Found problems: 85335
2015 239 Open Mathematical Olympiad, 6
Positive real numbers $a,b,c$ satisfy $$2a^3b+2b^3c+2c^3a=a^2b^2+b^2c^2+c^2a^2.$$
Prove that $$2ab(a-b)^2+2bc(b-c)^2+2ca(c-a)^2 \geq(ab+bc+ca)^2.$$
2019 Korea - Final Round, 3
Prove that there exist infinitely many positive integers $k$ such that the sequence $\{x_n\}$ satisfying
$$ x_1=1, x_2=k+2, x_{n+2}-(k+1)x_{n+1}+x_n=0(n \ge 0)$$
does not contain any prime number.
2022 May Olympiad, 3
Choose nine of the digits from $0$ to $9$ and place them in the boxes in the figure so that there are no repeated digits and the indicated sum is correct.
[img]https://cdn.artofproblemsolving.com/attachments/6/2/7f06575ec70eb9ddd58c6cf9dd3cb60d306e7c.png[/img]
Which digit was not used? You can fill in the boxes so that the unused digit is other?
2010 Regional Olympiad of Mexico Center Zone, 3
Let $a$, $b$ and $c$ be real positive numbers such that $\frac{1}{a} + \frac{1}{b} + \frac{1}{c} = 1$
Prove that:
$a^2+b^2+b^2 \ge 2a+2b+2c+9$
2023 LMT Fall, 5
In regular hexagon $ABCDEF$ with side length $2$, let $P$, $Q$, $R$, and $S$ be the feet of the altitudes from $A$ to $BC$, $EF$, $CF$, and $BE$, respectively. Find the area of quadrilateral $PQRS$.
VI Soros Olympiad 1999 - 2000 (Russia), 11.4
Let the line $L$ be perpendicular to the plane $P$. Three spheres touch each other in pairs so that each sphere touches the plane $P$ and the line $L$. The radius of the larger sphere is $1$. Find the minimum radius of the smallest sphere.
1984 AMC 12/AHSME, 2
If $x,y$ and $y - \frac{1}{x}$ are not 0, then \[\frac{x - \frac{1}{y}}{y - \frac{1}{x}}\] equals
$\textbf{(A) }1\qquad\textbf{(B) } \frac{x}{y}\qquad\textbf{(C) }\frac{y}{x}\qquad\textbf{(D) }\frac{x}{y} - \frac{y}{x}\qquad\textbf{(E) } xy - \frac{1}{xy}$
2024 ELMO Shortlist, A3
Find all functions $f : \mathbb{R}\to\mathbb{R}$ such that for all real numbers $x$ and $y$,
$$f(x+f(y))+xy=f(x)f(y)+f(x)+y.$$
[i]Andrew Carratu[/i]
2016 BMT Spring, 7
Consider the graph on $1000$ vertices $v_1, v_2, ...v_{1000}$ such that for all $1 \le i < j \le 1000$, $v_i$ is connected to $v_j$ if and only if $i$ divides $j$. Determine the minimum number of colors that must be used to color the vertices of this graph such that no two vertices sharing an edge are the same color.
1991 Tournament Of Towns, (308) 5
A $9 \times 9$ square is divided into $81$ unit cells. Some of the cells are coloured. The distance between the centres of any two coloured cells is more than $2$.
(a) Give an example of colouring with $17$ coloured cells.
(b) Prove that the numbers of coloured cells cannot exceed $17$.
(S. Fomin, Leningrad)
2013 Dutch Mathematical Olympiad, 2
Find all triples $(x, y, z)$ of real numbers satisfying:
$x + y - z = -1$ , $x^2 - y^2 + z^2 = 1$ and $- x^3 + y^3 + z^3 = -1$
1996 Chile National Olympiad, 5
Some time ago, on a radio program, a baker announced a special promotion in the purchase of two stuffed cakes. Each cake could contain up to five fillings of which had in the pastry. On the show, a lady said there were $1,048,576$ different possibilities to choose the two stuffed cakes. How many different fillings did the pastry chef have?
PEN E Problems, 15
Show that there exist two consecutive squares such that there are at least $1000$ primes between them.
1983 IMO Longlists, 18
Let $b \geq 2$ be a positive integer.
(a) Show that for an integer $N$, written in base $b$, to be equal to the sum of the squares of its digits, it is necessary either that $N = 1$ or that $N$ have only two digits.
(b) Give a complete list of all integers not exceeding $50$ that, relative to some base $b$, are equal to the sum of the squares of their digits.
(c) Show that for any base b the number of two-digit integers that are equal to the sum of the squares of their digits is even.
(d) Show that for any odd base $b$ there is an integer other than $1$ that is equal to the sum of the squares of its digits.
2014 BMT Spring, P1
Let a simple polygon be defined as a polygon in which no consecutive sides are parallel and no two non-consecutive sides share a common point. Given that all vertices of a simple polygon $P$ are lattice points (in a Cartesian coordinate system, each vertex has integer coordinates), and each side of $P$ has integer length, prove that the perimeter must be even.
1985 IMO Longlists, 92
Find a method by which one can compute the coefficients of $P(x) = x^6 + a_1x^5 + \cdots+ a_6$ from the roots of $P(x) = 0$ by performing not more than $15$ additions and $15$ multiplications.
2013 Princeton University Math Competition, 2
Let $\gamma$ be the incircle of $\triangle ABC$ (i.e. the circle inscribed in $\triangle ABC$) and $I$ be the center of $\gamma$. Let $D$, $E$ and $F$ be the feet of the perpendiculars from $I$ to $BC$, $CA$, and $AB$ respectively. Let $D'$ be the point on $\gamma$ such that $DD'$ is a diameter of $\gamma$. Suppose the tangent to $\gamma$ through $D$ intersects the line $EF$ at $P$. Suppose the tangent to $\gamma$ through $D'$ intersects the line $EF$ at $Q$. Prove that $\angle PIQ + \angle DAD' = 180^{\circ}$.
1986 IMO Longlists, 57
In a triangle $ABC$, the incircle touches the sides $BC, CA, AB$ in the points $A',B', C'$, respectively; the excircle in the angle $A$ touches the lines containing these sides in $A_1,B_1, C_1$, and similarly, the excircles in the angles $B$ and $C$ touch these lines in $A_2,B_2, C_2$ and $A_3,B_3, C_3$. Prove that the triangle $ABC$ is right-angled if and only if one of the point triples $(A',B_3, C'),$ $ (A_3,B', C_3), (A',B', C_2), (A_2,B_2, C'), (A_2,B_1, C_2), (A_3,B_3, C_1),$ $ (A_1,B_2, C_1), (A_1,B_1, C_3)$ is collinear.
2023 Middle European Mathematical Olympiad, 2
If $a, b, c, d>0$ and $abcd=1$, show that $$\frac{ab+1}{a+1}+\frac{bc+1}{b+1}+\frac{cd+1}{c+1}+\frac{da+1}{d+1} \geq 4. $$ When does equality hold?
2018 CCA Math Bonanza, I6
A lumberjack is building a non-degenerate triangle out of logs. Two sides of the triangle have lengths $\log 101$ and $\log 2018$. The last side of his triangle has side length $\log n$, where $n$ is an integer. How many possible values are there for $n$?
[i]2018 CCA Math Bonanza Individual Round #6[/i]
1986 Austrian-Polish Competition, 5
Find all real solutions of the system of equations
$$\begin{cases} x^2 + y^2 + u^2 + v^2 = 4 \\ xu + yv + xv + yu = 0 \\ xyu + yuv + uvx + vxy = - 2 \\ xyuv = -1 \end{cases}$$
1999 Junior Balkan Team Selection Tests - Moldova, 2
Let $ABC$ be an isosceles right triangle with $\angle A=90^o$. Point $D$ is the midpoint of the side $[AC]$, and point $E \in [AC]$ is so that $EC = 2AE$. Calculate $\angle AEB + \angle ADB$ .
2008 Swedish Mathematical Competition, 1
A rhombus is inscribed in a convex quadrilateral. The sides of the rhombus are parallel with the diagonals of the quadrilateral, which have the lengths $d_1$ and $d_2$. Calculate the length of side of the rhombus , expressed in terms of $d_1$ and $d_2$.
2020 CMIMC Combinatorics & Computer Science, 6
The nation of CMIMCland consists of 8 islands, none of which are connected. Each citizen wants to visit the other islands, so the government will build bridges between the islands. However, each island has a volcano that could erupt at any time, destroying that island and any bridges connected to it. The government wants to guarantee that after any eruption, a citizen from any of the remaining $7$ islands can go on a tour, visiting each of the remaining islands exactly once and returning to their home island (only at the end of the tour). What is the minimum number of bridges needed?
2014 France Team Selection Test, 4
Let $\mathbb{Z} _{>0}$ be the set of positive integers. Find all functions $f: \mathbb{Z} _{>0}\rightarrow \mathbb{Z} _{>0}$ such that
\[ m^2 + f(n) \mid mf(m) +n \]
for all positive integers $m$ and $n$.