Found problems: 85335
2017 Irish Math Olympiad, 3
A line segment $B_0B_n$ is divided into $n$ equal parts at points $B_1,B_2,...,B_{n-1} $ and $A$ is a point such that $\angle B_0AB_n$ is a right angle. Prove that :
$$\sum_{k=0}^{n} |AB_k|^{2} = \sum_{k=0}^{n} |B_0B_k|^2$$
1994 Taiwan National Olympiad, 4
Prove that there are infinitely many positive integers $n$ with the following property: For any $n$ integers $a_{1},a_{2},...,a_{n}$ which form in arithmetic progression, both the mean and the standard deviation of the set $\{a_{1},a_{2},...,a_{n}\}$ are integers.
[i]Remark[/i]. The mean and standard deviation of the set $\{x_{1},x_{2},...,x_{n}\}$ are defined by $\overline{x}=\frac{x_{1}+x_{2}+...+x_{n}}{n}$ and $\sqrt{\frac{\sum (x_{i}-\overline{x})^{2}}{n}}$, respectively.
1999 Cono Sur Olympiad, 5
Give a square of side $1$. Show that for each finite set of points of the sides of the square you can find a vertex of the square with the following property: the arithmetic mean of the squares of the distances from this vertex to the points of the set is greater than or equal to $3/4$.
2017 Purple Comet Problems, 26
The incircle of $\vartriangle ABC$ is tangent to sides $\overline{BC}, \overline{AC}$, and $\overline{AB}$ at $D, E$, and $F$, respectively. Point $G$ is the intersection of lines $AC$ and $DF$ as shown. The sides of $\vartriangle ABC$ have lengths $AB = 73, BC = 123$, and $AC = 120$. Find the length $EG$.
[img]https://cdn.artofproblemsolving.com/attachments/d/a/aede28071a1a6b94bbe3ad8e1e104822b89439.png[/img]
LMT Guts Rounds, 2018 F
[u]Round 5[/u]
[b]p13.[/b] Express the number $3024_8$ in base $2$.
[b]p14.[/b] $\vartriangle ABC$ has a perimeter of $10$ and has $AB = 3$ and $\angle C$ has a measure of $60^o$. What is the maximum area of the triangle?
[b]p15.[/b] A weighted coin comes up as heads $30\%$ of the time and tails $70\%$ of the time. If I flip the coin $25$ times, howmany tails am I expected to flip?
[u]Round 6[/u]
[b]p16.[/b] A rectangular box with side lengths $7$, $11$, and $13$ is lined with reflective mirrors, and has edges aligned with the coordinate axes. A laser is shot from a corner of the box in the direction of the line $x = y =
z$. Find the distance traveled by the laser before hitting a corner of the box.
[b]p17.[/b] The largest solution to $x^2 + \frac{49}{x^2}= 2018$ can be represented in the form $\sqrt{a}+\sqrt{b}$. Compute $a +b$.
[b]p18.[/b] What is the expected number of black cards between the two jokers of a $54$ card deck?
[u]Round 7[/u]
p19. Compute ${6 \choose 0} \cdot 2^0 + {6 \choose 1} \cdot 2^1+ {6 \choose 2} \cdot 2^2+ ...+ {6 \choose 6} \cdot 2^6$.
[b]p20.[/b] Define a sequence by $a_1 =5$, $a_{n+1} = a_n + 4 * n -1$ for $n\ge 1$. What is the value of $a_{1000}$?
[b]p21.[/b] Let $\vartriangle ABC$ be the triangle such that $\angle B = 15^o$ and $\angle C = 30^o$. Let $D$ be the point such that $\vartriangle ADC$ is an isosceles right triangle where $D$ is in the opposite side from $A$ respect to $BC$ and $\angle DAC = 90^o$. Find the $\angle ADB$.
[u]Round 8[/u]
[b]p22.[/b] Say the answer to problem $24$ is $z$. Compute $gcd (z,7z +24).$
[b]p23.[/b] Say the answer to problem $22$ is $x$. If $x$ is $1$, write down $1$ for this question. Otherwise, compute $$\sum^{\infty}_{k=1} \frac{1}{x^k}$$
[b]p24.[/b] Say the answer to problem $23$ is $y$. Compute $$\left \lfloor \frac{y^2 +1}{y} \right \rfloor$$
PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h3165983p28809209]here [/url] and 9-12 [url=https://artofproblemsolving.com/community/c3h3166045p28809814]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
2017 International Zhautykov Olympiad, 1
Let $ABC$ be a non-isosceles triangle with circumcircle $\omega$ and let $H, M$ be orthocenter and midpoint of $AB$ respectively. Let $P,Q$ be points on the arc $AB$ of $\omega$ not containing $C$ such that $\angle ACP=\angle BCQ < \angle ACQ$.Let $R,S$ be the foot of altitudes from $H$ to $CQ,CP$ respectively. Prove that thé points $P,Q,R,S$ are concyclic and $M$ is the center of this circle.
2016 CCA Math Bonanza, I4
The three digit number $n=CCA$ (in base $10$), where $C\neq A$, is divisible by $14$. How many possible values for $n$ are there?
[i]2016 CCA Math Bonanza Individual #4[/i]
2010 VJIMC, Problem 4
Let $f:[0,1]\to\mathbb R$ be a function satisfying
$$|f(x)-f(y)|\le|x-y|$$for every $x,y\in[0,1]$. Show that for every $\varepsilon>0$ there exists a countable family of rectangles $(R_i)$ of dimensions $a_i\times b_i$, $a_i\le b_i$ in the plane such that
$$\{(x,f(x)):x\in[0,1]\}\subset\bigcup_iR_i\text{ and }\sum_ia_i<\varepsilon.$$(The edges of the rectangles are not necessarily parallel to the coordinate axes.)
1996 AMC 12/AHSME, 21
Triangles $ABC$ and $ABD$ are isosceles with $AB =AC = BD$, and $BD$ intersects $AC$ at $E$. If $BD$ is perpendicular to $AC$, then $\angle C + \angle D$ is
[asy]
size(130);
defaultpen(linewidth(0.8) + fontsize(11pt));
pair A, B, C, D, E;
real angle = 70;
B = origin;
A = dir(angle);
D = dir(90-angle);
C = rotate(2*(90-angle), A) * B;
draw(A--B--C--cycle);
draw(B--D--A);
E = extension(B, D, C, A);
draw(rightanglemark(B, E, A, 1.5));
label("$A$", A, dir(90));
label("$B$", B, dir(210));
label("$C$", C, dir(330));
label("$D$", D, dir(0));
label("$E$", E, 1.5*dir(340));
[/asy]
$\textbf{(A)}\ 115^\circ \qquad \textbf{(B)}\ 120^\circ \qquad \textbf{(C)}\ 130^\circ \qquad \textbf{(D)}\ 135^\circ \qquad \textbf{(E)}\ \text{not uniquely determined}$
2016 Chile TST IMO, 2
There are 2016 points near a line such that the distance from each point to the line is less than 1 cm, and the distance between any two points is always greater than 2 cm. Prove that there exist two points whose distance is at least 17 meters.
2015 BMT Spring, 5
Let $x$ and $y$ be real numbers satisfying the equation $x^2-4x+y^2+3=0$. If the maximum and minimum values of $x^2+y^2$ are $M$ and $m$ respectively, compute the numerical value of $M-m$.
2022 Brazil Team Selection Test, 3
Show that $n!=a^{n-1}+b^{n-1}+c^{n-1}$ has only finitely many solutions in positive integers.
[i]Proposed by Dorlir Ahmeti, Albania[/i]
2014 Kyiv Mathematical Festival, 2
Can an $8\times8$ board be covered with 13 equal 5-celled figures?
It's alowed to rotate the figures or turn them over.
[size=85](Kyiv mathematical festival 2014)[/size]
2015 Greece Team Selection Test, 2
Consider $111$ distinct points which lie on or in the internal of a circle with radius 1.Prove that there are at least $1998$ segments formed by these points with length $\leq \sqrt{3}$
ICMC 7, 4
Let $(t_n)_{n\geqslant 1}$ be the sequence defined by $t_1=1, t_{2k}=-t_k$ and $t_{2k+1}=t_{k+1}$ for all $k\geqslant 1.$ Consider the series \[\sum_{n=1}^\infty\frac{t_n}{n^{1/2024}}.\]Prove that this series converges to a positive real number.
[i]Proposed by Dylan Toh[/i]
2009 Today's Calculation Of Integral, 429
Find the length of the curve expressed by the polar equation: $ r\equal{}1\plus{}\cos \theta \ (0\leq \theta \leq \pi)$.
2020 GQMO, 3
Let $A$ and $B$ be two distinct points in the plane. Let $M$ be the midpoint of the segment $AB$, and let $\omega$ be a circle that goes through $A$ and $M$. Let $T$ be a point on $\omega$ such that the line $BT$ is tangent to $\omega$. Let $X$ be a point (other than $B$) on the line $AB$ such that $TB = TX$, and let $Y$ be the foot of the perpendicular from $A$ onto the line $BT$.
Prove that the lines $AT$ and $XY$ are parallel.
[i]Navneel Singhal, India[/i]
1973 Miklós Schweitzer, 2
Let $ R$ be an Artinian ring with unity. Suppose that every idempotent element of $ R$ commutes with every element of $ R$ whose square is $ 0$. Suppose $ R$ is the sum of the ideals $ A$ and $ B$. Prove that $ AB\equal{}BA$.
[i]A. Kertesz[/i]
2006 Baltic Way, 6
Determine the maximal size of a set of positive integers with the following properties:
$1.$ The integers consist of digits from the set $\{ 1,2,3,4,5,6\}$.
$2.$ No digit occurs more than once in the same integer.
$3.$ The digits in each integer are in increasing order.
$4.$ Any two integers have at least one digit in common (possibly at different positions).
$5.$ There is no digit which appears in all the integers.
2019 Middle European Mathematical Olympiad, 3
There are $n$ boys and $n$ girls in a school class, where $n$ is a positive integer. The heights of all the children in this class are distinct. Every girl determines the number of boys that are taller than her, subtracts the number of girls that are taller than her, and writes the result on a piece of paper. Every boy determines the number of girls that are shorter than him, subtracts the number of boys that are shorter than him, and writes the result on a piece of paper.
Prove that the numbers written down by the girls are the same as the numbers written down by the boys (up to a permutation).
[i]Proposed by Stephan Wagner, Austria[/i]
2014 ASDAN Math Tournament, 7
Ben works quickly on his homework, but tires quickly. The first problem takes him $1$ minute to solve, and the second problem takes him $2$ minutes to solve. It takes him $N$ minutes to solve problem $N$ on his homework. If he works for an hour on his homework, compute the maximum number of problems he can solve.
1988 AMC 8, 9
An isosceles triangle is a triangle with two sides of equal length. How many of the five triangles on the square grid below are isosceles?
[asy]
for(int a=0; a<12; ++a)
{
draw((a,0)--(a,6));
}
for(int b=0; b<7; ++b)
{
draw((0,b)--(11,b));
}
draw((0,6)--(2,6)--(1,4)--cycle,linewidth(1));
draw((3,4)--(3,6)--(5,4)--cycle,linewidth(1));
draw((0,1)--(3,2)--(6,1)--cycle,linewidth(1));
draw((7,4)--(6,6)--(9,4)--cycle,linewidth(1));
draw((8,1)--(9,3)--(10,0)--cycle,linewidth(1));[/asy]
$ \text{(A)}\ 1\qquad\text{(B)}\ 2\qquad\text{(C)}\ 3\qquad\text{(D)}\ 4\qquad\text{(E)}\ 5 $
2014 Junior Balkan Team Selection Tests - Romania, 4
On each side of an equilateral triangle of side $n \ge 1$ consider $n - 1$ points that divide the sides into $n$ equal segments. Through these points draw parallel lines to the sides of the triangles, obtaining a net of equilateral triangles of side length $1$. On each of the vertices of the small triangles put a coin head up. A move consists in flipping over three mutually adjacent coins. Find all values of $n$ for which it is possible to turn all coins tail up after a finite number of moves.
Colombia 1997
2015 IMC, 1
For any integer $n\ge 2$ and two $n\times n$ matrices with real
entries $A,\; B$ that satisfy the equation
$$A^{-1}+B^{-1}=(A+B)^{-1}\;$$
prove that $\det (A)=\det(B)$.
Does the same conclusion follow for matrices with complex entries?
(Proposed by Zbigniew Skoczylas, Wroclaw University of Technology)
1951 Miklós Schweitzer, 6
In lawn-tennis the player who scores at least four points, while his opponent scores at least two points less, wins a game. The player who wins at least six games, while his opponent wins at least two games less, wins a set. What minimum percentage of all points does the winner have to score in a set?