Found problems: 85335
2004 AMC 12/AHSME, 21
If $ \displaystyle \sum_{n \equal{} 0}^{\infty} \cos^{2n} \theta \equal{} 5$, what is the value of $ \cos{2\theta}$?
$ \textbf{(A)}\ \frac15 \qquad \textbf{(B)}\ \frac25 \qquad \textbf{(C)}\ \frac {\sqrt5}{5}\qquad \textbf{(D)}\ \frac35 \qquad \textbf{(E)}\ \frac45$
1988 IMO Shortlist, 6
In a given tedrahedron $ ABCD$ let $ K$ and $ L$ be the centres of edges $ AB$ and $ CD$ respectively. Prove that every plane that contains the line $ KL$ divides the tedrahedron into two parts of equal volume.
2006 Korea - Final Round, 1
Given three distinct real numbers $a_{1}, a_{2}, a_{3}$ , define $b_{j}= (1+\frac{a_{j}a_{i}}{a_{j}-a_{i}})(1+\frac{a_{j}a_{k}}{a_{j}-a_{k}})$, where $\{i, j, k\}= \{1, 2, 3\}$.
Prove that $1+|a_{1}b_{1}+a_{2}b_{2}+a_{3}b_{3}| \leq (1+|a_{1}|)(1+|a_{2}|)(1+|a_{3}|)$ and find
the cases of equality.
1980 Miklós Schweitzer, 10
Suppose that the $ T_3$-space $ X$ has no isolated points and that in $ X$ any family of pairwise disjoint, nonempty, open sets
is countable. Prove that $ X$ can be covered by at most continuum many nowhere-dense sets.
[i]I. Juhasz[/i]
2022 May Olympiad, 2
There are nine cards that have the digits $1, 2, 3, 4, 5, 6, 7, 8$ and $9$ written on them, with one digit on each card. Using all the cards, some numbers are formed (for example, the numbers $8$, $213$, $94$, $65$ and $7$).
a) If all the numbers formed are prime, determine the smallest possible value of their sum.
b) If all formed numbers are composite, determine the smallest possible value of their sum.
Note: A number $p$ is prime if its only divisors are $1$ and $p$. A number is composite if it has more than two dividers. The number $1$ is neither prime nor composite.
Novosibirsk Oral Geo Oly VIII, 2023.7
A square with side $1$ is intersected by two parallel lines as shown in the figure. Find the sum of the perimeters of the shaded triangles if the distance between the lines is also $1$.
[img]https://cdn.artofproblemsolving.com/attachments/9/e/4e70610b80871325a72e923a0909eff06aebfa.png[/img]
2009 Singapore Senior Math Olympiad, 3
Suppose $ A $ is a subset of $ n $-elements taken from $ 1,2,3,4,...,2009 $ such that the difference of any two numbers in $ A $ is not a prime number. Find the largest value of $ n $ and the set $ A $ with this number of elements.
2022 Baltic Way, 9
Five elders are sitting around a large bonfire. They know that Oluf will put a hat of one of four colours (red, green, blue or yellow) on each elder’s head, and after a short time for silent reflection each elder will have to write down one of the four colours on a piece of paper. Each elder will only be able to see the colour of their two neighbours’ hats, not that of their own nor that of the remaining two elders’ hats, and they also cannot communicate after Oluf starts putting the hats on.
Show that the elders can devise a strategy ahead of time so that at most two elders will end up writing down the colour of their own hat
2014 Bosnia And Herzegovina - Regional Olympiad, 1
Find all possible values of $$\frac{(a+b-c)^2}{(a-c)(b-c)}+\frac{(b+c-a)^2}{(b-a)(c-a)}+\frac{(c+a-b)^2}{(c-b)(a-b)}$$
2000 Mexico National Olympiad, 6
Let $ABC$ be a triangle with $\angle B > 90^o$ such that there is a point $H$ on side $AC$ with $AH = BH$ and BH perpendicular to $BC$. Let $D$ and $E$ be the midpoints of $AB$ and $BC$ respectively. A line through $H$ parallel to $AB$ cuts $DE$ at $F$. Prove that $\angle BCF = \angle ACD$.
2016 USAJMO, 3
Let $X_1, X_2, \ldots, X_{100}$ be a sequence of mutually distinct nonempty subsets of a set $S$. Any two sets $X_i$ and $X_{i+1}$ are disjoint and their union is not the whole set $S$, that is, $X_i\cap X_{i+1}=\emptyset$ and $X_i\cup X_{i+1}\neq S$, for all $i\in\{1, \ldots, 99\}$. Find the smallest possible number of elements in $S$.
2016 Taiwan TST Round 2, 2
Find all function $f:\mathbb{Z}\rightarrow\mathbb{Z}$ such that
$f(f(x)+f(y))+f(x)f(y)=f(x+y)f(x-y)$
for all integer $x,y$
2005 Germany Team Selection Test, 3
A positive integer is called [i]nice[/i] if the sum of its digits in the number system with base $ 3$ is divisible by $ 3$.
Calculate the sum of the first $ 2005$ nice positive integers.
2020 BMT Fall, 6
Let $N$ be the number of non-empty subsets $T$ of $S = \{1,2, 3,4,...,2020\}$ satisfying $max (T) >1000$. Compute the largest integer $k$ such that $3^k$ divides $N$.
2004 Harvard-MIT Mathematics Tournament, 4
A horse stands at the corner of a chessboard, a white square. With each jump, the horse can move either two squares horizontally and one vertically or two vertically and one horizontally (like a knight moves). The horse earns two carrots every time it lands on a black square, but it must pay a carrot in rent to rabbit who owns the chessboard for every move it makes. When the horse reaches the square on which it began, it can leave. What is the maximum number of carrots the horse can earn without touching any square more than twice?
[img]https://cdn.artofproblemsolving.com/attachments/e/c/c817d92ead6cfb3868f9cb526fb4e1fd7ffe4d.png[/img]
2023 AMC 10, 24
What is the perimeter of the boundary of the region consisting of all points which can be expressed as $(2u-3w,v+4w)$ with $0 \le u \le 1$, $0 \le v \le 1$, and $0 \le w \le 1$? \\ \\
$\textbf{(A) } 10\sqrt{3} \qquad \textbf{(B) } 10 \qquad \textbf{(C) } 12 \qquad \textbf{(D) } 18 \qquad \textbf{(E) } 16$
2010 China Team Selection Test, 2
Let $ABCD$ be a convex quadrilateral. Assume line $AB$ and $CD$ intersect at $E$, and $B$ lies between $A$ and $E$. Assume line $AD$ and $BC$ intersect at $F$, and $D$ lies between $A$ and $F$. Assume the circumcircles of $\triangle BEC$ and $\triangle CFD$ intersect at $C$ and $P$. Prove that $\angle BAP=\angle CAD$ if and only if $BD\parallel EF$.
2019 China Team Selection Test, 4
Prove that there exist a subset $A$ of $\{1,2,\cdots,2^n\}$ with $n$ elements, such that for any two different non-empty subset of $A$, the sum of elements of one subset doesn't divide another's.
2022 Estonia Team Selection Test, 5
For each integer $n\ge 1,$ compute the smallest possible value of \[\sum_{k=1}^{n}\left\lfloor\frac{a_k}{k}\right\rfloor\] over all permutations $(a_1,\dots,a_n)$ of $\{1,\dots,n\}.$
[i]Proposed by Shahjalal Shohag, Bangladesh[/i]
Novosibirsk Oral Geo Oly VIII, 2023.5
One convex quadrilateral is inside another. Can it turn out that the sum of the lengths of the diagonals of the outer quadrilateral is less than the sum of the lengths of the diagonals of the inner?
1963 IMO, 6
Five students $ A, B, C, D, E$ took part in a contest. One prediction was that the contestants would finish in the order $ ABCDE$. This prediction was very poor. In fact, no contestant finished in the position predicted, and no two contestants predicted to finish consecutively actually did so. A second prediction had the contestants finishing in the order $ DAECB$. This prediction was better. Exactly two of the contestants finished in the places predicted, and two disjoint pairs of students predicted to finish consecutively actually did so. Determine the order in which the contestants finished.
1962 Vietnam National Olympiad, 2
Let $ f(x) \equal{} (1 \plus{} x)\cdot\sqrt{(2 \plus{} x^2)}\cdot\sqrt[3]{(3 \plus{} x^3)}$. Determine $ f'(1)$.
2020 Brazil Team Selection Test, 5
There are $2020$ positive integers written on a blackboard. Every minute, Zuming erases two of the numbers and replaces them by their sum, difference, product, or quotient. For example, if Zuming erases the numbers $6$ and $3$, he may replace them with one of the numbers in the set $\{6+3, 6-3, 3-6, 6\times 3, 6\div 3, 3\div 6\}$ $= \{9, 3, 3, 18, 2, \tfrac 12\}$. After $2019$ minutes, Zuming writes the single number $-2020$ on the blackboard. Show that it was possible for Zuming to have ended up with the single number $2020$ instead, using the same rules and starting with the same $2020$ integers.
[i]Proposed by Zhuo Qun (Alex) Song[/i]
2017 F = ma, 25
25) A planet orbits around a star S, as shown in the figure. The semi-major axis of the orbit is a. The perigee, namely the shortest distance between the planet and the star is 0.5a. When the planet passes point $P$ (on the line through the star and perpendicular to the major axis), its speed is $v_1$. What is its speed $v_2$ when it passes the perigee?
A) $v_2 = \frac{3}{\sqrt{5}}v_1$
B) $v_2 = \frac{3}{\sqrt{7}}v_1$
C) $v_2 = \frac{2}{\sqrt{3}}v_1$
D) $v_2 = \frac{\sqrt{7}}{\sqrt{3}}v_1$
E) $v_2 = 4v_1$
2015 ASDAN Math Tournament, 2
Find the sum of the squares of the roots of $x^2-5x-7$.