Found problems: 85335
2016 Middle European Mathematical Olympiad, 8
For a positive integer $n$, the equation $a^2 + b^2 + c^2 + n = abc$ is given in the positive integers.
Prove that:
1. There does not exist a solution $(a, b, c)$ for $n = 2017$.
2. For $n = 2016$, $a$ is divisible by $3$ for all solutions $(a, b, c)$.
3. There are infinitely many solutions $(a, b, c)$ for $n = 2016$.
1964 AMC 12/AHSME, 23
Two numbers are such that their difference, their sum, and their product are to one another as $1:7:24$. The product of the two numbers is:
$ \textbf{(A)}\ 6\qquad\textbf{(B)}\ 12\qquad\textbf{(C)}\ 24\qquad\textbf{(D)}\ 48\qquad\textbf{(E)}\ 96 $
1956 AMC 12/AHSME, 24
In the figure $ \overline{AB} \equal{} \overline{AC}$, angle $ BAD \equal{} 30^{\circ}$, and $ \overline{AE} \equal{} \overline{AD}$.
[asy]unitsize(20);
defaultpen(linewidth(.8pt)+fontsize(8pt));
pair A=(3,3),B=(0,0),C=(6,0),D=(2,0),E=(5,1);
draw(A--B--C--cycle);
draw(A--D--E);
label("$A$",A,N); label("$B$",B,W); label("$C$",C,E);
label("$D$",D,S); label("$E$",E,NE);[/asy]Then angle $ CDE$ equals:
$ \textbf{(A)}\ 7\frac {1}{2}^{\circ} \qquad\textbf{(B)}\ 10^{\circ} \qquad\textbf{(C)}\ 12\frac {1}{2}^{\circ} \qquad\textbf{(D)}\ 15^{\circ} \qquad\textbf{(E)}\ 20^{\circ}$
1988 All Soviet Union Mathematical Olympiad, 479
In the acute-angled triangle $ABC$, the altitudes $BD$ and $CE$ are drawn. Let $F$ and $G$ be the points of the line $ED$ such that $BF$ and $CG$ are perpendicular to $ED$. Prove that $EF = DG$.
2016 AMC 12/AHSME, 16
The graphs of $y=\log_3x$, $y=\log_x3$, $y=\log_{\frac13}x$, and $y=\log_x\frac13$ are plotted on the same set of axes. How many points in the plane with positive $x-$coordinates lie on two or more of the graphs?
$\textbf{(A) } 2 \qquad\textbf{(B) } 3 \qquad\textbf{(C) } 4 \qquad\textbf{(D) } 5 \qquad\textbf{(E) } 6$
2000 All-Russian Olympiad, 4
Some pairs of cities in a certain country are connected by roads, at least three roads going out of each city. Prove that there exists a round path consisting of roads whose number is not divisible by $3$.
1946 Moscow Mathematical Olympiad, 109
Solve the system of equations: $\begin{cases}
x_1 + x_2 + x_3 = 6 \\
x_2 + x_3 + x_4 = 9 \\
x_3 + x_4 + x_5 = 3 \\
x_4 + x_5 + x_6 = -3 \\
x_5 + x_6 + x_7 = -9 \\
x_6 + x_7 + x_8 = -6 \\
x_7 + x_8 + x_1 = -2 \\
x_8 + x_1 + x_2 = 2 \end{cases}$
1996 Canada National Olympiad, 3
We denote an arbitrary permutation of the integers $1$, $2$, $\ldots$, $n$ by $a_1$, $a_2$, $\ldots$, $a_n$. Let $f(n)$ denote the number of these permutations such that:
(1) $a_1 = 1$;
(2):$|a_i - a_{i+1}| \leq 2$, $i = 1, \ldots, n - 1$.
Determine whether $f(1996)$ is divisible by 3.
2017 Canadian Open Math Challenge, A1
Source: 2017 Canadian Open Math Challenge, Problem A1
-----
The average of the numbers $2$, $5$, $x$, $14$, $15$ is $x$. Determine the value of $x$ .
1995 AMC 12/AHSME, 14
If $f(x) = ax^4-bx^2+x+5$ and $f(-3) = 2$, then $f(3) =$
$\textbf{(A)}\ -5 \qquad
\textbf{(B)}\ -2 \qquad
\textbf{(C)}\ 1 \qquad
\textbf{(D)}\ 3 \qquad
\textbf{(E)}\ 8$
2023 Romania National Olympiad, 2
Determine the largest natural number $k$ such that there exists a natural number $n$ satisfying:
\[
\sin(n + 1) < \sin(n + 2) < \sin(n + 3) < \ldots < \sin(n + k).
\]
2001 Junior Balkan Team Selection Tests - Romania, 3
Let $n\ge 2$ be a positive integer. Find the positive integers $x$
\[\sqrt{x+\sqrt{x+\ldots +\sqrt{x}}}<n \]
for any number of radicals.
1972 Putnam, A5
Prove that there is no positive integer $n>1$ such that $n\mid2^{n} -1.$
2010 Paenza, 6
In space are given two tetrahedra with the same barycenter such that one of them contains the other.
For each tetrahedron, we consider the octahedron whose vertices are the midpoints of the tetrahedron's edges.
Prove that one of this octahedra contains the other.
2011 Moldova Team Selection Test, 2
Find all pairs of real number $x$ and $y$ which simultaneously satisfy the following 2 relations:
$x+y+4=\frac{12x+11y}{x^2+y^2}$
$y-x+3=\frac{11x-12y}{x^2+y^2}$
2012 NIMO Problems, 6
The positive numbers $a, b, c$ satisfy $4abc(a+b+c) = (a+b)^2(a+c)^2$. Prove that $a(a+b+c)=bc$.
[i]Proposed by Aaron Lin[/i]
2010 India Regional Mathematical Olympiad, 1
Let $ABCDEF$ be a convex hexagon in which diagonals $AD, BE, CF$ are concurrent at $O$. Suppose $[OAF]$ is geometric mean of $[OAB]$ and $[OEF]$ and $[OBC]$ is geometric mean of $[OAB]$ and $[OCD]$. Prove that $[OED]$ is the geometric mean of $[OCD]$ and $[OEF]$.
(Here $[XYZ]$ denotes are of $\triangle XYZ$)
2019 Portugal MO, 6
A metro network with $n \ge 2$ stations, where each station is connected to each of the others by a one-way line, is said to be [i]dispersed [/i]i f there are two stations $A$ and $B$ such that it is not possible to go from $A$ to $B$ through is from the network. If a network is [i]dispersed[/i], but it is possible to choose a station $A$ and reverse the direction of all lines to and from $A$ so that the new network is no longer dispersed, the network is said to be [i]correctable[/i]. Indicates all integers $n$ for which there is a network with $n$ stations, [i]dispersed [/i]and not [i]correctable[/i].
1965 All Russian Mathematical Olympiad, 069
A spy airplane flies on the circle with the centre $A$ and radius $10$ km. Its speed is $1000$ km/h. At a certain moment, a rocket , that has same speed with the airplane, is launched from point $A$ and moves along on the straight line connecting the airplane and point $A$.How long after launch will the rocket hit the plane?
2005 Germany Team Selection Test, 3
For an ${n\times n}$ matrix $A$, let $X_{i}$ be the set of entries in row $i$, and $Y_{j}$ the set of entries in column $j$, ${1\leq i,j\leq n}$. We say that $A$ is [i]golden[/i] if ${X_{1},\dots ,X_{n},Y_{1},\dots ,Y_{n}}$ are distinct sets. Find the least integer $n$ such that there exists a ${2004\times 2004}$ golden matrix with entries in the set ${\{1,2,\dots ,n\}}$.
2019 Dutch IMO TST, 2
Write $S_n$ for the set $\{1, 2,..., n\}$. Determine all positive integers $n$ for which there exist functions $f : S_n \to S_n$ and $g : S_n \to S_n$ such that for every $x$ exactly one of the equalities $f(g(x)) = x$ and $g(f(x)) = x$ holds.
1985 IMO Longlists, 50
From each of the vertices of a regular $n$-gon a car starts to move with constant speed along the perimeter of the $n$-gon in the same direction. Prove that if all the cars end up at a vertex $A$ at the same time, then they never again meet at any other vertex of the $n$-gon. Can they meet again at $A \ ?$
1986 Canada National Olympiad, 1
In the diagram line segments $AB$ and $CD$ are of length 1 while angles $ABC$ and $CBD$ are $90^\circ$ and $30^\circ$ respectively. Find $AC$.
[asy]
import geometry;
import graph;
unitsize(1.5 cm);
pair A, B, C, D;
B = (0,0);
D = (3,0);
A = 2*dir(120);
C = extension(B,dir(30),A,D);
draw(A--B--D--cycle);
draw(B--C);
draw(arc(B,0.5,0,30));
label("$A$", A, NW);
label("$B$", B, SW);
label("$C$", C, NE);
label("$D$", D, SE);
label("$30^\circ$", (0.8,0.2));
label("$90^\circ$", (0.1,0.5));
perpendicular(B,NE,C-B);
[/asy]
2020 DMO Stage 1, 4.
[b]Q.[/b] We paint the numbers $1,2,3,4,5$ with red or blue. Prove that the equation $x+y=z$ have a monocolor solution (that is, all the 3 unknown there are the same color . It not needed that $x, y, z$ must be different!)
[i]Proposed by TuZo[/i]
2019 PUMaC Team Round, 4
What is the sum of the leading (first) digits of the integers from $ 1$ to $2019$ when the integers are written in base $3$? Give your answer in base $10$.