Found problems: 85335
2011 International Zhautykov Olympiad, 1
Given is trapezoid $ABCD$, $M$ and $N$ being the midpoints of the bases of $AD$ and $BC$, respectively.
a) Prove that the trapezoid is isosceles if it is known that the intersection point of perpendicular bisectors of the lateral sides belongs to the segment $MN$.
b) Does the statement of point a) remain true if it is only known that the intersection point of perpendicular bisectors of the lateral sides belongs to the line $MN$?
2005 India IMO Training Camp, 3
A merida path of order $2n$ is a lattice path in the first quadrant of $xy$- plane joining $(0,0)$ to $(2n,0)$ using three kinds of steps $U=(1,1)$, $D= (1,-1)$ and $L= (2,0)$, i.e. $U$ joins $x,y)$ to $(x+1,y+1)$ etc... An ascent in a merida path is a maximal string of consecutive steps of the form $U$. If $S(n,k)$ denotes the number of merdia paths of order $2n$ with exactly $k$ ascents, compute $S(n,1)$ and $S(n,n-1)$.
2021 CMIMC, 2.4
What is the $101$st smallest integer which can represented in the form $3^a+3^b+3^c$, where $a,b,$ and $c$ are integers?
[i]Proposed by Dilhan Salgado[/i]
2020 Australian Maths Olympiad, 7
A $\emph{tetromino tile}$ is a tile that can be formed by gluing together four unit square tiles, edge to edge. For each positive integer $\emph{n}$, consider a bathroom whose floor is in the shape of a $2\times2 n$ rectangle. Let $T_n$ be the number of ways to tile this bathroom floor with tetromino tiles. For example, $T_2 = 4$ since there are four ways to tile a $2\times4$ rectangular bathroom floor with tetromino tiles, as shown below.
[click for diagram]
Prove that each of the numbers $T_1, T_2, T_3, ...$ is a perfect square.
2003 Bosnia and Herzegovina Team Selection Test, 3
Prove that for every positive integer $n$ holds:
$(n-1)^n+2n^n \leq (n+1)^{n} \leq 2(n-1)^n+2n^{n}$
2001 China Western Mathematical Olympiad, 1
The sequence $ \{x_n\}$ satisfies $ x_1 \equal{} \frac {1}{2}, x_{n \plus{} 1} \equal{} x_n \plus{} \frac {x_n^2}{n^2}$. Prove that $ x_{2001} < 1001$.
II Soros Olympiad 1995 - 96 (Russia), 9.9
There are $5$ ingots weighing $1$, $2$, $3$, $4$ and $5$ kg with an unknown copper content that varies in different ingots. Each ingot must be divided into $5$ parts and $5$ new ingots of the same mass of $1$, $2$, $3$, $4$ and $5$ kg must be made. This requires that the percentage of copper in all pieces be the same, regardless of what it was in the original pieces. What parts should each piece be divided into?
2023 Sinapore MO Open, P2
A grid of cells is tiled with dominoes such that every cell is covered by exactly one domino. A subset $S$ of dominoes is chosen. Is it true that at least one of the following 2 statements is false?
(1) There are $2022$ more horizontal dominoes than vertical dominoes in $S$.
(2) The cells covered by the dominoes in $S$ can be tiled completely and exactly by $L$-shaped tetrominoes.
2015 Switzerland - Final Round, 8
Let $ABCD$ be a trapezoid, where $AB$ and $CD$ are parallel. Let $P$ be a point on the side $BC$. Show that the parallels to $AP$ and $PD$ intersect through $C$ and $B$ to $DA$, respectively.
1996 South africa National Olympiad, 2
Find all real numbers for which $3^x+4^x=5^x$.
2022-23 IOQM India, 17
For a positive integer $n>1$, let $g(n)$ denote the largest positive proper divisor of $n$ and $f(n)=n-g(n)$. For example, $g(10)=5, f(10)=5$ and $g(13)=1,f(13)=12$. Let $N$ be the smallest positive integer such that $f(f(f(N)))=97$. Find the largest integer not exceeding $\sqrt{N}$
2022 Utah Mathematical Olympiad, 4
Alpha and Beta are playing a game on a $10\times 100$ grid of squares. At each turn, they can fold the grid along any of the interior horizontal or vertical gridlines, which creates a smaller (folded) grid of squares (on the first move, they can choose one of $9$ horizontal or $99$ vertical gridlines). The person who makes the last fold wins. If both players play optimally and Alpha starts, determine with proof who wins.
1999 IMO Shortlist, 6
Prove that for every real number $M$ there exists an infinite arithmetic progression such that:
- each term is a positive integer and the common difference is not divisible by 10
- the sum of the digits of each term (in decimal representation) exceeds $M$.
2022 Serbia National Math Olympiad, P4
Let $f(n)$ be number of numbers $x \in \{1,2,\cdots ,n\}$, $n\in\mathbb{N}$, such that $gcd(x, n)$ is either $1$ or prime. Prove
$$\sum_{d|n} f(d) + \varphi(n) \geq 2n$$
For which $n$ does equality hold?
2015 QEDMO 14th, 10
Find all prime numbers $p$ for which $p^3- p + 1$ is a perfect square .
2019 Turkey EGMO TST, 4
Let $\sigma (n)$ shows the number of positive divisors of $n$. Let $s(n)$ be the number of positive divisors of $n+1$ such that for every divisor $a$, $a-1$ is also a divisor of $n$. Find the maximum value of $2s(n)- \sigma (n) $.
1982 National High School Mathematics League, 7
Let $M=\{(x,y)||xy|=1,x>0\},N=\{(x,y)|\arctan x+\arctan y=\pi\}$. Which one is right?
$\text{(A)}M\cup N=\{(x,y)||xy|=1\}\qquad\text{(B)}M\cup N=M$
$\text{(C)}M\cup N=N\qquad\text{(D)}M\cup N=\{(x,y)||xy|=1,x,y\text{ cannot be negative the same time}\}$
2010 Today's Calculation Of Integral, 529
Prove that the following inequality holds for each natural number $ n$.
\[ \int_0^{\frac {\pi}{2}} \sum_{k \equal{} 1}^n \left(\frac {\sin kx}{k}\right)^2dx < \frac {61}{144}\pi\]
2021 Pan-American Girls' Math Olympiad, Problem 4
Lucía multiplies some positive one-digit numbers (not necessarily distinct) and obtains a number $n$ greater than 10. Then, she multiplies all the digits of $n$ and obtains an odd number. Find all possible values of the units digit of $n$.
$\textit{Proposed by Pablo Serrano, Ecuador}$
2003 Singapore Senior Math Olympiad, 1
It is given that n is a positive integer such that both numbers $2n + 1$ and $3n + 1$ are complete squares. Is it true that $n$ must be divisible by $40$ ? Justify your answer.
2013 Online Math Open Problems, 30
Let $P(t) = t^3+27t^2+199t+432$. Suppose $a$, $b$, $c$, and $x$ are distinct positive reals such that $P(-a)=P(-b)=P(-c)=0$, and \[
\sqrt{\frac{a+b+c}{x}} = \sqrt{\frac{b+c+x}{a}} + \sqrt{\frac{c+a+x}{b}} + \sqrt{\frac{a+b+x}{c}}. \] If $x=\frac{m}{n}$ for relatively prime positive integers $m$ and $n$, compute $m+n$.
[i]Proposed by Evan Chen[/i]
2005 Germany Team Selection Test, 3
For an ${n\times n}$ matrix $A$, let $X_{i}$ be the set of entries in row $i$, and $Y_{j}$ the set of entries in column $j$, ${1\leq i,j\leq n}$. We say that $A$ is [i]golden[/i] if ${X_{1},\dots ,X_{n},Y_{1},\dots ,Y_{n}}$ are distinct sets. Find the least integer $n$ such that there exists a ${2004\times 2004}$ golden matrix with entries in the set ${\{1,2,\dots ,n\}}$.
2011 Today's Calculation Of Integral, 707
In the $xyz$ space, consider a right circular cylinder with radius of base 2, altitude 4 such that
\[\left\{
\begin{array}{ll}
x^2+y^2\leq 4 &\quad \\
0\leq z\leq 4 &\quad
\end{array}
\right.\]
Let $V$ be the solid formed by the points $(x,\ y,\ z)$ in the circular cylinder satisfying
\[\left\{
\begin{array}{ll}
z\leq (x-2)^2 &\quad \\
z\leq y^2 &\quad
\end{array}
\right.\]
Find the volume of the solid $V$.
2024 CIIM, 2
Let $n$ be a positive integer, and let $M_n$ be the set of invertible matrices with integer entries and size $n \times n$.
(a) Find the largest possible value of $n$ such that there exists a symmetric matrix $A \in M_n$ satisfying
\[
\det(A^{20} + A^{24}) < 2024.
\]
(b) Prove that for every $n$, there exists a matrix $B \in M_n$ such that
\[
\det(B^{20} + B^{24}) < 2024.
\]
2011 Brazil Team Selection Test, 4
Let $ABCDE$ be a convex pentagon such that $BC \parallel AE,$ $AB = BC + AE,$ and $\angle ABC = \angle CDE.$ Let $M$ be the midpoint of $CE,$ and let $O$ be the circumcenter of triangle $BCD.$ Given that $\angle DMO = 90^{\circ},$ prove that $2 \angle BDA = \angle CDE.$
[i]Proposed by Nazar Serdyuk, Ukraine[/i]