Found problems: 85335
2011 District Olympiad, 4
Let be a nonzero real number $ a, $ and a natural number $ n. $ Prove the implication:
$$ \{ a \} +\left\{\frac{1}{a}\right\} =1 \implies \{ a^n \} +\left\{\frac{1}{a^n}\right\} =1 , $$
where $ \{\} $ is the fractional part.
2023 All-Russian Olympiad Regional Round, 9.5
Let $ABCD$ be a cyclic quadrilateral such that the circles with diameters $AB$ and $CD$ touch at $S$. If $M, N$ are the midpoints of $AB, CD$, prove that the perpendicular through $M$ to $MN$ meets $CS$ on the circumcircle of $ABCD$.
2014 Contests, 2
Let $D$ and $E$ be points in the interiors of sides $AB$ and $AC$, respectively, of a triangle $ABC$, such that $DB = BC = CE$. Let the lines $CD$ and $BE$ meet at $F$. Prove that the incentre $I$ of triangle $ABC$, the orthocentre $H$ of triangle $DEF$ and the midpoint $M$ of the arc $BAC$ of the circumcircle of triangle $ABC$ are collinear.
2009 Turkey MO (2nd round), 1
Find all prime numbers $p$ for which $p^3-4p+9$ is a perfect square.
2020 Durer Math Competition Finals, 3
Ann wrote down all the perfect squares from one to one million (all in a single line). However, at night, an evil elf erased one of the numbers. So the next day, Ann saw an empty space between the numbers $760384$ and $763876$. What is the sum of the digits of the erased number?
2010 Iran Team Selection Test, 5
Circles $W_1,W_2$ intersect at $P,K$. $XY$ is common tangent of two circles which is nearer to $P$ and $X$ is on $W_1$ and $Y$ is on $W_2$. $XP$ intersects $W_2$ for the second time in $C$ and $YP$ intersects $W_1$ in $B$. Let $A$ be intersection point of $BX$ and $CY$. Prove that if $Q$ is the second intersection point of circumcircles of $ABC$ and $AXY$
\[\angle QXA=\angle QKP\]
2019 ASDAN Math Tournament, 1
A square $ABCD$ and point $E$ are drawn in a plane such that lengths $DE < BE$ and $\vartriangle ACE$ is equilateral. Compute $\angle BAE$.
1975 All Soviet Union Mathematical Olympiad, 209
Denote the midpoints of the convex hexagon $A_1A_2A_3A_4A_5A_6$ diagonals $A_6A_2$, $A_1A_3$, $A_2A_4$, $A_3A_5$, $A_4A_6$, $A_5A_1$ as $B_1, B_2, B_3, B_4, B_5, B_6$ respectively. Prove that if the hexagon $B_1B_2B_3B_4B_5B_6$ is convex, than its area equals to the quarter of the initial hexagon.
2016 CHMMC (Fall), 5
Suppose you have $27$ identical unit cubes colored such that $3$ faces adjacent to a vertex are red and the other $3$ are colored blue. Suppose further that you assemble these $27$ cubes randomly into a larger cube with $3$ cubes to an edge (in particular, the orientation of each cube is random). The probability that the entire cube is one solid color can be written as $\frac{1}{2^n}$ for some positive integer $n$. Find $n$.
2012 AMC 10, 7
For a science project, Sammy observed a chipmunk and a squirrel stashing acorns in holes. The chipmunk hid $3$ acorns in each of the holes it dug. The squirrel hid $4$ acorns in each of the holes it dug. They each hid the same
number of acorns, although the squirrel needed $4$ fewer holes. How many acorns did the chipmunk hide?
${{ \textbf{(A)}\ 30\qquad\textbf{(B)}\ 36\qquad\textbf{(C)}\ 42\qquad\textbf{(D)}\ 48}\qquad\textbf{(E)}\ 54} $
1997 Singapore MO Open, 4
Let $n \ge 2$ be a positive integer. Suppose that $a_1,a_2,...,a_n$ and $b_1,b_2,...,b_n$ are 2n numbers such that $\sum_{i=1}^n a_i =\sum_{i=1}^n n_i= 1$ and $a_i\ge 0, 0 \le b_i\le \frac{n-1}{n}, i = 1, 2,..., n$. Show that
$$b_1a_2a_3...a_n+a_1b_2a_3...a_n+...+a_1a_2...a_{k-1}b_ka_{k+1}...a_n+ ...+ a_1a_2...a_{n-1}b_n \le \frac{1}{n(n-1)^{n-2}}$$
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 $AP$.