Found problems: 85335
2012 AMC 12/AHSME, 14
Bernado and Silvia play the following game. An integer between 0 and 999, inclusive, is selected and given to Bernado. Whenever Bernado receives a number, he doubles it and passes the result to Silvia. Whenever Silvia receives a number, she adds 50 to it and passes the result to Bernado. The winner is the last person who produces a number less than 1000. Let $N$ be the smallest initial number that results in a win for Bernado. What is the sum of the digits of $N$?
$\textbf{(A)}\ 7 \qquad\textbf{(B)}\ 8 \qquad\textbf{(C)}\ 9 \qquad\textbf{(D)}\ 10 \qquad\textbf{(E)}\ 11$
2019 Tuymaada Olympiad, 4
A quota of diplomas at the All-Russian Olympiad should be strictly less than $45\%$. More than $20$ students took part in the olympiad. After the olympiad the Authorities declared the results low because the quota of diplomas was significantly less than $45\%$. The Jury responded that the quota was already maximum possible on this olympiad or any other olympiad with smaller number of participants. Then the Authorities ordered to increase the number of participants for the next olympiad so that the quota of diplomas became at least two times closer to $45\%$. Prove that the number of participants should be at least doubled.
2017 IMC, 2
Let $f:\mathbb R\to(0,\infty)$ be a differentiabe function, and suppose that there exists a constant $L>0$ such that
$$|f'(x)-f'(y)|\leq L|x-y|$$
for all $x,y$. Prove that
$$(f'(x))^2<2Lf(x)$$
holds for all $x$.
1986 AMC 12/AHSME, 8
The population of the United States in 1980 was $226,504,825$. The area of the country is $3,615,122$ square miles. The are $(5280)^{2}$ square feet in one square mile. Which number below best approximates the average number of square feet per person?
$ \textbf{(A)}\ 5,000\qquad\textbf{(B)}\ 10,000\qquad\textbf{(C)}\ 50,000\qquad\textbf{(D)}\ 100,000\qquad\textbf{(E)}\ 500,000 $
2018 Finnish National High School Mathematics Comp, 1
Eve and Martti have a whole number of euros.
Martti said to Eve: ''If you give give me three euros, so I have $n$ times the money compared to you. '' Eve in turn said to Martti: ''If you give me $n$ euros then I have triple the amount of money compared to you'' . Suppose, that both claims are valid.
What values can a positive integer $n$ get?
1963 Polish MO Finals, 2
In space there are given four distinct points $ A $, $ B $, $ C $, $ D $. Prove that the three segments connecting the midpoints of the segments $ AB $ and $ CD $, $ AC $ and $ BD $, $ AD $ and $ BC $ have a common midpoint.
2019 India PRMO, 19
Let $AB$ be a diameter of a circle and let $C$ be a point on the segement $AB$ such that $AC : CB = 6 : 7$. Let $D$ be a point on the circle such that $DC$ is perpendicular to $AB$. Let $DE$ be the diameter through $D$. If $[XYZ]$ denotes the area of the triangle $XYZ$, find $[ABD]/[CDE]$ to the nearest integer.
2009 Tournament Of Towns, 5
Suppose that $X$ is an arbitrary point inside a tetrahedron. Through each vertex of the tetrahedron, draw a straight line that is parallel to the line segment connecting $X$ with the intersection point of the medians of the opposite face. Prove that these four lines meet at the same point.
2010 Germany Team Selection Test, 2
Given a cyclic quadrilateral $ABCD$, let the diagonals $AC$ and $BD$ meet at $E$ and the lines $AD$ and $BC$ meet at $F$. The midpoints of $AB$ and $CD$ are $G$ and $H$, respectively. Show that $EF$ is tangent at $E$ to the circle through the points $E$, $G$ and $H$.
[i]Proposed by David Monk, United Kingdom[/i]
2019 Jozsef Wildt International Math Competition, W. 48
Let $f : (0,+\infty) \to \mathbb{R}$ a convex function and $\alpha, \beta, \gamma > 0$. Then $$\frac{1}{6\alpha}\int \limits_0^{6\alpha}f(x)dx\ +\ \frac{1}{6\beta}\int \limits_0^{6\beta}f(x)dx\ +\ \frac{1}{6\gamma}\int \limits_0^{6\gamma}f(x)dx$$ $$\geq \frac{1}{3\alpha +2\beta +\gamma}\int \limits_0^{3\alpha +2\beta +\gamma}f(x)dx\ +\ \frac{1}{\alpha +3\beta +2\gamma}\int \limits_0^{\alpha +3\beta +2\gamma}f(x)dx\ $$ $$+\ \frac{1}{2\alpha +\beta +3\gamma}\int \limits_0^{2\alpha +\beta +3\gamma}f(x)dx$$
1979 Miklós Schweitzer, 11
Let $ \{\xi_{k \ell} \}_{k,\ell=1}^{\infty}$ be a double sequence of random variables such that
\[ \Bbb{E}( \xi_{ij} \xi_{k\ell})= \mathcal{O} \left(\frac{ \log(2|i-k|+2)}{ \log(2|j-\ell|+2)^{2}}\right) \;\;\;(i,j,k,\ell =1,2, \ldots ) \\\ .\]
Prove that with probability one,
\[ \frac{1}{mn} \sum_{k=1}^m \sum_{\ell=1}^n \xi_{k\ell} \rightarrow 0 \;\;\textrm{as} \; \max (m,n)\rightarrow \infty\ \\ .\]
[i]F. Moricz[/i]
2012 USAMTS Problems, 3
The $\textbf{symmetric difference}$, $\triangle$, of a pair of sets is the set of elements in exactly one set. For example, \[\{1,2,3\}\triangle\{2,3,4\}=\{1,4\}.\] There are fifteen nonempty subsets of $\{1,2,3,4\}$. Assign each subset to exactly one of the squares in the grid to the right so that the following conditions are satisfied.
(i) If $A$ and $B$ are in squares connected by a solid line then $A\triangle B$ has exactly one element.
(ii) If $A$ and $B$ are in squares connected by a dashed line then the largest element of $A$ is equal to the largest element of $B$.
You do not need to prove that your configuration is the only one possible; you merely need to find a configuration that satisfies the constraints above. (Note: In any other USAMTS problem, you need to provide a full proof. Only in this problem is an answer without justification acceptable.)
[asy]
size(150);
defaultpen(linewidth(0.8));
draw((0,1)--(0,3)--(3,3)^^(2,3)--(2,2)--(3,2)--(3,1)--(1,1)--(1,2)--(0,2)^^(2,1)--(2,0)--(0,0));
draw(origin--(0,1)^^(1,0)--(3,2)^^(1,1)--(0,2)^^(1,2)--(0,3)^^(1,3)--(2,2),linetype("4 4"));
real r=1/4;
path square=(r,r)--(r,-r)--(-r,-r)--(-r,r)--cycle;
int limit;
for(int i=0;i<=3;i=i+1)
{
if (i==0)
limit=2;
else
limit=3;
for(int j=0;j<=limit;j=j+1)
filldraw(shift(j,i)*square,white);
}
[/asy]
2022-IMOC, G1
The circumcenter and orthocenter of $ABC$ are $O$ and $H$, respectively. Let $XACH$ be a parallelogram. Show that if $OH$ is parallel to $BC$, then $OX$ and $AB$ intersect at some point on the perpendicular bisector of $AH$.
[i]proposed by USJL[/i]
2007 Princeton University Math Competition, 10
if $x$, $y$, and $z$ are real numbers such that $ x^2 + z^2 = 1 $ and $ y^2 + 2y \left( x + z \right) = 6 $, find the maximum value of $ y \left( z - x \right) $.
2008 Thailand Mathematical Olympiad, 10
On the sides of triangle $\vartriangle ABC$, $17$ points are added, so that there are $20$ points in total (including the vertices of $\vartriangle ABC$.) What is the maximum possible number of (nondegenerate) triangles that can be formed by these points.
2005 Bosnia and Herzegovina Team Selection Test, 2
If $a_1$, $a_2$ and $a_3$ are nonnegative real numbers for which $a_1+a_2+a_3=1$, then prove the inequality $a_1\sqrt{a_2}+a_2\sqrt{a_3}+a_3\sqrt{a_1}\leq \frac{1}{\sqrt{3}}$
1969 Yugoslav Team Selection Test, Problem 5
Prove that the product of the sines of two opposite dihedrals in a tetrahedron is proportional to the product of the lengths of the edges of these dihedrals.
1998 AMC 8, 16
Problems 15, 16, and 17 all refer to the following:
In the very center of the Irenic Sea lie the beautiful Nisos Isles. In 1998 the number of people on these islands is only 200, but the population triples every 25 years. Queen Irene has decreed that there must be at least 1.5 square miles for every person living in the Isles. The total area of the Nisos Isles is 24,900 square miles.
16. Estimate the year in which the population of Nisos will be approximately 6,000.
$ \text{(A)}\ 2050\qquad\text{(B)}\ 2075\qquad\text{(C)}\ 2100\qquad\text{(D)}\ 2125\qquad\text{(E)}\ 2150 $
2024 Regional Olympiad of Mexico West, 4
Let $\triangle ABC$ be a triangle and $\omega$ its circumcircle. The tangent to $\omega$ through $B$ cuts the parallel to $BC$ through $A$ at $P$. The line $CP$ cuts the circumcircle of $\triangle ABP$ again in $Q$ and line $AQ$ cuts $\omega$ at $R$. Prove that $BQCR$ is parallelogram if and only if $AC=BC$.
2021 Iberoamerican, 1
Let $P = \{p_1,p_2,\ldots, p_{10}\}$ be a set of $10$ different prime numbers and let $A$ be the set of all the integers greater than $1$ so that their prime decomposition only contains primes of $P$. The elements of $A$ are colored in such a way that:
[list]
[*] each element of $P$ has a different color,
[*] if $m,n \in A$, then $mn$ is the same color of $m$ or $n$,
[*] for any pair of different colors $\mathcal{R}$ and $\mathcal{S}$, there are no $j,k,m,n\in A$ (not necessarily distinct from one another), with $j,k$ colored $\mathcal{R}$ and $m,n$ colored $\mathcal{S}$, so that $j$ is a divisor of $m$ and $n$ is a divisor of $k$, simultaneously.
[/list]
Prove that there exists a prime of $P$ so that all its multiples in $A$ are the same color.
2016 Romania National Olympiad, 2
In a cube $ABCDA'B'C'D' $two points are considered, $M \in (CD')$ and $N \in (DA')$. Show that the $MN$ is common perpendicular to the lines $CD'$ and $DA'$ if and only if $$\frac{D'M}{D'C}=\frac{DN}{DA'} =\frac{1}{3}.$$
2014 Peru MO (ONEM), 3
a) Let $a, b, c$ be positive integers such that $ab + b + 1$, $bc + c + 1$ and $ca + a + 1$ are divisors of the number $abc - 1$, prove that $a = b = c$.
b) Find all triples $(a, b, c)$ of positive integers such that the product $$(ab - b + 1)(bc - c + 1)(ca - a + 1)$$ is a divisor of the number $(abc + 1)^2$.
2016 Kosovo Team Selection Test, 5
Let be $ABC$ an acute triangle with $|AB|>|AC|$ . Let be $D$ point in side $AB$ such that $\angle ACD=\angle CBD$ . Let be $E$ the midpoint of segment $BD$ and $S$ let be the circumcenter of triangle $BCD$ . Show that points $A,E,S$ and $C$ lie on a circle .
2022 Belarusian National Olympiad, 9.5
Given $n \geq 2$ distinct integers, which are bigger than $-10$. It turned out that the amount of odd numbers among them is equal to the biggest even number, and the amount of even to the biggest of odd.
a) Find the smallest $n$ possible
b) Find the biggest $n$ possible
2000 Croatia National Olympiad, Problem 1
Let $\mathcal P$ be the parabola $y^2=2px$, and let $T_0$ be a point on it. Point $T_0'$ is such that the midpoint of the segment $T_0T_0'$ lies on the axis of the parabola. For a variable point $T$ on $\mathcal P$, the perpendicular from $T_0'$ to the line $T_0T$ intersects the line through $T$ parallel to the axis of $\mathcal P$ at a point $T'$. Find the locus of $T'$.