Found problems: 85335
2003 All-Russian Olympiad, 4
A finite set of points $X$ and an equilateral triangle $T$ are given on a plane. Suppose that every subset $X'$ of $X$ with no more than $9$ elements can be covered by two images of $T$ under translations. Prove that the whole set $X$ can be covered by two images of $T$ under translations.
2010 Danube Mathematical Olympiad, 2
Given a triangle $ABC$, let $A',B',C'$ be the perpendicular feet dropped from the centroid $G$ of the triangle $ABC$ onto the sides $BC,CA,AB$ respectively. Reflect $A',B',C'$ through $G$ to $A'',B'',C''$ respectively. Prove that the lines $AA'',BB'',CC''$ are concurrent.
2017 China Western Mathematical Olympiad, 8
Let $a_1,a_2,\cdots,a_n>0$ $(n\geq 2)$. Prove that$$\sum_{i=1}^n max\{a_1,a_2,\cdots,a_i \} \cdot min
\{a_i,a_{i+1},\cdots,a_n\}\leq \frac{n}{2\sqrt{n-1}}\sum_{i=1}^n a^2_i$$
2015 Thailand TSTST, 2
Determine the least integer $n > 1$ such that the quadratic mean of the first $n$ positive integers is an integer.
[i]Note: the quadratic mean of $a_1, a_2, \dots , a_n$ is defined to be $\sqrt{\frac{a_1^2+a_2^2+\cdots+a_n^2}{n}}$.[/i]
2014 AMC 10, 5
Doug constructs a square window using $8$ equal-size panes of glass, as shown. The ratio of the height to width for each pane is $5:2$, and the borders around and between the panes are $2$ inches wide. In inches, what is the side length o the square window?
[asy]
fill((0,0)--(25,0)--(25,25)--(0,25)--cycle,grey);
for(int i = 0; i < 4; ++i){
for(int j = 0; j < 2; ++j){
fill((6*i+2,11*j+3)--(6*i+5,11*j+3)--(6*i+5,11*j+11)--(6*i+2,11*j+11)--cycle,white);
}
}[/asy]
$\textbf{(A) }26\qquad\textbf{(B) }28\qquad\textbf{(C) }30\qquad\textbf{(D) }32\qquad\textbf{(E) }34$
2016 Saudi Arabia IMO TST, 1
Call a positive integer $N \ge 2$ [i]special [/i] if for every k such that $2 \le k \le N, N$ can be expressed as a sum of $k$ positive integers that are relatively prime to $N$ (although not necessarily relatively prime to each other). Find all special positive integers.
2023 Malaysian IMO Training Camp, 1
Does there exist a positive integer, $x$, such that $(x+2)^{2023}-x^{2023}$ has exactly $2023^{2023}$ factors?
[i]Proposed by Wong Jer Ren[/i]
2008 Bulgarian Autumn Math Competition, Problem 12.1
Determine the values of the real parameter $a$, such that the solutions of the system of inequalities
$\begin{cases}
\log_{\frac{1}{3}}{(3^{x}-6a)}+\frac{2}{\log_{a}{3}}<x-3\\
\log_{\frac{1}{3}}{(3^{x}-18)}>x-5\\
\end{cases}$
form an interval of length $\frac{1}{3}$.
2020 Jozsef Wildt International Math Competition, W37
For all $x>0$ prove
$$\frac{\sin^2x-x}{\ln\left(\frac{\sin^2x}x\right)^{\sqrt x}}+\frac{\cos^2x-x}{\ln\left(\frac{\cos^2x}x\right)^{\sqrt x}}>|\sin x|+|\cos x|$$
[i]Proposed by Pirkulyiev Rovsen[/i]
2004 AMC 12/AHSME, 23
The polynomial $ x^3\minus{}2004x^2\plus{}mx\plus{}n$ has integer coefficients and three distinct positive zeros. Exactly one of these is an integer, and it is the sum of the other two. How many values of $ n$ are possible?
$ \textbf{(A)}\ 250,\!000 \qquad
\textbf{(B)}\ 250,\!250 \qquad
\textbf{(C)}\ 250,\!500 \qquad
\textbf{(D)}\ 250,\!750 \qquad
\textbf{(E)}\ 251,\!000$
1980 AMC 12/AHSME, 18
If $b>1$, $\sin x>0$, $\cos x>0$, and $\log_b \sin x = a$, then $\log_b \cos x$ equals
$\text{(A)} \ 2\log_b(1-b^{a/2}) ~~\text{(B)} \ \sqrt{1-a^2} ~~\text{(C)} \ b^{a^2} ~~\text{(D)} \ \frac 12 \log_b(1-b^{2a}) ~~\text{(E)} \ \text{none of these}$
2017 AMC 12/AHSME, 1
Pablo buys popsicles for his friends. The store sells single popsicles for $\$1$ each, 3-popsicle boxes for $\$2$, and 5-popsicle boxes for $\$3$. What is the greatest number of popsicles that Pablo can buy with $\$8$?
$\textbf{(A)}\ 8\qquad\textbf{(B)}\ 11\qquad\textbf{(C)}\ 12\qquad\textbf{(D)}\ 13\qquad\textbf{(E)}\ 15$
2017 Austria Beginners' Competition, 3
. Anthony denotes in sequence all positive integers which are divisible by $2$. Bertha denotes in sequence all positive integers which are divisible by $3$. Claire denotes in sequence all positive integers which are divisible by $4$. Orderly Dora denotes all numbers written by the other three. Thereby she puts them in order by size and does not repeat a number. What is the $2017th$ number in her list?
[i]¨Proposed by Richard Henner[/i]
1989 Swedish Mathematical Competition, 4
Let $ABCD$ be a regular tetrahedron. Find the positions of point $P$ on the edge $BD$ such that the edge $CD$ is tangent to the sphere with diameter $AP$.
2014 Contests, 4
Let $n$ be a positive integer. A 4-by-$n$ rectangle is divided into $4n$ unit squares in the usual way. Each unit square is colored black or white. Suppose that every white unit square shares an edge with at least one black unit square. Prove that there are at least $n$ black unit squares.
2024 Taiwan TST Round 3, 2
Let $I$ be the incenter of triangle $ABC$, and let $\omega$ be its incircle. Let $E$ and $F$ be the points of tangency of $\omega$ with $CA$ and $AB$, respectively. Let $X$ and $Y$ be the intersections of the circumcircle of $BIC$ and $\omega$. Take a point $T$ on $BC$ such that $\angle AIT$ is a right angle. Let $G$ be the intersection of $EF$ and $BC$, and let $Z$ be the intersection of $XY$ and $AT$. Prove that $AZ$, $ZG$, and $AI$ form an isosceles triangle.
[i]Proposed by Li4 and usjl.[/i]
2024 Caucasus Mathematical Olympiad, 8
Let $ABC$ be an acute triangle and let $X$ be a variable point on $AC$. The incircle of $\triangle ABX$ touches $AX, BX$ at $K, P$, respectively. The incircle of $\triangle BCX$ touches $CX, BX$ at $L, Q$, respectively. Find the locus of $KP \cap LQ$.
2015 Iran Team Selection Test, 5
Let $A$ be a subset of the edges of an $n\times n $ table. Let $V(A)$ be the set of vertices from the table which are connected to at least on edge from $A$ and $j(A)$ be the number of the connected components of graph $G$ which it's vertices are the set $V(A)$ and it's edges are the set $A$. Prove that for every natural number $l$:
$$\frac{l}{2}\leq min_{|A|\geq l}(|V(A)|-j(A)) \leq \frac{l}{2}+\sqrt{\frac{l}{2}}+1$$
2018 AMC 10, 22
Real numbers $x$ and $y$ are chosen independently and uniformly at random from the interval $[0,1]$. Which of the following numbers is closest to the probability that $x,y,$ and $1$ are the side lengths of an obtuse triangle?
$\textbf{(A)} \text{ 0.21} \qquad \textbf{(B)} \text{ 0.25} \qquad \textbf{(C)} \text{ 0.29} \qquad \textbf{(D)} \text{ 0.50} \qquad \textbf{(E)} \text{ 0.79}$
2020 Thailand TST, 3
Let $x_1, x_2, \dots, x_n$ be different real numbers. Prove that
\[\sum_{1 \leqslant i \leqslant n} \prod_{j \neq i} \frac{1-x_{i} x_{j}}{x_{i}-x_{j}}=\left\{\begin{array}{ll}
0, & \text { if } n \text { is even; } \\
1, & \text { if } n \text { is odd. }
\end{array}\right.\]
1990 AIME Problems, 8
In a shooting match, eight clay targets are arranged in two hanging columns of three targets each and one column of two targets. A marksman is to break all the targets according to the following rules:
1) The marksman first chooses a column from which a target is to be broken.
2) The marksman must then break the lowest remaining target in the chosen column.
If the rules are followed, in how many different orders can the eight targets be broken?
2006 Thailand Mathematical Olympiad, 7
Let $x, y, z$ be reals summing to $1$ which minimizes $2x^2 + 3y^2 + 4z^2$. Find $x$.
2018 Ramnicean Hope, 1
Solve in the real numbers the equation $ \sqrt[5]{2^x-2^{-1}} -\sqrt[5]{2^x+2^{-1}} =-1. $
[i]Mihai Neagu[/i]
1987 Vietnam National Olympiad, 2
Sequences $ (x_n)$ and $ (y_n)$ are constructed as follows: $ x_0 \equal{} 365$, $ x_{n\plus{}1} \equal{} x_n\left(x^{1986} \plus{} 1\right) \plus{} 1622$, and $ y_0 \equal{} 16$, $ y_{n\plus{}1} \equal{} y_n\left(y^3 \plus{} 1\right) \minus{} 1952$, for all $ n \ge 0$. Prove that $ \left|x_n\minus{} y_k\right|\neq 0$ for any positive integers $ n$, $ k$.
2019 Estonia Team Selection Test, 2
In an acute-angled triangle $ABC$, the altitudes intersect at point $H$, and point $K$ is the foot of the altitude drawn from the vertex $A$. Circle $c$ passing through points $A$ and $K$ intersects sides $AB$ and $AC$ at points $M$ and $N$, respectively. The line passing through point $A$ and parallel to line $BC$ intersects the circumcircles of triangles $AHM$ and $AHN$ for second time, respectively, at points $X$ and $Y$. Prove that $ | X Y | = | BC |$.