Found problems: 85335
2017 Harvard-MIT Mathematics Tournament, 3
A polyhedron has $7n$ faces. Show that there exist $n + 1$ of the polyhedron's faces that all have the same number of edges.
2020 Thailand TST, 2
Let $P$ be a point inside triangle $ABC$. Let $AP$ meet $BC$ at $A_1$, let $BP$ meet $CA$ at $B_1$, and let $CP$ meet $AB$ at $C_1$. Let $A_2$ be the point such that $A_1$ is the midpoint of $PA_2$, let $B_2$ be the point such that $B_1$ is the midpoint of $PB_2$, and let $C_2$ be the point such that $C_1$ is the midpoint of $PC_2$. Prove that points $A_2, B_2$, and $C_2$ cannot all lie strictly inside the circumcircle of triangle $ABC$.
(Australia)
1999 Federal Competition For Advanced Students, Part 2, 1
Ninety-nine points are given on one of the diagonals of a unit square. Prove that there is at most one vertex of the square such that the average squared distance from a given point to the vertex is less than or equal to $1/2$.
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.