Found problems: 85335
2016 Korea Winter Program Practice Test, 4
Let $x,y,z \ge 0$ be real numbers such that $(x+y-1)^2+(y+z-1)^2+(z+x-1)^2=27$.
Find the maximum and minimum of $x^4+y^4+z^4$
2024 239 Open Mathematical Olympiad, 5
Let $a, b, c$ be reals such that $$a^2(c^2-2b-1)+b^2(a^2-2c-1)+c^2(b^2-2a-1)=0.$$ Show that $$3(a^2+b^2+c^2)+4(a+b+c)+3 \geq 6abc.$$
2016 BMT Spring, 4
Let $ABC$ have side lengths $3$, $4$, and $5$. Let $P$ be a point inside $ABC$. What is the minimum sum of lengths of the altitudes from $P$ to the side lengths of $ABC$?
2007 South africa National Olympiad, 1
Determine whether $ \frac{1}{\sqrt{2}} \minus{} \frac{1}{\sqrt{6}}$ is less than or greater than $ \frac{3}{10}$.
2008 Harvard-MIT Mathematics Tournament, 1
How many different values can $ \angle ABC$ take, where $ A,B,C$ are distinct vertices of a cube?
2019-2020 Winter SDPC, 1
Six people sit at a circular table (in the shape of a regular hexagon) such that no two friends sit next to or across from each other. Find, with proof, the maximum number of unordered pairs of people that can be friends.
2013 Online Math Open Problems, 16
Let $S_1$ and $S_2$ be two circles intersecting at points $A$ and $B$. Let $C$ and $D$ be points on $S_1$ and $S_2$ respectively such that line $CD$ is tangent to both circles and $A$ is closer to line $CD$ than $B$. If $\angle BCA = 52^\circ$ and $\angle BDA = 32^\circ$, determine the degree measure of $\angle CBD$.
[i]Ray Li[/i]
1990 Romania Team Selection Test, 4
Let $M$ be a point on the edge $CD$ of a tetrahedron $ABCD$ such that the tetrahedra $ABCM$ and $ABDM$ have the same total areas. We denote by $\pi_{AB}$ the plane $ABM$. Planes $\pi_{AC},...,\pi_{CD}$ are analogously defined. Prove that the six planes $\pi_{AB},...,\pi_{CD}$ are concurrent in a certain point $N$, and show that $N$ is symmetric to the incenter $I$ with respect to the barycenter $G$.
2019 Romanian Master of Mathematics Shortlist, G3
Let $ABC$ be an acute-angled triangle with $AB \ne AC$, and let $I$ and $O$ be its incenter and circumcenter, respectively. Let the incircle touch $BC, CA$ and $AB$ at $D, E$ and $F$, respectively. Assume that the line through $I$ parallel to $EF$, the line through $D$ parallel to$ AO$, and the altitude from $A$ are concurrent. Prove that the concurrency point is the orthocenter of the triangle $ABC$.
Petar Nizic-Nikolac, Croatia
2020 Latvia Baltic Way TST, 3
Prove that for all positive integers $n$ the following inequality holds:
$$ \frac{1}{1^2 +2020}+\frac{1}{2^2+2020} + \ldots + \frac{1}{n^2+2020} < \frac{1}{22} $$
2016 China Western Mathematical Olympiad, 8
For any given integers $m,n$ such that $2\leq m<n$ and $(m,n)=1$. Determine the smallest positive integer $k$ satisfying the following condition: for any $m$-element subset $I$ of $\{1,2,\cdots,n\}$ if $\sum_{i\in I}i> k$, then there exists a sequence of $n$ real numbers $a_1\leq a_2 \leq \cdots \leq a_n$ such that
$$\frac1m\sum_{i\in I} a_i>\frac1n\sum_{i=1}^na_i$$
2014 Romania Team Selection Test, 1
Let $ABC$ be a triangle, let ${A}'$, ${B}'$, ${C}'$ be the orthogonal projections of the vertices $A$ ,$B$ ,$C$ on the lines $BC$, $CA$ and $AB$, respectively, and let $X$ be a point on the line $A{A}'$.Let $\gamma_{B}$ be the circle through $B$ and $X$, centred on the line $BC$, and let $\gamma_{C}$ be the circle through $C$ and $X$, centred on the line $BC$.The circle $\gamma_{B}$ meets the lines $AB$ and $B{B}'$ again at $M$ and ${M}'$, respectively, and the circle $\gamma_{C}$ meets the lines $AC$ and $C{C}'$ again at $N$ and ${N}'$, respectively.Show that the points $M$, ${M}'$, $N$ and ${N}'$ are collinear.
2021 Canadian Mathematical Olympiad Qualification, 4
Let $O$ be the centre of the circumcircle of triangle $ABC$ and let $I$ be the centre of the incircle of triangle $ABC$. A line passing through the point $I$ is perpendicular to the line $IO$ and passes through the incircle at points $P$ and $Q$. Prove that the diameter of the circumcircle is equal to the perimeter of triangle $OPQ$.
1998 Italy TST, 3
New license plates consist of two letters, three digits, and two letters (from the English alphabet of$ 26$ letters). What is the largest possible number of such license plates if it is required that every two of them differ at no less than two positions?
2011 India IMO Training Camp, 2
Suppose $a_1,\ldots,a_n$ are non-integral real numbers for $n\geq 2$ such that ${a_1}^k+\ldots+{a_n}^k$ is an integer for all integers $1\leq k\leq n$. Prove that none of $a_1,\ldots,a_n$ is rational.
2024 Romania National Olympiad, 1
Let $I \subset \mathbb{R}$ be an open interval and $f:I \to \mathbb{R}$ a twice differentiable function such that $f(x)f''(x)=0,$ for any $x \in I.$ Prove that $f''(x)=0,$ for any $x \in I.$
1999 Brazil Team Selection Test, Problem 2
If $a,b,c,d$ are Distinct Real no. such that
$a = \sqrt{4+\sqrt{5+a}}$
$b = \sqrt{4-\sqrt{5+b}}$
$c = \sqrt{4+\sqrt{5-c}}$
$d = \sqrt{4-\sqrt{5-d}}$
Then $abcd = $
2010 F = Ma, 8
A car attempts to accelerate up a hill at an angle $\theta$ to the horizontal. The coefficient of static friction between the tires and the hill is $\mu > \tan \theta$. What is the maximum acceleration the car can achieve (in the direction upwards along the hill)? Neglect the rotational inertia of the wheels.
(A) $g \tan \theta$
(B) $g(\mu \cos \theta - \sin \theta)$
(C) $g(\mu - \sin \theta)$
(D) $g \mu \cos \theta$
(E) $g(\mu \sin \theta - \cos \theta)$
2016 LMT, 8
How many lattice points $P$ in or on the circle $x^2+y^2=25$ have the property that there exists a unique line with rational slope through $P$ that divides the circle into two parts with equal areas?
[i]Proposed by Nathan Ramesh
2024 Romania EGMO TST, P4
Find the greatest positive integer $n$ such that there exist positive integers $a_1, a_2, ..., a_n$ for which the following holds $a_{k+2} = \dfrac{(a_{k+1}+a_k)(a_{k+1}+1)}{a_k}$ for all $1 \le k \le n-2$.
[i]Proposed by Mykhailo Shtandenko and Oleksii Masalitin[/i]
2023 MOAA, 8
Two consecutive positive integers $n$ and $n+1$ have the property that they both have $6$ divisors but a different number of distinct prime factors. Find the sum of the possible values of $n$.
[i]Proposed by Harry Kim[/i]
1958 AMC 12/AHSME, 36
The sides of a triangle are $ 30$, $ 70$, and $ 80$ units. If an altitude is dropped upon the side of length $ 80$, the larger segment cut off on this side is:
$ \textbf{(A)}\ 62\qquad
\textbf{(B)}\ 63\qquad
\textbf{(C)}\ 64\qquad
\textbf{(D)}\ 65\qquad
\textbf{(E)}\ 66$
2017 CCA Math Bonanza, I5
In the [i]magic square[/i] below, every integer from $1$ to $25$ can be filled in such that the sum in every row, column, and long diagonal is the same. Given that the number in the center square is $18$, what is the sum of the entries in the shaded squares?
[asy]
size(4cm);
for (int i = 0; i <= 5; ++i) {
draw((0,i)--(5,i));
}
for (int i = 0; i <= 5; ++i) {
draw((i,0)--(i,5));
}
for (int i = 0; i <= 4; ++i) {
for (int j = 0; j <= 4; ++j) {
if ((i+j)%6 == 1 || (i-j)%6 == 3) {
fill((i,j)--(i+1,j)--(i+1,j+1)--(i,j+1)--cycle, gray);
}
}
}
label("\Large{18}", (2.5,2.5));
[/asy]
[i]2017 CCA Math Bonanza Individual Round #5[/i]
Estonia Open Junior - geometry, 2009.2.1
A Christmas tree must be erected inside a convex rectangular garden and attached to the posts at the corners of the garden with four ropes running at the same height from the ground. At what point should the Christmas tree be placed, so that the sum of the lengths of these four cords is as small as possible?
VI Soros Olympiad 1999 - 2000 (Russia), 10.6
Points $A$ and $B$ are given on a circle. With the help of a compass and a ruler, construct on this circle the points $C,$ $D$, $E$ that lie on one side of the straight line $AB$ and for which the pentagon with vertices $A$, $B$, $C$, $D$, $E$ has the largest possible area