Found problems: 85335
2018 International Zhautykov Olympiad, 5
Find all real numbers $a$ such that there exist $f:\mathbb{R} \to \mathbb{R}$ with $$f(x-f(y))=f(x)+a[y]$$ for all $x,y\in \mathbb{R}$
2014 Iran Team Selection Test, 1
Consider a tree with $n$ vertices, labeled with $1,\ldots,n$ in a way that no label is used twice. We change the labeling in the following way - each time we pick an edge that hasn't been picked before and swap the labels of its endpoints. After performing this action $n-1$ times, we get another tree with its labeling a permutation of the first graph's labeling.
Prove that this permutation contains exactly one cycle.
2024 Auckland Mathematical Olympiad, 9
$100$ students came to a party. The students who did not have friends among other students left the party first. Then those with one friend among remaining students left. Then those with $2,3, \ldots 99$ friends among remaining students left. What is the maximal number of students that can still remain at the party after that? (If $A$ is a friend of $B$, then $B$ is a friend of $A$).
2003 JHMMC 8, 19
Two angles are supplementary, and one angle is $9$ times as large as the other. What is the number of
degrees in the measure of the larger angle?
1978 IMO Longlists, 33
A sequence $(a_n)^{\infty}_0$ of real numbers is called [i]convex[/i] if $2a_n\le a_{n-1}+a_{n+1}$ for all positive integers $n$. Let $(b_n)^{\infty}_0$ be a sequence of positive numbers and assume that the sequence $(\alpha^nb_n)^{\infty}_0$ is convex for any choice of $\alpha > 0$. Prove that the sequence $(\log b_n)^{\infty}_0$ is convex.
2008 Princeton University Math Competition, A5
If $f(x) = x^{x^{x^x}}$ , find the last two digits of $f(17) + f(18) + f(19) + f(20)$.
1981 IMO Shortlist, 13
Let $P$ be a polynomial of degree $n$ satisfying
\[P(k) = \binom{n+1}{k}^{-1} \qquad \text{ for } k = 0, 1, . . ., n.\]
Determine $P(n + 1).$
2019 Purple Comet Problems, 7
Find the number of real numbers $x$ that satisfy the equation $(3^x)^{x+2} + (4^x)^{x+2} - (6^x)^{x+2} = 1$
2008 China Team Selection Test, 2
Prove that for all $ n\geq 2,$ there exists $ n$-degree polynomial $ f(x) \equal{} x^n \plus{} a_{1}x^{n \minus{} 1} \plus{} \cdots \plus{} a_{n}$ such that
(1) $ a_{1},a_{2},\cdots, a_{n}$ all are unequal to $ 0$;
(2) $ f(x)$ can't be factorized into the product of two polynomials having integer coefficients and positive degrees;
(3) for any integers $ x, |f(x)|$ isn't prime numbers.
2007 Hanoi Open Mathematics Competitions, 8
Let a; b; c be positive integers. Prove that
$$ \frac{(b+c-a)^2}{(b+c)^2+a^2} + \frac{(c+a-b)^2}{(c+a)^2+b^2} + \frac{(a+b-c)^2}{(a+b)^2+c^2} \geq \frac{3}{5}$$
2023 AMC 12/AHSME, 19
What is the product of all the solutions to the equation $$\log_{7x}2023 \cdot \log_{289x} 2023 = \log_{2023x} 2023?$$
$\textbf{(A) }(\log_{2023}7 \cdot \log_{2023}289)^2 \qquad\textbf{(B) }\log_{2023}7 \cdot \log_{2023}289\qquad\textbf{(C) }1\qquad\textbf{(D) }\log_{7}2023 \cdot \log_{289}2023\qquad\textbf{(E) }(\log_{7}2023 \cdot \log_{289}2023)^2$
2003 Estonia National Olympiad, 5
The game [i]Clobber [/i] is played by two on a strip of $2k$ squares. At the beginning there is a piece on each square, the pieces of both players stand alternatingly. At each move the player shifts one of his pieces to the neighbouring square that holds a piece of his opponent and removes his opponent’s piece from the table. The moves are made in turn, the player whose opponent cannot move anymore is the winner.
Prove that if for some $k$ the player who does not start the game has the winning strategy, then for $k + 1$ and $k + 2$ the player who makes the first move has the winning strategy.
2003 Argentina National Olympiad, 6
Determine the positive integers $n$ such that the set of all positive divisors of $30^n$ can be divided into groups of three so that the product of the three numbers in each group is the same.
2025 Iran MO (2nd Round), 3
Point $P$ lies inside of scalene triangle $ABC$ with incenter $I$ such that $:$
$$ 2\angle ABP = \angle BCA , 2\angle ACP = \angle CBA $$
Lines $PB$ and $PC$ intersect line $AI$ respectively at $B'$ and $C'$. Line through $B'$ parallel to $AB$ intersects $BI$ at $X$ and line through $C'$ parallel to $AC$ intersects $CI$ at $Y$. Prove that triangles $PXY$ and $ABC$ are similar.
Kyiv City MO Juniors 2003+ geometry, 2014.85
Given an equilateral $\Delta ABC$, in which ${{A} _ {1}}, {{B} _ {1}}, {{C} _ {1}}$ are the midpoint of the sides $ BC, \, \, AC, \, \, AB$ respectively. The line $l$ passes through the vertex $A$, we denote by $P, Q$ the projection of the points $B, C$ on the line $l$, respectively (the line $ l $ and the point $Q, \, \, A, \, \, P$ are located as shown in fig.). Denote by $T $ the intersection point of the lines ${{B} _ {1}} P$ and ${{C} _ {1}} Q$. Prove that the line ${{A} _ {1}} T$ is perpendicular to the line $l$.
[img]https://cdn.artofproblemsolving.com/attachments/4/b/61f2f4ec9e6b290dfcd47e9351110bebd3bd43.png[/img]
(Serdyuk Nazar)
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 $\angle BCF = \angle ACD$.