Found problems: 85335
1988 IMO Longlists, 10
Let $ a$ be the greatest positive root of the equation $ x^3 \minus{} 3 \cdot x^2 \plus{} 1 \equal{} 0.$ Show that $ \left[a^{1788} \right]$ and $ \left[a^{1988} \right]$ are both divisible by 17. Here $ [x]$ denotes the integer part of $ x.$
2008 National Olympiad First Round, 1
Let $AD$ be a median of $\triangle ABC$ such that $m(\widehat{ADB})=45^{\circ}$ and $m(\widehat{ACB})=30^{\circ}$. What is the measure of $\widehat{ABC}$ in degrees?
$
\textbf{(A)}\ 75
\qquad\textbf{(B)}\ 90
\qquad\textbf{(C)}\ 105
\qquad\textbf{(D)}\ 120
\qquad\textbf{(E)}\ 135
$
2013 Bogdan Stan, 3
Let be four $ n\times n $ real matrices $ A,B,C,D $ having the property that $ C+D\sqrt{-1} $ is the inverse of $ A+B\sqrt{-1} . $
Show that $ \left| \det\left( A+B\sqrt{-1} \right) \right|^2\cdot\left| \det C \right| =\det A. $
[i]Vasile Pop[/i]
1990 IMO Longlists, 87
Let $m$ be an positive odd integer not divisible by $3$. Prove that $\left[4^m -(2+\sqrt 2)^m\right]$ is divisible by $112.$
2024 Assara - South Russian Girl's MO, 1
There is a set of $2024$ cards. Each card on both sides is colored in one of three colors — red, blue or white, and for each card its two sides are colored in different colors. The cards were laid out on the table. The card [i]lies beautifully[/i] if at least one of two conditions is met: its upper side — red; its underside is blue. It turned out that exactly $150$ cards are lying beautifully. Then all the cards were turned over. Now some of the cards are lying beautifully on the table. How many of them can there be?
[i]K.A.Sukhov[/i]
2000 AIME Problems, 1
The number \[ \frac 2{\log_4{2000^6}}+\frac 3{\log_5{2000^6}} \] can be written as $\frac mn$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$
2022 BMT, 5
Given a positive integer $n,$ let $s(n)$ denote the sum of the digits of $n.$ Compute the largest positive integer $n$ such that $n = s(n)^2 + 2s(n) - 2.$
KoMaL A Problems 2022/2023, A. 843
Let $N$ be the set of those positive integers $n$ for which $n\mid k^k-1$ implies $n\mid k-1$ for every positive integer $k$. Prove that if $n_1,n_2\in N$, then their greatest common divisor is also in $N$.
2021 Kosovo National Mathematical Olympiad, 1
There are $9$ point in the Cartezian plane with coordinates
$(0,0),(0,1),(0,2),(1,0),(1,1),(1,2),(2,0),(2,1),(2,2).$
Some points are coloured in red and the others in blue. Prove that for any colouring of the points we can always find a right isosceles triangle whose vertexes have the same colour.
2007 Balkan MO, 2
Find all real functions $f$ defined on $ \mathbb R$, such that \[f(f(x)+y) = f(f(x)-y)+4f(x)y ,\] for all real numbers $x,y$.
2021 USAMTS Problems, 3
Let $n$ be a positive integer. Let $S$ be the set of $n^2$ cells in an $n\times n$ grid. Call a subset $T$ of $S$ a [b]double staircase [/b] if
[list]
[*] $T$ can be partitioned into $n$ horizontal nonoverlapping rectangles of dimensions $1 \times 1,
1 \times 2, ..., 1 \times n,$ and
[*]$T$ can also be partitioned into $n$ vertical nonoverlapping rectangles of dimensions $1\times1,
2 \times 1, ..., n \times 1$.
[/list]
In terms of $n$, how many double staircases are there? (Rotations and reflections are considered distinct.)
An example of a double staircase when $n = 3$ is shown below.
[asy]
unitsize(1cm);
for (int i = 0; i <= 3; ++i)
{
draw((0,i)--(3,i),linewidth(0.2));
draw((i,0)--(i,3),linewidth(0.2));
}
filldraw((0,0)--(1,0)--(1,1)--(0,1)--cycle, lightgray, linewidth(0.2));
filldraw((1,0)--(2,0)--(2,1)--(1,1)--cycle, lightgray, linewidth(0.2));
filldraw((2,0)--(3,0)--(3,1)--(2,1)--cycle, lightgray, linewidth(0.2));
filldraw((0,1)--(1,1)--(1,2)--(0,2)--cycle, lightgray, linewidth(0.2));
filldraw((1,1)--(2,1)--(2,2)--(1,2)--cycle, lightgray, linewidth(0.2));
filldraw((1,2)--(2,2)--(2,3)--(1,3)--cycle, lightgray, linewidth(0.2));
[/asy]
2017 Korea Winter Program Practice Test, 1
Let $\gamma_1, \gamma_2, \gamma_3$ be mutually externally tangent circles and $\Gamma_1, \Gamma_2, \Gamma_3$ also be mutually externally tangent circles. For each $1 \le i \le 3$, $\gamma_i$ and $\Gamma_{i+1}$ are externally tangent at $A_i$, $\gamma_i$ and $\Gamma_{i+2}$ are externally tangent at $B_i$, and $\gamma_i$ and $\Gamma_i$ do not meet. Show that the six points $A_1, A_2, A_3, B_1, B_2, B_3$ lie on either a line or a circle.
2014 VTRMC, Problem 5
Let $n\ge1$ and $r\ge2$ be positive integers. Prove that there is no integer $m$ such that $n(n+1)(n+2)=m^r$.
2017 Israel National Olympiad, 4
Three rational number $x,p,q$ satisfy $p^2-xq^2$=1. Prove that there are integers $a,b$ such that $p=\frac{a^2+xb^2}{a^2-xb^2}$ and $q=\frac{2ab}{a^2-xb^2}$.
1999 Turkey MO (2nd round), 1
Find the number of ordered quadruples $(x,y,z,w)$ of integers with $0\le x,y,z,w\le 36$ such that ${{x}^{2}}+{{y}^{2}}\equiv {{z}^{3}}+{{w}^{3}}\text{ (mod 37)}$.
2020 DMO Stage 1, 3.
[b]Q.[/b] Prove that:
$$\sum_{\text{cyc}}\tan (\tan A) - 2 \sum_{\text{cyc}} \tan \left(\cot \frac{A}{2}\right) \geqslant -3 \tan (\sqrt 3)$$where $A, B$ and $C$ are the angles of an acute-angled $\triangle ABC$.
[i]Proposed by SA2018[/i]
2020 Stanford Mathematics Tournament, 10
Three circles with radii $23$, $46$, and $69$ are tangent to each other as shown in the figure below (figure is not drawn to scale). Find the radius of the largest circle that can fit inside the shaded region.
[img]https://cdn.artofproblemsolving.com/attachments/6/d/158abc178e4ddd72541580958a4ee2348b2026.png[/img]
2019 Purple Comet Problems, 19
Rectangle $ABCD$ has sides $AB = 10$ and $AD = 7$. Point $G$ lies in the interior of $ABCD$ a distance $2$ from side $\overline{CD}$ and a distance $2$ from side $\overline{BC}$. Points $H, I, J$, and $K$ are located on sides $\overline{BC}, \overline{AB}, \overline{AD}$, and $\overline{CD}$, respectively, so that the path $GHIJKG$ is as short as possible. Then $AJ = \frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
2021 Science ON grade VII, 4
Take $k\in \mathbb{Z}_{\ge 1}$ and the sets $A_1,A_2,\dots, A_k$ consisting of $x_1,x_2,\dots ,x_k$ positive integers, respectively. For any two sets $A$ and $B$, define $A+B=\{a+b~|~a\in A,~b\in B\}$.
Find the least and greatest number of elements the set $A_1+A_2+\dots +A_k$ may have.
[i] (Andrei Bâra)[/i]
2018 BAMO, A
Twenty-five people of different heights stand in a $5\times 5$ grid of squares, with one person in each square. We know that each row has a shortest person, suppose Ana is the tallest of these five people. Similarly, we know that each column has a tallest person, suppose Bev is the shortest of these five people.
Assuming Ana and Bev are not the same person, who is taller: Ana or Bev?
Prove that your answer is always correct.
2003 China Team Selection Test, 2
Find all functions $f,g$:$R \to R$ such that $f(x+yg(x))=g(x)+xf(y)$ for $x,y \in R$.
2004 Germany Team Selection Test, 3
Let $n \geq 2$ be a natural number, and let $\left( a_{1};\;a_{2};\;...;\;a_{n}\right)$ be a permutation of $\left(1;\;2;\;...;\;n\right)$. For any integer $k$ with $1 \leq k \leq n$, we place $a_k$ raisins on the position $k$ of the real number axis. [The real number axis is the $x$-axis of a Cartesian coordinate system.]
Now, we place three children A, B, C on the positions $x_A$, $x_B$, $x_C$, each of the numbers $x_A$, $x_B$, $x_C$ being an element of $\left\{1;\;2;\;...;\;n\right\}$. [It is not forbidden to place different children on the same place!]
For any $k$, the $a_k$ raisins placed on the position $k$ are equally handed out to those children whose positions are next to $k$. [So, if there is only one child lying next to $k$, then he gets the raisin. If there are two children lying next to $k$ (either both on the same position or symmetric with respect to $k$), then each of them gets one half of the raisin. Etc..]
After all raisins are distributed, a child is unhappy if he could have received more raisins than he actually has received if he had moved to another place (while the other children would rest on their places).
For which $n$ does there exist a configuration $\left( a_{1};\;a_{2};\;...;\;a_{n}\right)$ and numbers $x_A$, $x_B$, $x_C$, such that all three children are happy?
2016 VJIMC, 4
Find the value of sum $\sum_{n=1}^\infty A_n$, where $$A_n=\sum_{k_1=1}^\infty\cdots\sum_{k_n=1}^\infty \frac{1}{k_1^2}\frac{1}{k_1^2+k_2^2}\cdots\frac{1}{k_1^2+\cdots+k_n^2}.$$
2024 Durer Math Competition Finals, 3
We have a stick of length $2n{}$ and a machine which cuts sticks of length $k\in\mathbb{N}$ with $k>1$ into two sticks with arbitrary positive integer lengths. What is the smallest number of cuts after which we can always find some sticks whose lengths sum up to $n{}$?
2023 AMC 12/AHSME, 15
Suppose $a$, $b$, and $c$ are positive integers such that \[\frac{a}{14}+\frac{b}{15}=\frac{c}{210}.\] Which of the following statements are necessarily true?
I. If $\gcd(a,14)=1$ or $\gcd(b,15)=1$ or both, then $\gcd(c,210)=1$.
II. If $\gcd(c,210)=1$, then $\gcd(a,14)=1$ or $\gcd(b,15)=1$ or both.
III. $\gcd(c,210)=1$ if and only if $\gcd(a,14)=\gcd(b,15)=1$.
$\textbf{(A)}~\text{I, II, and III}\qquad\textbf{(B)}~\text{I only}\qquad\textbf{(C)}~\text{I and II only}\qquad\textbf{(D)}~\text{III only}\qquad\textbf{(E)}~\text{II and III only}$