Found problems: 85335
1968 Poland - Second Round, 4
Prove that if the numbers $ a, b, c $, are the lengths of the sides of a triangle and the sum of the numbers $x,y,z$ is zero, then $$a^2yz + b^2zx + c^2xy \leq 0.$$
2007 F = Ma, 5
A crate of toys remains at rest on a sleigh as the sleigh is pulled up a hill with an increasing speed. The crate is not fastened down to the sleigh. What force is responsible for the crate’s increase in speed up the hill?
$\textbf{(A)} \ \text{the force of static friction of the sleigh on the crate}$
$ \textbf{(B)} \ \text{the contact force (normal force) of the ground on the sleigh}$
$ \textbf{(C)} \ \text{the contact force (normal force) of the sleigh on the crate}$
$ \textbf{(D)} \ \text{the gravitational force acting on the sleigh}$
$ \textbf{(E)} \ \text{no force is needed}$
2006 Harvard-MIT Mathematics Tournament, 8
Solve for all complex numbers $z$ such that $z^4+4z^2+6=z$.
2017 Iranian Geometry Olympiad, 3
Let $O$ be the circumcenter of triangle $ABC$. Line $CO$ intersects the altitude from $A$ at point $K$. Let $P,M$ be the midpoints of $AK$, $AC$ respectively. If $PO$ intersects $BC$ at $Y$, and the circumcircle of triangle $BCM$ meets $AB$ at $X$, prove that $BXOY$ is cyclic.
[i]Proposed by Ali Daeinabi - Hamid Pardazi[/i]
1974 Czech and Slovak Olympiad III A, 1
Let $\left(a_k\right)_{k=1}^\infty$ be a sequence of positive numbers such that \[a_{k-1}a_{k+1}\ge a_k^2\] for all $k>1.$ For $n\ge1$ denote \[b_n=\left(a_1a_2\cdots a_n\right)^{1/n}.\] Show that also the inequality \[b_{n-1}b_{n+1}\ge b_n^2\] holds for every $n>1.$
2022 IFYM, Sozopol, 8
Let $x$ be a real number. Find the greatest possible value of the following expression:
$\frac{47^x}{\sqrt{43}}+\frac{43^x}{\sqrt{47}}-2021^x$.
2016 ASDAN Math Tournament, 2
Simplify the expression
$$\sqrt[3]{20+14\sqrt{2}}+\sqrt[3]{20-14\sqrt{2}}.$$
Mathematical Minds 2023, P3
Let $ABC$ be a triangle. It is known that the triangle formed by the midpoints of the medians of $ABC$ is equilateral. Prove that $ABC$ is equilateral as well.
PEN A Problems, 29
For which positive integers $k$, is it true that there are infinitely many pairs of positive integers $(m, n)$ such that \[\frac{(m+n-k)!}{m! \; n!}\] is an integer?
2018 Portugal MO, 4
Let $[ABC]$ be any triangle and let $D, E$ and $F$ be the symmetrics of the circumcenter wrt the three sides. Prove that the triangles $[ABC]$ and $[DEF]$ are congruent.
[img]https://cdn.artofproblemsolving.com/attachments/c/6/45bd929dfff87fb8deb09eddb59ef46e0dc0f4.png[/img]
2015 HMNT, 2
Let $a$ and $b$ be real numbers randomly (and independently) chosen from the range $[0,1]$. Find the probability that $a, b$ and $1$ form the side lengths of an obtuse triangle.
2022 239 Open Mathematical Olympiad, 2
Five edges of a tetrahedron are tangent to a sphere. Prove that there are another five edges from this tetrahedron that are also tangent to a $($not necessarily the same$)$ sphere.
2007 Germany Team Selection Test, 3
Points $ A_{1}$, $ B_{1}$, $ C_{1}$ are chosen on the sides $ BC$, $ CA$, $ AB$ of a triangle $ ABC$ respectively. The circumcircles of triangles $ AB_{1}C_{1}$, $ BC_{1}A_{1}$, $ CA_{1}B_{1}$ intersect the circumcircle of triangle $ ABC$ again at points $ A_{2}$, $ B_{2}$, $ C_{2}$ respectively ($ A_{2}\neq A, B_{2}\neq B, C_{2}\neq C$). Points $ A_{3}$, $ B_{3}$, $ C_{3}$ are symmetric to $ A_{1}$, $ B_{1}$, $ C_{1}$ with respect to the midpoints of the sides $ BC$, $ CA$, $ AB$ respectively. Prove that the triangles $ A_{2}B_{2}C_{2}$ and $ A_{3}B_{3}C_{3}$ are similar.
2000 China Team Selection Test, 1
Let $ABC$ be a triangle such that $AB = AC$. Let $D,E$ be points on $AB,AC$ respectively such that $DE = AC$. Let $DE$ meet the circumcircle of triangle $ABC$ at point $T$. Let $P$ be a point on $AT$. Prove that $PD + PE = AT$ if and only if $P$ lies on the circumcircle of triangle $ADE$.
1960 AMC 12/AHSME, 39
To satisfy the equation $\frac{a+b}{a}=\frac{b}{a+b}$, $a$ and $b$ must be:
$ \textbf{(A)}\ \text{both rational} \qquad\textbf{(B)}\ \text{both real but not rational} \qquad\textbf{(C)}\ \text{both not real}\qquad$
$\textbf{(D)}\ \text{one real, one not real}\qquad\textbf{(E)}\ \text{one real, one not real or both not real} $
2006 Junior Balkan MO, 4
Consider a $2n \times 2n$ board. From the $i$th line we remove the central $2(i-1)$ unit squares. What is the maximal number of rectangles $2 \times 1$ and $1 \times 2$ that can be placed on the obtained figure without overlapping or getting outside the board?
2010 Harvard-MIT Mathematics Tournament, 4
Suppose that there exist nonzero complex numbers $a$, $b$, $c$, and $d$ such that $k$ is a root of both the equations $ax^3+bx^2+cx+d=0$ and $bx^3+cx^2+dx+a=0$. Find all possible values of $k$ (including complex values).
MOAA Team Rounds, 2019.3
For how many ordered pairs of positive integers $(a, b)$ such that $a \le 50$ is it true that $x^2 - ax + b$ has integer roots?
2006 Germany Team Selection Test, 3
Suppose we have a $n$-gon. Some $n-3$ diagonals are coloured black and some other $n-3$ diagonals are coloured red (a side is not a diagonal), so that no two diagonals of the same colour can intersect strictly inside the polygon, although they can share a vertex. Find the maximum number of intersection points between diagonals coloured differently strictly inside the polygon, in terms of $n$.
[i]Proposed by Alexander Ivanov, Bulgaria[/i]
1995 Irish Math Olympiad, 3
Points $ A,X,D$ lie on a line in this order, point $ B$ is on the plane such that $ \angle ABX>120^{\circ}$, and point $ C$ is on the segment $ BX$. Prove the inequality:
$ 2AD \ge \sqrt{3} (AB\plus{}BC\plus{}CD)$.
2001 China Team Selection Test, 3
Given $a$, $b$ are positive integers greater than $1$, and for every positive integer $n$, $b^{n}-1$ divides $a^{n}-1$. Define the polynomial $p_{n}(x)$ as follows: $p_0{x}=-1$, $p_{n+1}(x)=b^{n+1}(x-1)p_{n}(bx)-a(b^{n+1}-1)p_{n}(x)$, for $n \ge 0$. Prove that there exist integers $C$ and positive integer $k$ such that $p_{k}(x)=Cx^k$.
2006 Estonia Math Open Senior Contests, 1
All the streets in a city run in one of two perpendicular directions, forming unit
squares. Organizers of a car race want to mark down a closed race track in the city in such a way that it would not go through any of the crossings twice and that the track would turn 90◦ right or left at every crossing. Find all possible values of the length of the track.
1949 Putnam, A2
We consider three vectors drawn from the same initial point $O,$ of lengths $a,b$ and $c$, respectively. Let $E$ be the parallelepiped with vertex $O$ of which the given vectors are the edges and $H$ the parallelepiped with vertex $O$ of which the given vectors are the altitudes. Show that the product of the volumes of $E$ and $H$ equals $(abc)^{2}$ and generalize this result to $n$ dimensions.
2006 Oral Moscow Geometry Olympiad, 6
In an acute-angled triangle, one of the angles is $60^o$. Prove that the line passing through the center of the circumcircle and the intersection point of the medians of the triangle cuts off an equilateral triangle from it.
(A. Zaslavsky)
2007 Kurschak Competition, 1
We have placed $n>3$ cards around a circle, facing downwards. In one step we may perform the following operation with three consecutive cards. Calling the one on the center $B$, the two on the ends $A$ and $C$, we put card $C$ in the place of $A$, then move $A$ and $B$ to the places originally occupied by $B$ and $C$, respectively. Meanwhile, we flip the cards $A$ and $B$.
Using a number of these steps, is it possible to move each card to its original place, but facing upwards?