Found problems: 85335
2001 Estonia Team Selection Test, 2
Point $X$ is taken inside a regular $n$-gon of side length $a$. Let $h_1,h_2,...,h_n$ be the distances from $X$ to the lines defined by the sides of the $n$-gon. Prove that $\frac{1}{h_1}+\frac{1}{h_2}+...+\frac{1}{h_n}>\frac{2\pi}{a}$
2017 AMC 8, 4
When 0.000315 is multiplied by 7,928,564 the product is closest to which of the following?
$\textbf{(A) }210\qquad\textbf{(B) }240\qquad\textbf{(C) }2100\qquad\textbf{(D) }2400\qquad\textbf{(E) }24000$
2009 Romanian Master of Mathematics, 3
Given four points $ A_1, A_2, A_3, A_4$ in the plane, no three collinear, such that
\[ A_1A_2 \cdot A_3 A_4 \equal{} A_1 A_3 \cdot A_2 A_4 \equal{} A_1 A_4 \cdot A_2 A_3,
\]
denote by $ O_i$ the circumcenter of $ \triangle A_j A_k A_l$ with $ \{i,j,k,l\} \equal{} \{1,2,3,4\}.$ Assuming $ \forall i A_i \neq O_i ,$ prove that the four lines $ A_iO_i$ are concurrent or parallel.
[i]Nikolai Ivanov Beluhov, Bulgaria[/i]
2013 European Mathematical Cup, 2
Palindrome is a sequence of digits which doesn't change if we reverse the order of its digits. Prove that a sequence $(x_n)^{\infty}_{n=0}$ defined as
$x_n=2013+317n$
contains infinitely many numbers with their decimal expansions being palindromes.
2024 AMC 10, 6
What is the minimum number of successive swaps of adjacent letters in the string ABCDEF that are needed to change the string to FEDCBA? (For example, 3 swaps are required to change ABC to CBA; one such sequence of swaps is ABC $\rightarrow$ BAC $\rightarrow$ BCA $\rightarrow$ CBA.)
$
\textbf{(A) }6 \qquad
\textbf{(B) }10 \qquad
\textbf{(C) }12 \qquad
\textbf{(D) }15 \qquad
\textbf{(E) }24 \qquad
$
2020 Israel Olympic Revenge, P1
Find all functions $f:\mathbb{R}\to \mathbb{R}$ such that for all $x,y\in \mathbb{R}$ one has
\[f(f(x)+y)=f(x+f(y))\]
and in addition the set $f^{-1}(a)=\{b\in \mathbb{R}\mid f(b)=a\}$ is a finite set for all $a\in \mathbb{R}$.
1966 IMO Longlists, 5
Prove the inequality
\[\tan \frac{\pi \sin x}{4\sin \alpha} + \tan \frac{\pi \cos x}{4\cos \alpha} >1\]
for any $x, \alpha$ with $0 \leq x \leq \frac{\pi }{2}$ and $\frac{\pi}{6} < \alpha < \frac{\pi}{3}.$
2019 MIG, 25
Each day John's mother sends him to the store with $\$1$ to buy widgets and gadgets, each of which cost a whole number of cents. On the first day John comes back with $4$ widgets, $5$ gadgets, and $35$ cents in change. On the second day, John comes back with $5$ widgets, $4$ gadgets, and $39$ cents in change. On the third day, John comes back with only $c$ cents in change. He hands his mother the change, telling her that he had tripped coming home and broken all the widgets and gadgets. His mother, thinking for a moment, begins yelling at him for lying, as she noticed that there was no way he could have received exactly $c$ cents in change given the price of widgets and gadgets. What is the sum of the digits of the least possible value of $c$?
$\textbf{(A) }10\qquad\textbf{(B) }13\qquad\textbf{(C) }15\qquad\textbf{(D) }18\qquad\textbf{(E) }\text{impossible to determine}$
1983 Bulgaria National Olympiad, Problem 5
Can the polynomials $x^{5}-x-1$ and $x^{2}+ax+b$ , where $a,b\in Q$, have common complex roots?
1950 Miklós Schweitzer, 6
Consider an arc of a planar curve; let the radius of curvature at any point of the arc be a differentiable function of the arc length and its derivative be everywhere different from zero; moreover, let the total curvature be less than $ \frac{\pi}{2}$. Let $ P_1,P_2,P_3,P_4,P_5$ and $ P_6$ be any points on this arc, subject to the only condition that the radius of curvature at $ P_k$ is greater than at $ P_j$ if $ j<k$.
Prove that the radius of the circle passing through the points $ P_1,P_3$ and $ P_5$ is less than the radius of the circle through $ P_2,P_4$ and $ P_6$
2005 Swedish Mathematical Competition, 5
Every cell of a $2005 \times 2005$ square board is colored white or black so that every $2 \times 2$ subsquare contains an odd number of black cells.
Show that among the corner cells there is an even number of black ones. How many ways are there to color the board in this manner?
2013 Oral Moscow Geometry Olympiad, 4
Let $ABC$ be a triangle. On the extensions of sides $AB$ and $CB$ towards $B$, points $C_1$ and $A_1$ are taken, respectively, so that $AC = A_1C = AC_1$. Prove that circumscribed circles of triangles $ABA_1$ and $CBC_1$ intersect on the bisector of angle $B$.
2014 Saint Petersburg Mathematical Olympiad, 5
Incircle $\omega$ of $ABC$ touch $AC$ at $B_1$. Point $E,F$ on the $\omega$ such that $\angle AEB_1=\angle B_1FC=90$. Tangents to $\omega$ at $E,F$ intersects in $D$, and $B$ and $D$ are on different sides for line $AC$. $M$- midpoint of $AC$.
Prove, that $AE,CF,DM$ intersects at one point.
1970 IMO Longlists, 34
In connection with a convex pentagon $ABCDE$ we consider the set of ten circles, each of which contains three of the vertices of the pentagon on its circumference. Is it possible that none of these circles contains the pentagon? Prove your answer.
2022 Brazil National Olympiad, 3
Let $ABC$ be a triangle with incenter $I$ and let $\Gamma$ be its circumcircle. Let $M$ be the midpoint of $BC$, $K$ the midpoint of the arc $BC$ which does not contain $A$, $L$ the midpoint of the arc $BC$ which contains $A$ and $J$ the reflection of $I$ by the line $KL$. The line $LJ$ intersects $\Gamma$ again at the point $T\neq L$. The line $TM$ intersects $\Gamma$ again at the point $S\neq T$. Prove that $S, I, M, K$ lie on the same circle.
2005 iTest, 31
Let $X = 123456789$. Find the sum of the tens digits of all integral multiples of $11$ that can be obtained by interchanging two digits of $X$.
2021 CHMMC Winter (2021-22), 5
Find all functions $f : R \to R$ such that
$$f(f(x) + f(y)^2) = f(x)^2 +y^2f(y)^3.$$
Here $R$ denotes the usual real numbers.
2012 Kazakhstan National Olympiad, 1
Solve the equation $p+\sqrt{q^{2}+r}=\sqrt{s^{2}+t}$ in prime numbers.
Russian TST 2018, P2
Let $\mathcal{F}$ be a finite family of subsets of some set $X{}$. It is known that for any two elements $x,y\in X$ there exists a permutation $\pi$ of the set $X$ such that $\pi(x)=y$, and for any $A\in\mathcal{F}$ \[\pi(A):=\{\pi(a):a\in A\}\in\mathcal{F}.\]A bear and crocodile play a game. At a move, a player paints one or more elements of the set $X$ in his own color: brown for the bear, green for the crocodile. The first player to fully paint one of the sets in $\mathcal{F}$ in his own color loses. If this does not happen and all the elements of $X$ have been painted, it is a draw. The bear goes first. Prove that he doesn't have a winning strategy.
2014 Chile TST IMO, 2
Given \(n, k \in \mathbb{N}\), prove that \((n-1)^2\) divides \(n^k - 1\) if and only if \(n-1 \mid k\).
2006 Iran Team Selection Test, 4
Let $x_1,x_2,\ldots,x_n$ be real numbers. Prove that
\[ \sum_{i,j=1}^n |x_i+x_j|\geq n\sum_{i=1}^n |x_i| \]
Cono Sur Shortlist - geometry, 2005.G4.2
Let $ABC$ be an acute-angled triangle and let $AN$, $BM$ and $CP$ the altitudes with respect to the sides $BC$, $CA$ and $AB$, respectively. Let $R$, $S$ be the pojections of $N$ on the sides $AB$, $CA$, respectively, and let $Q$, $W$ be the projections of $N$ on the altitudes $BM$ and $CP$, respectively.
(a) Show that $R$, $Q$, $W$, $S$ are collinear.
(b) Show that $MP=RS-QW$.
2008 Abels Math Contest (Norwegian MO) Final, 1
Let $s(n) = \frac16 n^3 - \frac12 n^2 + \frac13 n$.
(a) Show that $s(n)$ is an integer whenever $n$ is an integer.
(b) How many integers $n$ with $0 < n \le 2008$ are such that $s(n)$ is divisible by $4$?
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.
2004 Oral Moscow Geometry Olympiad, 1
In a convex quadrilateral $ABCD$, $E$ is the midpoint of $CD$, $F$ is midpoint of $AD$, $K$ is the intersection point of $AC$ with $BE$. Prove that the area of triangle $BKF$ is half the area of triangle $ABC$.