Found problems: 85335
Kvant 2024, M2821
Peter and Basil take turns drawing roads on a plane, Peter starts. The road is either horizontal or a vertical line along which one can drive in only one direction (that direction is determined when the road is drawn). Can Basil always act in such a way that after each of his moves one could drive according to the rules between any two constructed crossroads, regardless of Peter's actions?
Alexandr Perepechko
2006 Polish MO Finals, 2
Find all positive integers $k$ for which number $3^k+5^k$ is a power of some integer with exponent greater than $1$.
1976 Spain Mathematical Olympiad, 2
Consider the set $C$ of all $r$ -tuple whose components are $1$ or $-1$. Calculate the sum of all the components of all the elements of $C$ excluding the $ r$ -tuple $(1, 1, 1, . . . , 1)$.
2017 AMC 12/AHSME, 7
The functions $\sin(x)$ and $\cos(x)$ are periodic with least period $2\pi$. What is the least period of the function $\cos(\sin(x))$?
$\textbf{(A)}\ \frac{\pi}{2}\qquad\textbf{(B)}\ \pi\qquad\textbf{(C)}\ 2\pi\qquad\textbf{(D)}\ 4\pi\qquad\textbf{(E)}$ It's not periodic.
2002 China Team Selection Test, 3
For positive integers $a,b,c$ let $ \alpha, \beta, \gamma$ be pairwise distinct positive integers such that
\[ \begin{cases}{c} \displaystyle a &= \alpha + \beta + \gamma, \\
b &= \alpha \cdot \beta + \beta \cdot \gamma + \gamma \cdot \alpha, \\
c^2 &= \alpha\beta\gamma. \end{cases} \]
Also, let $ \lambda$ be a real number that satisfies the condition
\[\lambda^4 -2a\lambda^2 + 8c\lambda + a^2 - 4b = 0.\]
Prove that $\lambda$ is an integer if and only if $\alpha, \beta, \gamma$ are all perfect squares.
1988 IMO Longlists, 14
Let $ a$ and $ b$ be two positive integers such that $ a \cdot b \plus{} 1$ divides $ a^{2} \plus{} b^{2}$. Show that $ \frac {a^{2} \plus{} b^{2}}{a \cdot b \plus{} 1}$ is a perfect square.
2016 Hanoi Open Mathematics Competitions, 4
A monkey in Zoo becomes lucky if he eats three different fruits. What is the largest number of monkeys one can make lucky, by having $20$ oranges, $30$ bananas, $40$ peaches and $50$ tangerines? Justify your answer.
(A): $30$ (B): $35$ (C): $40$ (D): $45$ (E): None of the above.
2001 IMO Shortlist, 5
Find all finite sequences $(x_0, x_1, \ldots,x_n)$ such that for every $j$, $0 \leq j \leq n$, $x_j$ equals the number of times $j$ appears in the sequence.
1995 Tournament Of Towns, (477) 1
If P is a point inside a convex quadrilateral $ABCD$, let the angle bisectors of $\angle APB$, $\angle BPC$, $\angle CPD$ and $\angle DPA$ meet $AB$, $BC$, $CD$ and $DA$ at $K$, $L$, $M$ and $N$ respectively.
(a) Find a point $P$ such that $KLMN$ is a parallelogram.
(b) Find the locus of all such points $P$.
(S Tokarev)
2007 Bosnia and Herzegovina Junior BMO TST, 3
Is it possible to place some circles inside a square side length $1$, such that no two circles intersect and the sum of their radii is $2007$?
2012 Romania National Olympiad, 4
[color=darkred]Find all differentiable functions $f\colon [0,\infty)\to [0,\infty)$ for which $f(0)=0$ and $f^{\prime}(x^2)=f(x)$ for any $x\in [0,\infty)$ .[/color]
2024 Philippine Math Olympiad, P8
Find all positive integers $n$ for wich $\phi(\phi (n))$ divides $n$.
2002 China Team Selection Test, 2
$ m$ and $ n$ are positive integers. In a $ 8 \times 8$ chessboard, $ (m,n)$ denotes the number of grids a Horse can jump in a chessboard ($ m$ horizontal $ n$ vertical or $ n$ horizontal $ m$ vertical ). If a $ (m,n) \textbf{Horse}$ starts from one grid, passes every grid once and only once, then we call this kind of Horse jump route a $ \textbf{H Route}$. For example, the $ (1,2) \textbf{Horse}$ has its $ \textbf{H Route}$. Find the smallest positive integer $ t$, such that from any grid of the chessboard, the $ (t,t\plus{}1) \textbf{Horse}$ does not has any $ \textbf{H Route}$.
1999 IMO Shortlist, 4
For a triangle $T = ABC$ we take the point $X$ on the side $(AB)$ such that $AX/AB=4/5$, the point $Y$ on the segment $(CX)$ such that $CY = 2YX$ and, if possible, the point $Z$ on the ray ($CA$ such that $\widehat{CXZ} = 180 - \widehat{ABC}$. We denote by $\Sigma$ the set of all triangles $T$ for which
$\widehat{XYZ} = 45$. Prove that all triangles from $\Sigma$ are similar and find the measure of their smallest angle.
2021/2022 Tournament of Towns, P4
What is the minimum $k{}$ for which among any three nonzero real numbers there are two numbers $a{}$ and $b{}$ such that either $|a-b|\leqslant k$ or $|1/a-1/b|\leqslant k$?
[i]Maxim Didin[/i]
1996 May Olympiad, 2
Considering the three-digit natural numbers, how many of them, when adding two of their digits, are double of their remainder? Justify your answer.
2022 Serbia Team Selection Test, P6
Let $ABCD$ be a trapezoid with bases $AB,CD$ such that $CD=k \cdot AB$ ($0<k<1$). Point $P$ is such that $\angle PAB=\angle CAD$ and $\angle PBA=\angle DBC$. Prove that $PA+PB \leq \dfrac{1}{\sqrt{1-k^2}} \cdot AB$.
1993 Italy TST, 1
Let $x_1,x_2,...,x_n$ ($n \ge 2$) be positive numbers with the sum $1$. Prove that
$$\sum_{i=1}^{n} \frac{1}{\sqrt{1-x_i}} \ge n\sqrt{\frac{n}{n-1}} $$
2010 Contests, 3
A triangle $ ABC$ is inscribed in a circle $ C(O,R)$ and has incenter $ I$. Lines $ AI,BI,CI$ meet the circumcircle $ (O)$ of triangle $ ABC$ at points $ D,E,F$ respectively. The circles with diameter $ ID,IE,IF$ meet the sides $ BC,CA, AB$ at pairs of points $ (A_1,A_2), (B_1, B_2), (C_1, C_2)$ respectively.
Prove that the six points $ A_1,A_2, B_1, B_2, C_1, C_2$ are concyclic.
Babis
1988 IMO, 3
A function $ f$ defined on the positive integers (and taking positive integers values) is given by:
$ \begin{matrix} f(1) \equal{} 1, f(3) \equal{} 3 \\
f(2 \cdot n) \equal{} f(n) \\
f(4 \cdot n \plus{} 1) \equal{} 2 \cdot f(2 \cdot n \plus{} 1) \minus{} f(n) \\
f(4 \cdot n \plus{} 3) \equal{} 3 \cdot f(2 \cdot n \plus{} 1) \minus{} 2 \cdot f(n), \end{matrix}$
for all positive integers $ n.$ Determine with proof the number of positive integers $ \leq 1988$ for which $ f(n) \equal{} n.$
2010 Putnam, A2
Find all differentiable functions $f:\mathbb{R}\to\mathbb{R}$ such that
\[f'(x)=\frac{f(x+n)-f(x)}n\]
for all real numbers $x$ and all positive integers $n.$
1983 IMO Longlists, 67
The altitude from a vertex of a given tetrahedron intersects the opposite face in its orthocenter. Prove that all four altitudes of the tetrahedron are concurrent.
1999 Argentina National Olympiad, 5
A rectangle-shaped puzzle is assembled with $2000$ pieces that are all equal rectangles, and similar to the large rectangle, so that the sides of the small rectangles are parallel to those of the large one. The shortest side of each piece measures $1$. Determine what is the minimum possible value of the area of the large rectangle.
2023 Novosibirsk Oral Olympiad in Geometry, 7
Squares $ABCD$ and $BEFG$ are located as shown in the figure. It turned out that points $A, G$ and $E$ lie on the same straight line. Prove that then the points $D, F$ and $E$ also lie on the same line.
[img]https://cdn.artofproblemsolving.com/attachments/4/2/9faf29a399d3a622c84f5d4a3cfcf5e99539c0.png[/img]
2019 India IMO Training Camp, P1
Let $\mathbb{Q}_{>0}$ denote the set of all positive rational numbers. Determine all functions $f:\mathbb{Q}_{>0}\to \mathbb{Q}_{>0}$ satisfying $$f(x^2f(y)^2)=f(x)^2f(y)$$ for all $x,y\in\mathbb{Q}_{>0}$