Found problems: 85335
1986 AMC 12/AHSME, 8
The population of the United States in 1980 was $226,504,825$. The area of the country is $3,615,122$ square miles. The are $(5280)^{2}$ square feet in one square mile. Which number below best approximates the average number of square feet per person?
$ \textbf{(A)}\ 5,000\qquad\textbf{(B)}\ 10,000\qquad\textbf{(C)}\ 50,000\qquad\textbf{(D)}\ 100,000\qquad\textbf{(E)}\ 500,000 $
2017 Purple Comet Problems, 14
Find the sum of all integers $n$ for which $n - 3$ and $n^2 + 4$ are both perfect cubes.
2007 Tournament Of Towns, 3
A triangle with sides $a, b, c$ is folded along a line $\ell$ so that a vertex $C$ is on side $c$. Find the segments on which point $C$ divides $c$, given that the angles adjacent to $\ell$ are equal.
[i](2 points)[/i]
2007 Today's Calculation Of Integral, 237
Calculate $ \int \frac {dx}{x^{2008}(1 \minus{} x)}$
2006 VJIMC, Problem 1
Given real numbers $0=x_1<x_2<\ldots<x_{2n}<x_{2n+1}=1$ such that $x_{i+1}-x_i\le h$ for $1\le i\le2n$, show that
$$\frac{1-h}2<\sum_{i=1}^nx_{2i}(x_{2i+1}-x_{2i-1})<\frac{1+h}2.$$
2011 F = Ma, 4
Rank the [i]magnitudes[/i] of the [i]distance[/i] traveled during the ten second interval.
(A) $\text{I} > \text{II} > \text{III}$
(B) $\text{II} > \text{I} > \text{III}$
(C) $\text{III} > \text{II} > \text{I}$
(D) $\text{I} > \text{II = III}$
(E) $\text{I = II = III}$
2022 Oral Moscow Geometry Olympiad, 2
Angle bisectors from vertices $B$ and $C$ and the perpendicular bisector of side $BC$ are drawn in a non-isosceles triangle $ABC$. Next, three points of pairwise intersection of these three lines were marked (remembering which point is which), and the triangle itself was erased. Restore it according to the marked points using a compass and ruler.
(Yu. Blinkov)
2007 Turkey Team Selection Test, 2
A number $n$ is satisfying the conditions below
i) $n$ is a positive odd integer;
ii) there are some odd integers such that their squares' sum is equal to $n^{4}$.
Find all such numbers.
2008 Sharygin Geometry Olympiad, 14
(V.Protasov, 9--10) The Euler line of a non-isosceles triangle is parallel to the bisector of one of its angles. Determine this
angle (There was an error in published condition of this problem).
1987 Romania Team Selection Test, 11
Let $P(X,Y)=X^2+2aXY+Y^2$ be a real polynomial where $|a|\geq 1$. For a given positive integer $n$, $n\geq 2$ consider the system of equations: \[ P(x_1,x_2) = P(x_2,x_3) = \ldots = P(x_{n-1},x_n) = P(x_n,x_1) = 0 . \] We call two solutions $(x_1,x_2,\ldots,x_n)$ and $(y_1,y_2,\ldots,y_n)$ of the system to be equivalent if there exists a real number $\lambda \neq 0$, $x_1=\lambda y_1$, $\ldots$, $x_n= \lambda y_n$. How many nonequivalent solutions does the system have?
[i]Mircea Becheanu[/i]
2002 AMC 10, 12
For $f_n(x)=x^n$ and $a\neq 1$ consider
I. $(f_{11}(a)f_{13}(a))^{14}$
II. $f_{11}(a)f_{13}(a)f_{14}(a)$
III. $(f_{11}(f_{13}(a)))^{14}$
IV. $f_{11}(f_{13}(f_{14}(a)))$
Which of these equal $f_{2002}(a)$?
$\textbf{(A) }\text{I and II only}\qquad\textbf{(B) }\text{II and III only}\qquad\textbf{(C) }\text{III and IV only}\qquad\textbf{(D) }\text{II, III, and IV only}\qquad\textbf{(E) }\text{all of them}$
2022 BMT, 2
The equation
$$4^x -5 \cdot 2^{x+1} +16 = 0$$
has two integer solutions for $x.$ Find their sum.
2022 Greece JBMO TST, 3
The real numbers $x,y,z$ are such that $x+y+z=4$ and $0 \le x,y,z \le 2$. Find the minimun value of the expression $$A=\sqrt{2+x}+\sqrt{2+y}+\sqrt{2+z}+\sqrt{x+y}+\sqrt{y+z}+\sqrt{z+x}$$.
2013 Kurschak Competition, 2
Consider the closed polygonal discs $P_1$, $P_2$, $P_3$ with the property that for any three points $A\in P_1$, $B\in P_2$, $C\in P_3$, we have $[\triangle ABC]\le 1$. (Here $[X]$ denotes the area of polygon $X$.)
(a) Prove that $\min\{[P_1],[P_2],[P_3]\}<4$.
(b) Give an example of polygons $P_1,P_2,P_3$ with the above property such that $[P_1]>4$ and $[P_2]>4$.
2020 LMT Fall, A3
Find the value of $\left\lfloor \frac{1}{6}\right\rfloor+\left\lfloor\frac{4}{6}\right\rfloor+\left\lfloor\frac{9}{6}\right\rfloor+\dots+\left\lfloor\frac{1296}{6}\right\rfloor$.
[i]Proposed by Zachary Perry[/i]
2018 Nordic, 3
Let $ABC$ be a triangle with $AB < AC$. Let $D$ and $E$ be on the lines $CA$ and $BA$, respectively, such that $CD = AB$, $BE = AC$, and $A$, $D$ and $E$ lie on the same side of $BC$. Let $I$ be the incenter of triangle $ABC$, and let $H$ be the orthocenter of triangle $BCI$. Show that $D$, $E$, and $H$ are collinear.
1997 All-Russian Olympiad Regional Round, 11.3
Let us denote by $S(m)$ the sum of the digits of the natural number $m$. Prove that there are infinitely many positive integers $n$ such that $$S(3^n) \ge S(3^{n+1}).$$
2014 Contests, 2
Let $k\ge 2$, $n\ge 1$, $a_1, a_2,\dots, a_k$ and $b_1, b_2, \dots, b_n$ be integers such that $1<a_1<a_2<\dots <a_k<b_1<b_2<\dots <b_n$. Prove that if $a_1+a_2+\dots +a_k>b_1+b_2+\dots + b_n$, then $a_1\cdot a_2\cdot \ldots \cdot a_k>b_1\cdot b_2 \cdot \ldots \cdot b_n$.
2005 Putnam, B3
Find all differentiable functions $f: (0,\infty)\mapsto (0,\infty)$ for which there is a positive real number $a$ such that
\[ f'\left(\frac ax\right)=\frac x{f(x)} \]
for all $x>0.$
2008 China National Olympiad, 3
Find all triples $(p,q,n)$ that satisfy
\[q^{n+2} \equiv 3^{n+2} (\mod p^n) ,\quad p^{n+2} \equiv 3^{n+2} (\mod q^n)\]
where $p,q$ are odd primes and $n$ is an positive integer.
1965 Vietnam National Olympiad, 1
At a time $t = 0$, a navy ship is at a point $O$, while an enemy ship is at a point $A$ cruising with speed $v$ perpendicular to $OA = a$. The speed and direction of the enemy ship do not change. The strategy of the navy ship is to travel with constant speed $u$ at a angle $0 < \phi < \pi /2$ to the line $OA$.
1) Let $\phi$ be chosen. What is the minimum distance between the two ships? Under what conditions will the distance vanish?
2) If the distance does not vanish, what is the choice of $\phi$ to minimize the distance? What are directions of the two ships when their distance is minimum?
2014 Bosnia Herzegovina Team Selection Test, 2
It is given regular $n$-sided polygon, $n \geq 6$. How many triangles they are inside the polygon such that all of their sides are formed by diagonals of polygon and their vertices are vertices of polygon?
2014 Lithuania Team Selection Test, 4
(a) Is there a natural number $n$ such that the number $2^n$ has last digit $6$ and the sum of the other digits is $2$?
b) Are there natural numbers $a$ and $m\ge 3$ such that the number $a^m$ has last digit $6$ and the sum of the other digits is 3?
2018 India PRMO, 15
Let $a$ and $b$ be natural numbers such that $2a-b$, $a-2b$ and $a+b$ are all distinct squares. What is the smallest possible value of $b$ ?
2017 Bosnia and Herzegovina EGMO TST, 1
It is given sequence wih length of $2017$ which consists of first $2017$ positive integers in arbitrary order (every number occus exactly once). Let us consider a first term from sequence, let it be $k$. From given sequence we form a new sequence of length 2017, such that first $k$ elements of new sequence are same as first $k$ elements of original sequence, but in reverse order while other elements stay unchanged. Prove that if we continue transforming a sequence, eventually we will have sequence with first element $1$.