Found problems: 85335
2017 India IMO Training Camp, 2
Define a sequence of integers $a_0=m, a_1=n$ and $a_{k+1}=4a_k-5a_{k-1}$ for all $k \ge 1$. Suppose $p>5$ is a prime with $p \equiv 1 \pmod{4}$. Prove that it is possible to choose $m,n$ such that $p \nmid a_k$ for any $k \ge 0$.
2018 EGMO, 5
Let $\Gamma $ be the circumcircle of triangle $ABC$. A circle $\Omega$ is tangent to the line segment $AB$ and is tangent to $\Gamma$ at a point lying on the same side of the line $AB$ as $C$. The angle bisector of $\angle BCA$ intersects $\Omega$ at two different points $P$ and $Q$.
Prove that $\angle ABP = \angle QBC$.
2014 India IMO Training Camp, 1
In a triangle $ABC$, let $I$ be its incenter; $Q$ the point at which the incircle touches the line $AC$; $E$ the midpoint of $AC$ and $K$ the orthocenter of triangle $BIC$. Prove that the line $KQ$ is perpendicular to the line $IE$.
2004 Estonia National Olympiad, 4
Prove that the number $n^n-n$ is divisible by $24$ for any odd integer $n$.
2006 Mediterranean Mathematics Olympiad, 4
Let $0\le x_{i,j} \le 1$, where $i=1,2, \ldots m$ and $j=1,2, \ldots n$. Prove the inequality
\[ \prod_{j=1}^n\left(1-\prod_{i=1}^mx_{i,j} \right)+ \prod_{i=1}^m\left(1-\prod_{j=1}^n(1-x_{i,j}) \right) \ge 1 \]
2018 AMC 12/AHSME, 24
Let $\lfloor x\rfloor$ denote the greatest integer less than or equal to $x$. How many real numbers $x$ satisfy the equation $x^2 + 10{,}000\lfloor x \rfloor = 10{,}000x$?
$\textbf{(A) } 197 \qquad \textbf{(B) } 198 \qquad \textbf{(C) } 199 \qquad \textbf{(D) } 200 \qquad \textbf{(E) } 201$
2021 Stanford Mathematics Tournament, 10
In acute $\vartriangle ABC$, let points $D$, $E,$ and $F$ be the feet of the altitudes of the triangle from $A$, $B$,and $C$, respectively. The area of $\vartriangle AEF$ is $1$, the area of $\vartriangle CDE$ is $2$, and the area of $\vartriangle BF D$ is $2 -\sqrt3$. What is the area of $\vartriangle DEF$?
2012 Rioplatense Mathematical Olympiad, Level 3, 3
Let $T$ be a non-isosceles triangle and $n \ge 4$ an integer . Prove that you can divide $T$ in $n$ triangles and draw in each of them an inner bisector so that those $n$ bisectors are parallel.
1978 Romania Team Selection Test, 1
In a convex quadrilateral $ ABCD, $ let $ A’,B’ $ be the orthogonal projections to $ CD $ of $ A, $ respectively, $ B. $
[b]a)[/b] Assuming that $ BB’\le AA’ $ and that the perimeter of $ ABCD $ is $ (AB+CD)\cdot BB’, $ is $ ABCD $ necessarily a trapezoid?
[b]b)[/b] The same question with the addition that $ \angle BAD $ is obtuse.
2021 Harvard-MIT Mathematics Tournament., 4
Let $S = \{1, 2, \dots, 9\}.$ Compute the number of functions $f : S \rightarrow S$ such that, for all $s \in S, f(f(f(s))) =s$ and $f(s) - s$ is not divisible by $3$.
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
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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:
$x+y+4=\frac{12x+11y}{x^2+y^2}$
$y-x+3=\frac{11x-12y}{x^2+y^2}$