Found problems: 85335
2009 Estonia Team Selection Test, 3
Find all natural numbers $n$ for which there exists a convex polyhedron satisfying the following conditions:
(i) Each face is a regular polygon.
(ii) Among the faces, there are polygons with at most two different numbers of edges.
(iii) There are two faces with common edge that are both $n$-gons.
1996 Moldova Team Selection Test, 9
Let $x_1,x_2,...,x_n \in [0;1]$ prove that
$x_1(1-x_2)+x_2(1-x_3)+...+x_{n-1}(1-x_n)+x_n(1-x_1) \leq [\frac{n}{2}]$
2018 Dutch Mathematical Olympiad, 1
We call a positive integer a [i]shuffle[/i] number if the following hold:
(1) All digits are nonzero.
(2) The number is divisible by $11$.
(3) The number is divisible by $12$. If you put the digits in any other order, you again have a number that is divisible by $12$.
How many $10$-digit [i]shuffle[/i] numbers are there?
1956 Moscow Mathematical Olympiad, 333
Let $O$ be the center of the circle circumscribed around $\vartriangle ABC$, let $A_1, B_1, C_1$ be symmetric to $O$ through respective sides of $\vartriangle ABC$. Prove that all altitudes of $\vartriangle A_1B_1C_1$ pass through $O$, and all altitudes of $\vartriangle ABC$ pass through the center of the circle circumscribed around $\vartriangle A_1B_1C_1$.
2018 All-Russian Olympiad, 5
On the table, there're $1000$ cards arranged on a circle. On each card, a positive integer was written so that all $1000$ numbers are distinct. First, Vasya selects one of the card, remove it from the circle, and do the following operation: If on the last card taken out was written positive integer $k$, count the $k^{th}$ clockwise card not removed, from that position, then remove it and repeat the operation. This continues until only one card left on the table. Is it possible that, initially, there's a card $A$ such that, no matter what other card Vasya selects as first card, the one that left is always card $A$?
1955 AMC 12/AHSME, 48
Given triangle $ ABC$ with medians $ AE$, $ BF$, $ CD$; $ FH$ parallel and equal to $ AE$; $ BH$ and $ HE$ are drawn; $ FE$ extended meets $ BH$ in $ G$. Which one of the following statements is not necessarily correct?
$ \textbf{(A)}\ AEHF \text{ is a parallelogram} \qquad
\textbf{(B)}\ HE\equal{}HG \\
\textbf{(C)}\ BH\equal{}DC \qquad
\textbf{(D)}\ FG\equal{}\frac{3}{4}AB \qquad
\textbf{(E)}\ FG\text{ is a median of triangle }BFH$
2009 China Team Selection Test, 3
Let nonnegative real numbers $ a_{1},a_{2},a_{3},a_{4}$ satisfy $ a_{1} \plus{} a_{2} \plus{} a_{3} \plus{} a_{4} \equal{} 1.$ Prove that
$ max\{\sum_{1}^4{\sqrt {a_{i}^2 \plus{} a_{i}a_{i \minus{} 1} \plus{} a_{i \minus{} 1}^2 \plus{} a_{i \minus{} 1}a_{i \minus{} 2}}},\sum_{1}^4{\sqrt {a_{i}^2 \plus{} a_{i}a_{i \plus{} 1} \plus{} a_{i \plus{} 1}^2 \plus{} a_{i \plus{} 1}a_{i \plus{} 2}}}\}\ge 2.$
Where for all integers $ i, a_{i \plus{} 4} \equal{} a_{i}$ holds.
2023 IFYM, Sozopol, 8
Given an acute triangle $ABC$ with altitudes $AA_1$, $BB_1$, and $CC_1$ ($A_1 \in BC$, $B_1 \in AC$, $C_1 \in AB$) and circumcircle $k$, the rays $B_1A_1$, $C_1B_1$, and $A_1C_1$ meet $k$ at points $A_2$, $B_2$, and $C_2$, respectively. Find the maximum possible value of
\[
\sin \angle ABB_2 \cdot \sin \angle BCC_2 \cdot \sin \angle CAA_2
\]
and all acute triangles $ABC$ for which it is achieved.
2016 Switzerland Team Selection Test, Problem 9
Find all functions $f : \mathbb{R} \mapsto \mathbb{R} $ such that
$$ \left(f(x)+y\right)\left(f(x-y)+1\right)=f\left(f(xf(x+1))-yf(y-1)\right)$$
for all $x,y \in \mathbb{R}$
2014 IMAC Arhimede, 1
The function $f: N \to N_0$ is such that $f (2) = 0, f (3)> 0, f (6042) = 2014$ and $f (m + n)- f (m) - f (n) \in\{0,1\}$ for all $m,n \in N$. Determine $f (2014)$. $N_0=\{0,1,2,...\}$
2017 Iran Team Selection Test, 6
In triangle $ABC$ let $O$ and $H$ be the circumcenter and the orthocenter. The point $P$ is the reflection of $A$ with respect to $OH$. Assume that $P$ is not on the same side of $BC$ as $A$. Points $E,F$ lie on $AB,AC$ respectively such that $BE=PC \ , CF=PB$. Let $K$ be the intersection point of $AP,OH$. Prove that $\angle EKF = 90 ^{\circ}$
[i] Proposed by Iman Maghsoudi[/i]
2008 Estonia Team Selection Test, 3
Let $ n$ be a positive integer, and let $ x$ and $ y$ be a positive real number such that $ x^n \plus{} y^n \equal{} 1.$ Prove that
\[ \left(\sum^n_{k \equal{} 1} \frac {1 \plus{} x^{2k}}{1 \plus{} x^{4k}} \right) \cdot \left( \sum^n_{k \equal{} 1} \frac {1 \plus{} y^{2k}}{1 \plus{} y^{4k}} \right) < \frac {1}{(1 \minus{} x) \cdot (1 \minus{} y)}.
\]
[i]Author: Juhan Aru, Estonia[/i]
1963 Miklós Schweitzer, 6
Show that if $ f(x)$ is a real-valued, continuous function on the half-line $ 0\leq x < \infty$, and \[ \int_0^{\infty} f^2(x)dx
<\infty\] then the function \[ g(x)\equal{}f(x)\minus{}2e^{\minus{}x}\int_0^x e^tf(t)dt\] satisfies \[ \int _0^{\infty}g^2(x)dx\equal{}\int_0^{\infty}f^2(x)dx.\] [B. Szokefalvi-Nagy]
2016 South East Mathematical Olympiad, 6
Toss the coin $n$ times, assume that each time, only appear only head or tail
Let $a(n)$ denote number of way that head appear in multiple of $3$ times among $n$ times
Let $b(n)$ denote numbe of way that head appear in multiple of $6$ times among $n$ times
$(1)$ Find $a(2016)$ and $b(2016)$
$(2)$ Find the number of positive integer $n\leq 2016$ that $2b(n)-a(n)\geq 0$
2025 Olympic Revenge, 5
DK plays the following game in a simple graph: in each round, he does one of the two operations:
([i]i[/i]) Choose a vertex of odd degree and delete it. Before doing that, DK changes the relation between every two
neighbors of the chosen vertex (that is, if they were connected by an edge, then remove this edge, and, if this edge did not exist, then put this edge on the graph).
([i]ii[/i]) Choose a vertex of even degree and change the relation between every two neighbors of it (note that the chosen vertex is not deleted).
DK plays this game until there are no more edges on the graph. Show that the number of remaining vertices does not depend on the chosen operations.
LMT Accuracy Rounds, 2021 F3
Two circles with radius $2$, $\omega_1$ and $\omega_2$, are centered at $O_1$ and $O_2$ respectively. The circles $\omega_1$ and $\omega_2$ are externally tangent to each other and internally tangent to a larger circle $\omega$ centered at $O$ at points $A$ and $B$, respectively. Let $M$ be the midpoint of minor arc $AB$. Let $P$ be the intersection of $\omega_1$ and $O_1M$, and let $Q$ be the intersection of $\omega_2$ and $O_2M$. Given that there is a point $R$ on $\omega$ such that $\triangle PQR$ is equilateral, the radius of $\omega$ can be written as $\frac{a+\sqrt{b}}{c}$ where $a$, $b$, and $c$ are positive integers and $a$ and $c$ are relatively prime. Find $a+b+c$.
2019 AIME Problems, 6
In a Martian civilization, all logarithms whose bases are not specified are assumed to be base $b$, for some fixed $b \geq 2$. A Martian student writes down
\begin{align*}3 \log(\sqrt{x}\log x) &= 56\\\log_{\log (x)}(x) &= 54
\end{align*}
and finds that this system of equations has a single real number solution $x > 1$. Find $b$.
1990 IMO Longlists, 14
We call a set $S$ on the real line $R$ "superinvariant", if for any stretching $A$ of the set $S$ by the transformation taking $x$ to $A(x) = x_0 + a(x - x_0)$, where $a > 0$, there exists a transformation $B, B(x) = x + b$, such that the images of $S$ under $A$ and $B$ agree; i.e., for any $x \in S$, there is $y \in S$ such that $A(x) = B(y)$, and for any $t \in S$, there is a $u \in S$ such that $B(t) = A(u).$ Determine all superinvariant sets.
PEN O Problems, 57
Prove that every selection of $1325$ integers from $M=\{1, 2, \cdots, 1987 \}$ must contain some three numbers $\{a, b, c\}$ which are pairwise relatively prime, but that it can be avoided if only $1324$ integers are selected.
2002 Moldova Team Selection Test, 2
Let $S= \{ a_1, \ldots, a_n\}$ be a set of $n\geq 1$ positive real numbers. For each nonempty subset of $S$ the sum of its elements is written down. Show that all written numbers can be divided into $n$ classes such that in each class the ratio of the greatest number to the smallest number is not greater than $2$.
2013 Hanoi Open Mathematics Competitions, 3
The largest integer not exceeding $[(n+1)a]-[na]$ where $n$ is a natural number, $a=\frac{\sqrt{2013}}{\sqrt{2014}}$ is:
(A): $1$, (B): $2$, (C): $3$, (D): $4$, (E) None of the above.
2018 All-Russian Olympiad, 8
Initially, on the lower left and right corner of a $2018\times 2018$ board, there're two horses, red and blue, respectively. $A$ and $B$ alternatively play their turn, $A$ start first. Each turn consist of moving their horse ($A$-red, and $B$-blue) by, simultaneously, $20$ cells respect to one coordinate, and $17$ cells respect to the other; while preserving the rule that the horse can't occupied the cell that ever occupied by any horses in the game. The player who can't make the move loss, who has the winning strategy?
2000 Harvard-MIT Mathematics Tournament, 7
$8712$ is an integral multiple of its reversal, $2178$, as $8712=4 * 2178$. Find another $4$-digit number which is a non-trivial integral multiple of its reversal.
2016 Tournament Of Towns, 4
There are $2016$ red and $2016$ blue cards each having a number written on it. For some $64$ distinct positive real numbers, it is known that the set of numbers on cards of a particular color happens to be the set of their pairwise sums and the other happens to be the set of their pairwise products. Can we necessarily determine which color corresponds to sum and which to product?
[i](B. Frenkin)[/i]
(Translated from [url=http://sasja.shap.homedns.org/Turniry/TG/index.html]here.[/url])
1996 Bosnia and Herzegovina Team Selection Test, 1
$a)$ Let $a$, $b$ and $c$ be positive real numbers. Prove that for all positive integers $m$ holds: $$(a+b)^m+(b+c)^m+(c+a)^m \leq 2^m(a^m+b^m+c^m)$$
$b)$ Does inequality $a)$ holds for
$1)$ arbitrary real numbers $a$, $b$ and $c$
$2)$ any integer $m$