Found problems: 85335
1958 Poland - Second Round, 2
Six equal disks are placed on a plane so that their centers lie at the vertices of a regular hexagon with sides equal to the diameter of the disks. How many revolutions will a seventh disk of the same size make when rolling in the same plane externally over the disks before returning to its initial position?
2020 Greece JBMO TST, 3
Find all pairs $(a,b)$ of prime positive integers $a,b$ such that number $A=3a^2b+16ab^2$ equals to a square of an integer.
2017 Romania National Olympiad, 4
Let be a function $ f $ of class $ \mathcal{C}^1[a,b] $ whose derivative is positive. Prove that there exists a real number $ c\in (a,b) $ such that
$$ f(f(b))-f(f(a))=(f'(c))^2(b-a) . $$
2008 Romania National Olympiad, 3
Let $ p,q,r$ be 3 prime numbers such that $ 5\leq p <q<r$. Knowing that $ 2p^2\minus{}r^2 \geq 49$ and $ 2q^2\minus{}r^2\leq 193$, find $ p,q,r$.
2024-IMOC, N8
Find all integers $(a,b)$ satisfying: there is an integer $k>1$ such that
$$a^k+b^k-1\ |\ a^n+b^n-1$$
holds for all integer $n\geq k$ (we define that $0|0$)
2023-IMOC, N2
Find all pairs of positive integers $(a, b)$ such that $a^b+b^a=a!+b^2+ab+1$.
VI Soros Olympiad 1999 - 2000 (Russia), 10.4
Solve the equation $$16x^3 = (11x^2 + x -1)\sqrt{x^2 - x + 1}.$$
2003 Bosnia and Herzegovina Team Selection Test, 1
Board has written numbers: $5$, $7$ and $9$. In every step we do the following: for every pair $(a,b)$, $a>b$ numbers from the board, we also write the number $5a-4b$. Is it possible that after some iterations, $2003$ occurs at the board ?
2011 Math Prize For Girls Problems, 3
The figure below shows a triangle $ABC$ with a semicircle on each of its three sides.
[asy]
unitsize(5);
pair A = (0, 20 * 21) / 29.0;
pair B = (-20^2, 0) / 29.0;
pair C = (21^2, 0) / 29.0;
draw(A -- B -- C -- cycle);
label("$A$", A, S);
label("$B$", B, S);
label("$C$", C, S);
filldraw(arc((A + C)/2, C, A)--cycle, gray);
filldraw(arc((B + C)/2, C, A)--cycle, white);
filldraw(arc((A + B)/2, A, B)--cycle, gray);
filldraw(arc((B + C)/2, A, B)--cycle, white);
[/asy]
If $AB = 20$, $AC = 21$, and $BC = 29$, what is the area of the shaded region?
2016 Saudi Arabia Pre-TST, 1.3
A lock has $16$ keys arranged in a $4\times 4$ array, each key oriented either horizontally or vertically. In order to open it, all the keys must be vertically oriented. When a key is switched to another position, all the other keys in the same row and column automatically switch their positions too. Show that no matter what the starting positions are, it is always
possible to open this lock. (Only one key at a time can be switched.)
1978 Romania Team Selection Test, 6
[b]a)[/b] Prove that $ 0=\inf\{ |x\sqrt 2+y\sqrt 3+y\sqrt 5|\big| x,y,z\in\mathbb{Z} ,x^2+y^2+z^2>0 \} $
[b]b)[/b] Prove that there exist three positive rational numbers $ a,b,c $ such that the expression $ E(x,y,z):=xa+yb+zc $ vanishes for infinitely many integer triples $ (x,y,z), $ but it doesn´t get arbitrarily close to $ 0. $
2015 Math Prize for Girls Problems, 19
Sabrina has a fair tetrahedral die whose faces are numbered 1, 2, 3, and 4, respectively. She creates a sequence by rolling the die and recording the number on its bottom face. However, she discards (without recording) any roll such that appending its number to the sequence would result in two consecutive terms that sum to 5. Sabrina stops the moment that all four numbers appear in the sequence. Find the expected (average) number of terms in Sabrina's sequence.
2009 Cono Sur Olympiad, 4
Andrea and Bruno play a game on a table with $11$ rows and $9$ columns. First Andrea divides the table in $33$ zones. Each zone is formed by $3$ contiguous cells aligned vertically or horizontally, as shown in the figure.
[code]
._
|_|
|_| _ _ _
|_| |_|_|_|
[/code]
Then, Bruno writes one of the numbers $0, 1, 2, 3, 4, 5$ in each cell in such a way that the sum of the numbers in each zone is equal to $5$. Bruno wins if the sum of the numbers written in each of the $9$ columns of the table is a prime number. Otherwise, Andrea wins. Show that Bruno always has a winning strategy.
2022 Olimphíada, 4
Let $a_1,a_2,\dots$ be a sequence of integers satisfying $a_1=2$ and:
$$a_n=\begin{cases}a_{n-1}+1, & \text{ if }n\ne a_k \text{ for some }k=1,2,\dots,n-1; \\ a_{n-1}+2, & \text{ if } n=a_k \text{ for some }k=1,2,\dots,n-1. \end{cases}$$
Find the value of $a_{2022!}$.
1959 Czech and Slovak Olympiad III A, 2
Let $a, b, c$ be real numbers such that $a+b+c > 0$, $ab+bc+ca > 0$, $abc > 0$. Show that $a, b, c$ are all positive.
2007 Poland - Second Round, 2
We are given a cyclic quadrilateral $ABCD \quad AB\not=CD$. Quadrilaterals $AKDL$ and $CMBN$ are rhombuses with equal sides. Prove, that $KLMN$ is cyclic
2002 Austria Beginners' Competition, 1
We calculate the sum of $7$ natural consecutive pairs (e.g. $2+4+6+8+10+12+14$) and we will call the result $A$, then the sum of the next $7$ consecutive pairs (in the example, $16+ 18+...$) and its result we will call $B$, and finally we calculate the sum of the following $7$ consecutive pairs and its result we will call $C$. Can the product $ABC$ be $2002^3$?
2002 AIME Problems, 15
Circles $\mathcal{C}_{1}$ and $\mathcal{C}_{2}$ intersect at two points, one of which is $(9,6),$ and the product of the radii is $68.$ The x-axis and the line $y=mx$, where $m>0,$ are tangent to both circles. It is given that $m$ can be written in the form $a\sqrt{b}/c,$ where $a,$ $b,$ and $c$ are positive integers, $b$ is not divisible by the square of any prime, and $a$ and $c$ are relatively prime. Find $a+b+c.$
2009 Today's Calculation Of Integral, 454
Let $ a$ be positive constant number. Evaluate $ \int_{ \minus{} a}^a \frac {x^2\cos x \plus{} e^{x}}{e^{x} \plus{} 1}\ dx.$
2021 USA IMO Team Selection Test, 2
Points $A$, $V_1$, $V_2$, $B$, $U_2$, $U_1$ lie fixed on a circle $\Gamma$, in that order, and such that $BU_2 > AU_1 > BV_2 > AV_1$.
Let $X$ be a variable point on the arc $V_1 V_2$ of $\Gamma$ not containing $A$ or $B$. Line $XA$ meets line $U_1 V_1$ at $C$, while line $XB$ meets line $U_2 V_2$ at $D$. Let $O$ and $\rho$ denote the circumcenter and circumradius of $\triangle XCD$, respectively.
Prove there exists a fixed point $K$ and a real number $c$, independent of $X$, for which $OK^2 - \rho^2 = c$ always holds regardless of the choice of $X$.
[i]Proposed by Andrew Gu and Frank Han[/i]
2016 Peru IMO TST, 14
Determine all positive integers $M$ such that the sequence $a_0, a_1, a_2, \cdots$ defined by \[ a_0 = M + \frac{1}{2} \qquad \textrm{and} \qquad a_{k+1} = a_k\lfloor a_k \rfloor \quad \textrm{for} \, k = 0, 1, 2, \cdots \] contains at least one integer term.
2004 Bundeswettbewerb Mathematik, 3
Given two circles $k_1$ and $k_2$ which intersect at two different points $A$ and $B$. The tangent to the circle $k_2$ at the point $A$ meets the circle $k_1$ again at the point $C_1$. The tangent to the circle $k_1$ at the point $A$ meets the circle $k_2$ again at the point $C_2$. Finally, let the line $C_1C_2$ meet the circle $k_1$ in a point $D$ different from $C_1$ and $B$.
Prove that the line $BD$ bisects the chord $AC_2$.
1998 Federal Competition For Advanced Students, Part 2, 3
In a parallelogram $ABCD$ with the side ratio $AB : BC = 2 : \sqrt 3$ the normal through $D$ to $AC$ and the normal through $C$ to $AB$ intersects in the point $E$ on the line $AB$. What is the relationship between the lengths of the diagonals $AC$ and $BD$?
2011 AIME Problems, 2
On square $ABCD$, point $E$ lies on side $\overline{AD}$ and point $F$ lies on side $\overline{BC}$, so that $BE=EF=FD=30$. Find the area of square $ABCD$.
2003 AMC 8, 11
Business is a little slow at Lou's Fine Shoes, so Lou decides to have a sale. On Friday, Lou increases all of Thursday's prices by $10$ percent. Over the weekend, Lou advertises the sale: "Ten percent off the listed price. Sale starts Monday." How much does a pair of shoes cost on Monday that cost $40$ dollars on Thursday?
$\textbf{(A)}\ 36 \qquad
\textbf{(B)}\ 39.60 \qquad
\textbf{(C)}\ 40 \qquad
\textbf{(D)}\ 40.40 \qquad
\textbf{(E)}\ 44$