Found problems: 85335
2024 Nigerian MO Round 2, Problem 1
Given a number $\overline{abcd}$, where $a$, $b$, $c$, and $d$, represent the digits of $\overline{abcd}$, find the minimum value of
\[\frac{\overline{abcd}}{a+b+c+d}\]
where $a$, $b$, $c$, and $d$ are distinct
[hide=Answer]$\overline{abcd}=1089$, minimum value of $\dfrac{\overline{abcd}}{a+b+c+d}=60.5$[/hide]
2017 ITAMO, 2
Let $n\geq 2$ be an integer. Consider the solutions of the system
$$\begin{cases}
n=a+b-c \\
n=a^2+b^2-c^2
\end{cases}$$
where $a,b,c$ are integers. Show that there is at least one solution and that the solutions are finitely many.
2001 Bulgaria National Olympiad, 2
Suppose that $ABCD$ is a parallelogram such that $DAB>90$. Let the point $H$ to be on $AD$ such that $BH$ is perpendicular to $AD$. Let the point $M$ to be the midpoint of $AB$. Let the point $K$ to be the intersecting point of the line $DM$ with the circumcircle of $ADB$. Prove that $HKCD$ is concyclic.
1975 USAMO, 4
Two given circles intersect in two points $ P$ and $ Q$. Show how to construct a segment $ AB$ passing through $ P$ and terminating on the circles such that $ AP \cdot PB$ is a maximum.
2023 Grosman Mathematical Olympiad, 1
An arithmetic progression of natural numbers of length $10$ and with difference $11$ is given. Prove that the product of the numbers in this progression is divisible by $10!$.
1991 All Soviet Union Mathematical Olympiad, 552
$p(x)$ is the cubic $x^3 - 3x^2 + 5x$. If $h$ is a real root of $p(x) = 1$ and $k$ is a real root of $p(x) = 5$, find $h + k$.
1966 IMO Longlists, 2
Given $n$ positive numbers $a_{1},$ $a_{2},$ $...,$ $a_{n}$ such that $a_{1}\cdot a_{2}\cdot ...\cdot a_{n}=1.$ Prove \[ \left( 1+a_{1}\right) \left( 1+a_{2}\right) ...\left(1+a_{n}\right) \geq 2^{n}.\]
2018 India National Olympiad, 6
Let $\mathbb N$ denote set of all natural numbers and let $f:\mathbb{N}\to\mathbb{N}$ be a function such that
$\text{(a)} f(mn)=f(m).f(n)$ for all $m,n \in\mathbb{N}$;
$\text{(b)} m+n$ divides $f(m)+f(n)$ for all $m,n\in \mathbb N$.
Prove that, there exists an odd natural number $k$ such that $f(n)= n^k$ for all $n$ in $\mathbb{N}$.
2016 Israel Team Selection Test, 1
A square $ABCD$ is given. A point $P$ is chosen inside the triangle $ABC$ such that $\angle CAP = 15^\circ = \angle BCP$. A point $Q$ is chosen such that $APCQ$ is an isosceles trapezoid: $PC \parallel AQ$, and $AP=CQ, AP\nparallel CQ$. Denote by $N$ the midpoint of $PQ$. Find the angles of the triangle $CAN$.
2023 Auckland Mathematical Olympiad, 1
A single section at a stadium can hold either $7$ adults or $11$ children. When $N$ sections are completely lled, an equal number of adults and children will be seated in them. What is the least possible value of $N$?
2021 MOAA, 8
Andrew chooses three (not necessarily distinct) integers $a$, $b$, and $c$ independently and uniformly at random from $\{1,2,3,4,5,6,7\}$. Let $p$ be the probability that $abc(a+b+c)$ is divisible by $4$. If $p$ can be written as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$, then compute $m+n$.
[i]Proposed by Andrew Wen[/i]
2005 Morocco TST, 1
Find all the positive primes $p$ for which there exist integers $m,n$ satisfying :
$p=m^2+n^2$ and $m^3+n^3-4$ is divisible by $p$.
2000 All-Russian Olympiad, 5
Let $M$ be a finite sum of numbers, such that among any three of its elements there are two whose sum belongs to $M$. Find the greatest possible number of elements of $M$.
1994 AIME Problems, 11
Ninety-four bricks, each measuring $4''\times10''\times19'',$ are to stacked one on top of another to form a tower 94 bricks tall. Each brick can be oriented so it contribues $4''$ or $10''$ or $19''$ to the total height of the tower. How many differnt tower heights can be achieved using all 94 of the bricks?
1991 China Team Selection Test, 3
All edges of a polyhedron are painted with red or yellow. For an angle of a facet, if the edges determining it are of different colors, then the angle is called [i]excentric[/i]. The[i] excentricity [/i]of a vertex $A$, namely $S_A$, is defined as the number of excentric angles it has. Prove that there exist two vertices $B$ and $C$ such that $S_B + S_C \leq 4$.
2023 AMC 10, 6
Let $L_1=1$, $L_2=3$, and $L_{n+2}=L_{n+1}+L_n$ for $n\geq1$. How many terms in the sequence $L_1, L_2, L_3, \dots, L_{2023}$ are even?
$\textbf{(A) }673\qquad\textbf{(B) }1011\qquad\textbf{(C) }675\qquad\textbf{(D) }1010\qquad\textbf{(E) }674$
2004 Abels Math Contest (Norwegian MO), 2
(a) Prove that $(x+y+z)^2 \le 3(x^2 +y^2 +z^2)$ for any real numbers $x,y,z$.
(b) If positive numbers $a,b,c$ satisfy $a+b+c \ge abc$, prove that $a^2 +b^2 +c^2 \ge \sqrt3 abc$
1962 All Russian Mathematical Olympiad, 022
The $M$ point is the midpoint of the base $[AC]$ of an isosceles triangle $ABC$. $[MH]$ is orthogonal to $[BC]$ side. Point $P$ is the midpoint of the segment $[MH]$. Prove that $[AH]$ is orthogonal to $[BP]$.
1988 IMO Longlists, 23
In a right-angled triangle $ ABC$ let $ AD$ be the altitude drawn to the hypotenuse and let the straight line joining the incentres of the triangles $ ABD, ACD$ intersect the sides $ AB, AC$ at the points $ K,L$ respectively. If $ E$ and $ E_1$ dnote the areas of triangles $ ABC$ and $ AKL$ respectively, show that
\[ \frac {E}{E_1} \geq 2.
\]
1997 AMC 8, 12
$\angle 1 + \angle 2 = 180^\circ $
$\angle 3 = \angle 4$
Find $\angle 4.$
[asy]pair H,I,J,K,L;
H = (0,0); I = 10*dir(70); J = I + 10*dir(290); K = J + 5*dir(110); L = J + 5*dir(0);
draw(H--I--J--cycle);
draw(K--L--J);
draw(arc((0,0),dir(70),(1,0),CW)); label("$70^\circ$",dir(35),NE);
draw(arc(I,I+dir(250),I+dir(290),CCW)); label("$40^\circ$",I+1.25*dir(270),S);
label("$1$",J+0.25*dir(162.5),NW); label("$2$",J+0.25*dir(17.5),NE);
label("$3$",L+dir(162.5),WNW); label("$4$",K+dir(-52.5),SE);
[/asy]
$\textbf{(A)}\ 20^\circ \qquad \textbf{(B)}\ 25^\circ \qquad \textbf{(C)}\ 30^\circ \qquad \textbf{(D)}\ 35^\circ \qquad \textbf{(E)}\ 40^\circ$
2008 VJIMC, Problem 3
Find all $c\in\mathbb R$ for which there exists an infinitely differentiable function $f:\mathbb R\to\mathbb R$ such that for all $n\in\mathbb N$ and $x\in\mathbb R$ we have
$$f^{(n+1)}(x)>f^{(n)}(x)+c.$$
2019 MMATHS, 4
The continuous function $f(x)$ satisfies $c^2f(x + y) = f(x)f(y)$ for all real numbers $x$ and $y,$ where $c > 0$ is a constant. If $f(1) = c$, find $f(x)$ (with proof).
2002 District Olympiad, 4
Consider a function $f:\mathbb{R}\rightarrow \mathbb{R}$ such that:
1. $f$ has one-side limits in any $a\in \mathbb{R}$ and $f(a-0)\le f(a)\le f(a+0)$.
2. for any $a,b\in \mathbb{R},\ a<b$, we have $f(a-0)<f(b-0)$.
Prove that $f$ is strictly increasing.
[i]Mihai Piticari & Sorin Radulescu[/i]
2009 Stanford Mathematics Tournament, 6
Rhombus $ABCD$ has side length $ 1$. The size of $\angle A$ (in degrees) is randomly selected from all real numbers between $0$ and $90$. Find the expected value of the area of $ABCD$.
1966 AMC 12/AHSME, 40
[asy]draw(Circle((0,0), 1));
dot((0,0));
label("$O$", (0,0), S);
label("$A$", (-1,0), W);
label("$B$", (1,0), E);
label("$a$", (-0.5,0), S);
draw((-1,-1.25)--(-1,1.25));
draw((1,-1.25)--(1,1.25));
draw((-1,0)--(1,0));
draw((-1,0)--(-1,0)+2.3*dir(30));
label("$C$", (-1,0)+2.3*dir(30), E);
label("$D$", (-1,0)+1.8*dir(30), N);
dot((-1,0)+.4*dir(30));
label("$E$", (-1,0)+.4*dir(30), N);
[/asy]
In this figure $AB$ is a diameter of a circle, centered at $O$, with radius $a$. A chord $AD$ is drawn and extended to meet the tangent to the circle at $B$ in point $C$. Point $E$ is taken on $AC$ so that $AE=DC$. Denoting the distances of $E$ from the tangent through $A$ and from the diameter $AB$ by $x$ and $y$, respectively, we can deduce the relation:
$\text{(A)}\ y^2=\dfrac{x^3}{2a-x} \qquad
\text{(B)}\ y^2=\frac{x^3}{2a+x}\qquad
\text{(C)}\ y^4=\frac{x^2}{2-x}\qquad\\
\text{(D)}\ x^2=\dfrac{y^2}{2a-x}\qquad
\text{(E)}\ x^2=\frac{y^2}{2a+x}$