midpoint of the segment $MM_1$.
Prove that $M_1$ lies on the segment $BH_1$.
Kvant 2023, M2742
Given an integer $h > 1$. Let's call a positive common fraction (not necessarily irreducible) [i]good[/i] if the sum of its numerator and denominator is equal to $h$. Let's say that a number $h$ is [i]remarkable[/i] if every positive common fraction whose denominator is less than $h$ can be expressed in terms of good fractions (not necessarily various) using the operations of addition and subtraction.
Prove that $h$ is remarkable if and only if it is prime.
(Recall that an common fraction has an integer numerator and a natural denominator.)
2010 Sharygin Geometry Olympiad, 8
Let $AH$ be the altitude of a given triangle $ABC.$ The points $I_b$ and $I_c$ are the incenters of the triangles $ABH$ and $ACH$ respectively. $BC$ touches the incircle of the triangle $ABC$ at a point $L.$ Find $\angle LI_bI_c.$
2010 BAMO, 3
All vertices of a polygon $P$ lie at points with integer coordinates in the plane, and all sides of $P$ have integer lengths. Prove that the perimeter of $P$ must be an even number.
2024 Bulgarian Winter Tournament, 11.4
Let $n, k$ be positive integers with $k \geq 3$. The edges of of a complete graph $K_n$ are colored in $k$ colors, such that for any color $i$ and any two vertices, there exists a path between them, consisting only of edges in color $i$. Prove that there exist three vertices $A, B, C$ of $K_n$, such that $AB, BC$ and $CA$ are all distinctly colored.
2002 IMO Shortlist, 8
Let two circles $S_{1}$ and $S_{2}$ meet at the points $A$ and $B$. A line through $A$ meets $S_{1}$ again at $C$ and $S_{2}$ again at $D$. Let $M$, $N$, $K$ be three points on the line segments $CD$, $BC$, $BD$ respectively, with $MN$ parallel to $BD$ and $MK$ parallel to $BC$. Let $E$ and $F$ be points on those arcs $BC$ of $S_{1}$ and $BD$ of $S_{2}$ respectively that do not contain $A$. Given that $EN$ is perpendicular to $BC$ and $FK$ is perpendicular to $BD$ prove that $\angle EMF=90^{\circ}$.
2018 District Olympiad, 1
Find all strictly increasing functions $f : \mathbb{N} \to \mathbb{N} $ such that $\frac {f(x) + f(y)}{1 + f(x + y)}$ is a non-zero natural number, for all $x, y\in\mathbb{N}$.
2007 Croatia Team Selection Test, 8
Positive integers $x>1$ and $y$ satisfy an equation $2x^2-1=y^{15}$. Prove that 5 divides $x$.
2017 NIMO Problems, 7
Call a pair of integers $(a,b)$ [i]primitive[/i] if there exists a positive integer $\ell$ such that $(a+bi)^\ell$ is real. Find the smallest positive integer $n$ such that less than $1\%$ of the pairs $(a, b)$ with $0 \le a, b \le n$ are primitive.
[i]Proposed by Mehtaab Sawhney[/i]
2015 Math Prize for Girls Olympiad, 2
A tetrahedron $T$ is inside a cube $C$. Prove that the volume of $T$ is at most one-third the volume of $C$.
2006 USA Team Selection Test, 2
In acute triangle $ABC$ , segments $AD; BE$ , and $CF$ are its altitudes, and $H$ is its orthocenter. Circle $\omega$, centered at $O$, passes through $A$ and $H$ and intersects sides $AB$ and $AC$ again at $Q$ and $P$ (other than $A$), respectively. The circumcircle of triangle $OPQ$ is tangent to segment $BC$ at $R$. Prove that $\frac{CR}{BR}=\frac{ED}{FD}.$
Durer Math Competition CD Finals - geometry, 2017.C+1
Given a plane with two circles, one with points $A$ and $B$, and the other with points $C$ and $D$ are shown in the figure. The line $AB$ passes through the center of the first circle and touches the second circle while the line $CD$ passes through the center of the second circle and touches the first circle. Prove that the lines $AD$ and $BC$ are parallel.
[img]https://cdn.artofproblemsolving.com/attachments/e/e/92f7b57751e7828a6487a052d4869e27e658b2.png[/img]
2009 Pan African, 2
Find all functions $f:\mathbb{N}_0\to\mathbb{N}_0$ for which $f(0)=0$ and
\[f(x^2-y^2)=f(x)f(y) \]
for all $x,y\in\mathbb{N}_0$ with $x>y$.
2016 Peru Cono Sur TST, P4
Let $n$ be a positive integer. Andrés has $n+1$ cards and each of them has a positive integer written, in such a way that the sum of the $n+1$ numbers is $3n$. Show that Andrés can place one or more cards in a red box and one or more cards in a blue box in such a way that the sum of the numbers of the cards in the red box is equal to twice the sum of the numbers of the cards in the blue box.
Clarification: Some of Andrés's letters can be left out of the boxes.