Found problems: 85335
1976 Bulgaria National Olympiad, Problem 4
Let $0<x_1\le x_2\le\ldots\le x_n$. Prove that
$$\frac{x_1}{x_2}+\frac{x_2}{x_3}+\ldots+\frac{x_{n-1}}{x_n}+\frac{x_n}{x_1}\ge\frac{x_2}{x_1}+\frac{x_3}{x_2}+\ldots+\frac{x_n}{x_{n-1}}+\frac{x_1}{x_n}$$
[i]I. Tonov[/i]
2010 Contests, 1
Triangle $ABC$ is given. Circle $ \omega $ passes through $B$, touch $AC$ in $D$ and intersect sides $AB$ and $BC$ at $P$ and $Q$ respectively. Line $PQ$ intersect $BD$ and $AC$ at $M$ and $N$ respectively. Prove that $ \omega $, circumcircle of $DMN$ and circle, touching $PQ$ in $M$ and passes through B, intersects in one point.
LMT Team Rounds 2021+, A10
Pieck the Frog hops on Pascal's Triangle, where she starts at the number $1$ at the top. In a hop, Pieck can hop to one of the two numbers directly below the number she is currently on with equal probability. Given that the expected value of the number she is on after $7$ hops is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, find $m+n$.
[i]Proposed by Steven Yu[/i]
2000 Czech And Slovak Olympiad IIIA, 3
In the plane are given $2000$ congruent triangles of area $1$, which are all images of one triangle under translations. Each of these triangles contains the centroid of every other triangle. Prove that the union of these triangles has area less than $22/9$.
1990 AMC 8, 9
The grading scale shown is used at Jones Junior High. The fifteen scores in Mr. Freeman's class were:
\[ \begin{tabular}[t]{lllllllll}89, & 72, & 54, & 97, & 77, & 92, & 85, & 74, & 75,\\ 63, & 84, & 78, & 71, & 80, & 90. & & &\\ \end{tabular} \]
In Mr. Freeman's class, what percent of the students received a grade of C?
\[ \boxed{\begin{tabular}[t]{cc}A: & 93-100\\ B: & 85-92\\ C: & 75-84\\ D: & 70-74\\ F: & 0-69\end{tabular}} \]
$ \text{(A)}\ 20\%\qquad\text{(B)}\ 25\%\qquad\text{(C)}\ 30\%\qquad\text{(D)}\ 33\frac{1}{3}\%\qquad\text{(E)}\ 40\% $
2003 All-Russian Olympiad, 3
In a triangle $ABC, O$ is the circumcenter and $I$ the incenter. The excircle $\omega_a$ touches rays $AB,AC$ and side $BC$ at $K,M,N$, respectively. Prove that if the midpoint $P$ of $KM$ lies on the circumcircle of $\triangle ABC$, then points $O,N, I$ lie on a line.
2016 MMATHS, 1
Let unit blocks be unit squares in the coordinate plane with vertices at lattice points (points $(a, b)$ such that $a$ and $b$ are both integers). Prove that a circle with area $\pi$ can cover parts of no more than $9$ unit blocks. The circle below covers part of $8$ unit blocks.
[img]https://cdn.artofproblemsolving.com/attachments/4/4/43da9abed06d0feba94012ba68c177e3c2835b.png[/img]
2006 Germany Team Selection Test, 1
For any positive integer $n$, let $w\left(n\right)$ denote the number of different prime divisors of the number $n$. (For instance, $w\left(12\right)=2$.) Show that there exist infinitely many positive integers $n$ such that $w\left(n\right)<w\left(n+1\right)<w\left(n+2\right)$.
2010 Bundeswettbewerb Mathematik, 3
On the sides of a triangle $XYZ$ to the outside construct similar triangles $YDZ, EXZ ,YXF$ with circumcenters $K, L ,M$ respectively. Here are $\angle ZDY = \angle ZXE = \angle FXY$ and $\angle YZD = \angle EZX = \angle YFX$. Show that the triangle $KLM$ is similar to the triangles .
[img]https://cdn.artofproblemsolving.com/attachments/e/f/fe0d0d941015d32007b6c00b76b253e9b45ca5.png[/img]
2018 India IMO Training Camp, 1
Let $\Delta ABC$ be an acute triangle. $D,E,F$ are the touch points of incircle with $BC,CA,AB$ respectively. $AD,BE,CF$ intersect incircle at $K,L,M$ respectively. If,$$\sigma = \frac{AK}{KD} + \frac{BL}{LE} + \frac{CM}{MF}$$ $$\tau = \frac{AK}{KD}.\frac{BL}{LE}.\frac{CM}{MF}$$
Then prove that $\tau = \frac{R}{16r}$. Also prove that there exists integers $u,v,w$ such that, $uvw \neq 0$, $u\sigma + v\tau +w=0$.
2018 China Team Selection Test, 5
Given positive integers $n, k$ such that $n\ge 4k$, find the minimal value $\lambda=\lambda(n,k)$ such that for any positive reals $a_1,a_2,\ldots,a_n$, we have
\[ \sum\limits_{i=1}^{n} {\frac{{a}_{i}}{\sqrt{{a}_{i}^{2}+{a}_{{i}+{1}}^{2}+{\cdots}{{+}}{a}_{{i}{+}{k}}^{2}}}}
\le \lambda\]
Where $a_{n+i}=a_i,i=1,2,\ldots,k$
2004 Junior Balkan Team Selection Tests - Romania, 3
A finite set of positive integers is called [i]isolated [/i]if the sum of the numbers in any given proper subset is co-prime with the sum of the elements of the set.
a) Prove that the set $A=\{4,9,16,25,36,49\}$ is isolated;
b) Determine the composite numbers $n$ for which there exist the positive integers $a,b$ such that the set
\[ A=\{(a+b)^2, (a+2b)^2,\ldots, (a+nb)^2\}\] is isolated.
2013 Singapore MO Open, 2
Let $ABC$ be an acute-angled triangle and let $D$, $E$, and $F$ be the midpoints of $BC$, $CA$, and $AB$ respectively. Construct a circle, centered at the orthocenter of triangle $ABC$, such that triangle $ABC$ lies in the interior of the circle. Extend $EF$ to intersect the circle at $P$, $FD$ to intersect the circle at $Q$ and $DE$ to intersect the circle at $R$. Show that $AP=BQ=CR$.
1987 Romania Team Selection Test, 7
Determine all positive integers $n$ such that $n$ divides $3^n - 2^n$.
2017 Bulgaria National Olympiad, 2
Let $m>1$ be a natural number and $N=m^{2017}+1$. On a blackboard, left to right, are written the following numbers:
\[N, N-m, N-2m,\dots, 2m+1,m+1, 1.\]
On each move, we erase the most left number, written on the board, and all its divisors (if any). This procces continues till all numbers are deleted.
Which numbers will be deleted on the last move.
1970 IMO Longlists, 45
Let $M$ be an interior point of tetrahedron $V ABC$. Denote by $A_1,B_1, C_1$ the points of intersection of lines $MA,MB,MC$ with the planes $VBC,V CA,V AB$, and by $A_2,B_2, C_2$ the points of intersection of lines $V A_1, VB_1, V C_1$ with the sides $BC,CA,AB$.
[b](a)[/b] Prove that the volume of the tetrahedron $V A_2B_2C_2$ does not exceed one-fourth of the volume of $V ABC$.
[b](b)[/b] Calculate the volume of the tetrahedron $V_1A_1B_1C_1$ as a function of the volume of $V ABC$, where $V_1$ is the point of intersection of the line $VM$ with the plane $ABC$, and $M$ is the barycenter of $V ABC$.
2016 HMNT, 19-21
19. Let $S$ be the set of all positive integers whose prime factorizations only contain powers of the primes $2$ and $2017$ ($1$, powers of $2$, and powers of $2017$ are thus contained in $S$). Compute $\sum_{s\in S}\frac1s$.
20. Let $\mathcal{V}$ be the volume enclosed by the graph
$$x^ {2016} + y^{2016} + z^2 = 2016$$
Find $\mathcal{V}$ rounded to the nearest multiple of ten.
21. Zlatan has $2017$ socks of various colours. He wants to proudly display one sock of each of the
colours, and he counts that there are $N$ ways to select socks from his collection for display. Given this
information, what is the maximum value of $N$?
2025 Benelux, 1
Does there exist a function $f:\mathbb{R}\to \mathbb{R}$ such that $$f(x^2+f(y))=f(x)^2-y$$ for all $x, y\in \mathbb{R}$?
2014 AMC 12/AHSME, 20
For how many positive integers $x$ is $\log_{10}{(x-40)} + \log_{10}{(60-x)} < 2$?
${ \textbf{(A)}\ 10\qquad\textbf{(B)}\ 18\qquad\textbf{(C)}\ 19\qquad\textbf{(D)}}\ 20\qquad\textbf{(E)}\ \text{infinitely many} $
2023 AMC 12/AHSME, 2
Carlos went to a sports store to buy running shoes. Running shoes were on sale, with prices reduced by 20% on every pair of shoes. Carlos also knew that he had to pay a 7.5% sales tax on the discounted price. He had 43 dollars. What is the original (before discount) price of the most expensive shoes he could afford to buy?
A)$46$ B)$50$ C)$48$ D)$47$ E)$49$
2014 Turkey EGMO TST, 4
Let $x,y,z$ be positive real numbers such that $x+y+z \ge x^2+y^2+z^2$. Show that;
$$\dfrac{x^2+3}{x^3+1}+\dfrac{y^2+3}{y^3+1}+\dfrac{z^2+3}{z^3+1}\ge6$$
2020 Online Math Open Problems, 4
An alien from the planet OMO Centauri writes the first ten prime numbers in arbitrary order as U, W, XW, ZZ, V, Y, ZV, ZW, ZY, and X. Each letter represents a nonzero digit. Each letter represents the same digit everywhere it appears, and different letters represent different digits. Also, the alien is using a base other than base ten. The alien writes another number as UZWX. Compute this number (expressed in base ten, with the usual, human digits).
[i]Proposed by Luke Robitaille & Eric Shen[/i]
2015 AIME Problems, 4
Point $B$ lies on line segment $\overline{AC}$ with $AB=16$ and $BC=4$. Points $D$ and $E$ lie on the same side of line $AC$ forming equilateral triangles $\triangle ABD$ and $\triangle BCE$. Let $M$ be the midpoint of $\overline{AE}$, and $N$ be the midpoint of $\overline{CD}$. The area of $\triangle BMN$ is $x$. Find $x^2$.
1973 AMC 12/AHSME, 27
Cars A and B travel the same distance. Care A travels half that [i]distance[/i] at $ u$ miles per hour and half at $ v$ miles per hour. Car B travels half the [i]time[/i] at $ u$ miles per hour and half at $ v$ miles per hour. The average speed of Car A is $ x$ miles per hour and that of Car B is $ y$ miles per hour. Then we always have
$ \textbf{(A)}\ x \leq y\qquad
\textbf{(B)}\ x \geq y \qquad
\textbf{(C)}\ x\equal{}y \qquad
\textbf{(D)}\ x<y\qquad
\textbf{(E)}\ x>y$
2005 Purple Comet Problems, 5
In January Jeff’s investment went up by three quarters. In February it went down by one quarter. In March it went up by one third. In April it went down by one fifth. In May it went up by one seventh. In June Jeff’s investment fell by $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. If Jeff’s investment was worth the same amount at the end of June as it had been at the beginning of January, find $m + n$.