Found problems: 85335
1962 IMO, 3
Consider the cube $ABCDA'B'C'D'$ ($ABCD$ and $A'B'C'D'$ are the upper and lower bases, repsectively, and edges $AA', BB', CC', DD'$ are parallel). The point $X$ moves at a constant speed along the perimeter of the square $ABCD$ in the direction $ABCDA$, and the point $Y$ moves at the same rate along the perimiter of the square $B'C'CB$ in the direction $B'C'CBB'$. Points $X$ and $Y$ begin their motion at the same instant from the starting positions $A$ and $B'$, respectively. Determine and draw the locus of the midpionts of the segments $XY$.
2008 Balkan MO Shortlist, A7
Let $x,y,z,t \in \mathbb{R}_{\geq 0}$. Show
\begin{align*} \sqrt{xy}+\sqrt{xz}+\sqrt{xt}+\sqrt{yz}+\sqrt{yt}+\sqrt{zt} \geq 3 \sqrt[3]{xyz+xyt+xzt+yzt} \end{align*}
and determine the equality cases.
2005 Tournament of Towns, 2
Two ants crawl along the perimeter of a polygonal table, so that the distance between them is always 10 cm. Each side of the table is more than 1 meter long. At the initial moment both ants are on the same side of the table.
(a) [i](2 points)[/i] Suppose that the table is a convex polygon. Is it always true that both ants can visit each point on the perimeter?
(b) [i](3 points)[/i] Is it always true (this time without assumption of convexity) that each point on the perimeter can be visited by at least one ant?
2008 Germany Team Selection Test, 3
Find all real polynomials $ f$ with $ x,y \in \mathbb{R}$ such that
\[ 2 y f(x \plus{} y) \plus{} (x \minus{} y)(f(x) \plus{} f(y)) \geq 0.
\]
2019 New Zealand MO, 4
Show that the number $122^n - 102^n - 21^n$ is always one less than a multiple of $2020$, for any positive integer $n$.
1957 AMC 12/AHSME, 12
Comparing the numbers $ 10^{\minus{}49}$ and $ 2\cdot 10^{\minus{}50}$ we may say:
$ \textbf{(A)}\ \text{the first exceeds the second by }{8\cdot 10^{\minus{}1}}\qquad\\
\textbf{(B)}\ \text{the first exceeds the second by }{2\cdot 10^{\minus{}1}}\qquad \\
\textbf{(C)}\ \text{the first exceeds the second by }{8\cdot 10^{\minus{}50}}\qquad \\
\textbf{(D)}\ \text{the second is five times the first}\qquad \\
\textbf{(E)}\ \text{the first exceeds the second by }{5}$
2017 IOM, 2
In a country there are two-way non-stopflights between some pairs of cities. Any city can be reached from any other by a sequence of at most $100$ flights. Moreover, any city can be reached from any other by a sequence of an even number of flights. What is the smallest $d$ for which one can always claim that any city can be reached from any other by a sequence of an even number of flights not exceeding $d$?
1984 Putnam, A4
A convex pentagon $P=ABCDE$ is inscribed in a circle of radius $1$. Find the maximum area of $P$ subject to the condition that the chords $AC$ and $BD$ are perpendicular.
2017 239 Open Mathematical Olympiad, 5
A school has three classes. Some pairs of children from different classes are enemies (there are no enemies in a class). It is known that every child from the first class has as many enemies in the second class as in the third; the same is true for other classes. Prove that the number of pairs of children from classes having a common enemy is not less than the number of pairs of children being enemies.
2023 Pan-American Girls’ Mathematical Olympiad, 2
In each cell of an \(n \times n\) grid, one of the numbers \(0\), \(1,\) or \(2\) must be written. Determine all positive integers \(n\) for which there exists a way to fill the \(n \times n\) grid such that, when calculating the sum of the numbers in each row and each column, the numbers \(1, 2, \ldots, 2n\) are obtained in some order.
2022 Baltic Way, 17
Let $n$ be a positive integer such that the sum of its positive divisors is at least $2022n$. Prove that $n$ has at least $2022$ distinct prime factors.
2017 Hanoi Open Mathematics Competitions, 5
Write $2017$ following numbers on the blackboard: $-\frac{1008}{1008}, -\frac{1007}{1008}, ..., -\frac{1}{1008}, 0,\frac{1}{1008},\frac{2}{1008}, ... ,\frac{1007}{1008},\frac{1008}{1008}$ .
One processes some steps as: erase two arbitrary numbers $x, y$ on the blackboard and then write on it the number $x + 7xy + y$. After $2016$ steps, there is only one number. The last one on the blackboard is
(A): $-\frac{1}{1008}$ (B): $0$ (C): $\frac{1}{1008}$ (D): $-\frac{144}{1008}$ (E): None of the above
2004 Romania National Olympiad, 3
Let $H$ be the orthocenter of the acute triangle $ABC$. Let $BB'$ and $CC'$ be altitudes of the triangle ($B^{\prime} \in AC$, $C^{\prime} \in AB$). A variable line $\ell$ passing through $H$ intersects the segments $[BC']$ and $[CB']$ in $M$ and $N$. The perpendicular lines of $\ell$ from $M$ and $N$ intersect $BB'$ and $CC'$ in $P$ and $Q$. Determine the locus of the midpoint of the segment $[ PQ]$.
[i]Gheorghe Szolosy[/i]
2001 Tournament Of Towns, 2
At the end of the school year it became clear that for any arbitrarily chosen group of no less than 5 students, 80% of the marks “F” received by this group were given to no more than 20% of the students in the group. Prove that at least 3/4 of all “F” marks were given to the same student.
2020 Purple Comet Problems, 19
Right $\vartriangle ABC$ has side lengths $6, 8$, and $10$. Find the positive integer $n$ such that the area of the region inside the circumcircle but outside the incircle of $\vartriangle ABC$ is $n\pi$.
[img]https://cdn.artofproblemsolving.com/attachments/d/1/cb112332069c09a3b370343ca8a2ef21102fe2.png[/img]
2016 PUMaC Number Theory A, 5
Let $k = 2^6 \cdot 3^5 \cdot 5^2 \cdot 7^3 \cdot 53$.
Let $S$ be the sum of $\frac{gcd(m,n)}{lcm(m,n)}$ over all ordered pairs of positive integers $(m, n)$ where $mn = k$.
If $S$ can be written in simplest form as $\frac{r}{s}$, compute $r + s$.
2016 Romania National Olympiad, 2
Let be a function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ satisfying the conditions:
$$ \left\{\begin{matrix} f(x+y) &\le & f(x)+f(y) \\ f(tx+(1-t)y) &\le & t(f(x)) +(1-t)f(y) \end{matrix}\right. , $$
for all real numbers $ x,y,t $ with $ t\in [0,1] . $
Prove that:
[b]a)[/b] $ f(b)+f(c)\le f(a)+f(d) , $ for any real numbers $ a,b,c,d $ such that $ a\le b\le c\le d $ and $ d-c=b-a. $
[b]b)[/b] for any natural number $ n\ge 3 $ and any $ n $ real numbers $ x_1,x_2,\ldots ,x_n, $ the following inequality holds.
$$ f\left( \sum_{1\le i\le n} x_i \right) +(n-2)\sum_{1\le i\le n} f\left( x_i \right)\ge \sum_{1\le i<j\le n} f\left( x_i+x_j \right) $$
2008 Brazil Undergrad MO, 3
Prove that there are real numbers $ a_1, a_2, ..$ such that:
i) For all real numbers x, the serie $ f(x) \equal{} \sum_{n \equal{} 1}^\infty a_nx^n$ converge;
ii) f is a bijection of R to R;
iii) f'(x) >0;
iv) f(Q) = A, where Q is the set of rational numbers and A is the set of algebraic numbers.
2022-IMOC, N5
Find all solution $(p,r)$ of the "Pythagorean-Euler Theorem" $$p^p+(p+1)^p+\cdots+(p+r)^p=(p+r+1)^p$$Where $p$ is a prime and $r$ is a positive integer.
[i]Proposed by Li4 and Untro368[/i]
2014 India Regional Mathematical Olympiad, 3
let $m,n$ be natural number with $m>n$ . find all such pairs of $(m,n) $
such that
$gcd(n+1,m+1)=gcd(n+2,m+2) =..........=gcd(m, 2m-n) = 1 $
2015 Bulgaria National Olympiad, 3
The sequence $a_1, a_2,...$ is de fined by the equalities $a_1 = 2, a_2 = 12$ and $a_{n+1} = 6a_n-a_{n-1}$ for every positive integer $n \ge 2$. Prove that no member of this sequence is equal to a perfect power (greater than one) of a positive integer.
2020 European Mathematical Cup, 4
Let \(a,b,c\) be positive real numbers such that \(ab+bc+ac = a+b+c\). Prove the following inequality:
\[\sqrt{a+\frac{b}{c}} + \sqrt{b+\frac{c}{a}} + \sqrt{c+\frac{a}{b}} \leq \sqrt{2} \cdot \min \left\{ \frac{a}{b}+\frac{b}{c}+\frac{c}{a},\ \frac{b}{a}+\frac{c}{b}+\frac{a}{c} \right\}.\] \\ \\ [i]Proposed by Dorlir Ahmeti.[/i]
2023 Yasinsky Geometry Olympiad, 4
Let $C$ be one of the two points of intersection of circles $\omega_1$ and $\omega_2$ with centers at points $O_1$ and $O_2$, respectively. The line $O_1O_2$ intersects the circles at points $A$ and $B$ as shown in the figure. Let $K$ be the second point of intersection of line $AC$ with circle $\omega_2$, $L$ be the second point of intersection of line $BC$ with circle $\omega_1$. Lines $AL$ and $BK$ intersect at point $D$. Prove that $AD=BD$.
(Yurii Biletskyi)
[img]https://cdn.artofproblemsolving.com/attachments/6/4/2cdccb43743fcfcb155e846a0e05ec79ba90e4.png[/img]
2022 Junior Balkan Team Selection Tests - Moldova, 6
The non-negative numbers $x,y,z$ satisfy the relation $x + y+ z = 3$. Find the smallest possible numerical value and the largest possible numerical value for the expression
$$E(x,y, z) = \sqrt{x(y + 3)} + \sqrt{y(z + 3)} + \sqrt{z(x + 3)} .$$
1997 Nordic, 2
Let $ABCD$ be a convex quadrilateral. We assume that there exists a point $P$ inside the quadrilateral such that
the areas of the triangles $ABP, BCP, CDP$, and $DAP$ are equal. Show that at least one of the diagonals of the
quadrilateral bisects the other diagonal.