Found problems: 85335
2001 Romania National Olympiad, 1
Let $A$ be a set of real numbers which verifies:
\[ a)\ 1 \in A \\ b) \ x\in A\implies x^2\in A\\ c)\ x^2-4x+4\in A\implies x\in A \]
Show that $2000+\sqrt{2001}\in A$.
2021 Bolivian Cono Sur TST, 2
Find all posible pairs of positive integers $x,y$ such that $$\text{lcm}(x,y+3001)=\text{lcm}(y,x+3001)$$
1969 IMO Longlists, 4
$(BEL 4)$ Let $O$ be a point on a nondegenerate conic. A right angle with vertex $O$ intersects the conic at points $A$ and $B$. Prove that the line $AB$ passes through a fixed point located on the normal to the conic through the point $O.$
2010 Hanoi Open Mathematics Competitions, 8
If $n$ and $n^3+2n^2+2n+4$ are both perfect squares, fi nd $n$.
2010 Saudi Arabia IMO TST, 3
Find all primes $p$ for which $p^2 - p + 1$ is a perfect cube.
1997 All-Russian Olympiad Regional Round, 8.6
The numbers from 1 to 37 are written in a line so that the sum of any first several numbers is divided by the number following them. What number is worth in third place, if the number 37 is written in the first place, and in the second, 1?
2022/2023 Tournament of Towns, P5
On the sides of a regular nonagon $ABCDEFGHI$, triangles $XAB, YBC, ZCD$ and $TDE$ are constructed outside the nonagon. The angles at $X, Y, Z, T$ in these triangles are each $20^\circ$. The angles $XAB, YBC, ZCD$ and $TDE$ are such that (except for the first angle) each angle is $20^\circ$ greater than the one listed before it. Prove that the points $X, Y , Z, T$ lie on the same circle.
2006 Vietnam Team Selection Test, 1
Prove that for all real numbers $x,y,z \in [1,2]$ the following inequality always holds:
\[ (x+y+z)(\frac{1}{x}+\frac{1}{y}+\frac{1}{z})\geq 6(\frac{x}{y+z}+\frac{y}{z+x}+\frac{z}{x+y}). \]
When does the equality occur?
2006 Switzerland Team Selection Test, 2
Let $n\ge5$ be an integer. Find the biggest integer $k$ such that there always exists a $n$-gon with exactly $k$ interior right angles. (Find $k$ in terms of $n$).
2017 Saudi Arabia JBMO TST, 5
Let $a,b,c>0$ and $a+b+c=6$ . Prove that $$ \frac{1}{a^2b+16}+\frac{1}{b^2c+16}+\frac{1}{c^2a+16}
\ge \frac{1}{8}.$$
2011 CIIM, Problem 3
Let $f(x)$ be a rational function with complex coefficients whose denominator does not have multiple roots. Let $u_0, u_1,... , u_n$ be the complex roots of $f$ and $w_1, w_2,..., w_m$ be the roots of $f'$. Suppose that $u_0$ is a simple root of $f$. Prove that
\[ \sum_{k=1}^m \frac{1}{w_k - u_0} = 2\sum_{k = 1}^n\frac{1}{u_k - u_0}.\]
2003 AMC 10, 5
Moe uses a mower to cut his rectangular $ 90$-foot by $ 150$-foot lawn. The swath he cuts is $ 28$ inches wide, but he overlaps each cut by $ 4$ inches to make sure that no grass is missed. He walks at the rate of $ 5000$ feet per hour while pushing the mower. Which of the following is closest to the number of hours it will take Moe to mow his lawn?
$ \textbf{(A)}\ 0.75 \qquad
\textbf{(B)}\ 0.8 \qquad
\textbf{(C)}\ 1.35 \qquad
\textbf{(D)}\ 1.5 \qquad
\textbf{(E)}\ 3$
2023 Kazakhstan National Olympiad, 2
$a,b,c$ are positive real numbers such that $a+b+c\ge 3$ and $a^2+b^2+c^2=2abc+1$. Prove that $$a+b+c\le 2\sqrt{abc}+1$$
2016 AMC 12/AHSME, 5
Goldbach's conjecture states that every even integer greater than 2 can be written as the sum of two prime numbers (for example, $2016=13+2003$). So far, no one has been able to prove that the conjecture is true, and no one has found a counterexample to show that the conjecture is false. What would a counterexample consist of?
$ \textbf{(A)}\ \text{an odd integer greater than } 2 \text{ that can be written as the sum of two prime numbers}$\\
$\textbf{(B)}\ \text{an odd integer greater than } 2 \text{ that cannot be written as the sum of two prime numbers}$\\
$\textbf{(C)}\ \text{an even integer greater than } 2 \text{ that can be written as the sum of two numbers that are not prime}$\\
$\textbf{(D)}\ \text{an even integer greater than } 2 \text{ that can be written as the sum of two prime numbers}$\\
$\textbf{(E)}\ \text{an even integer greater than } 2 \text{ that cannot be written as the sum of two prime numbers}$
2008 Vietnam Team Selection Test, 1
Let $ m$ and $ n$ be positive integers. Prove that $ 6m | (2m \plus{} 3)^n \plus{} 1$ if and only if $ 4m | 3^n \plus{} 1$
1993 All-Russian Olympiad Regional Round, 10.3
Solve in positive numbers the system
$ x_1\plus{}\frac{1}{x_2}\equal{}4, x_2\plus{}\frac{1}{x_3}\equal{}1, x_3\plus{}\frac{1}{x_4}\equal{}4, ..., x_{99}\plus{}\frac{1}{x_{100}}\equal{}4, x_{100}\plus{}\frac{1}{x_1}\equal{}1$
2023 Belarusian National Olympiad, 9.1
Real numbers $a,b,c,d$ satisfy the equality
$$\frac{1-ab}{a+b}=\frac{bc-1}{b+c}=\frac{1-cd}{c+d}=\sqrt{3}$$
Find all possible values of $ad$.
2000 Saint Petersburg Mathematical Olympiad, 11.7
It is known that for irrational numbers $\alpha$, $\beta$, $\gamma$, $\delta$ and for any positive integer $n$ the following is true:
$$[n\alpha]+[n\beta]=[n\gamma]+[n\delta]$$
Does this mean that sets $\{\alpha,\beta\}$ and $\{\gamma,\delta\}$ are equal? (As usual $[x]$ means the greatest integer not greater than $x$).
2003 Tournament Of Towns, 3
Can one cover a cube by three paper triangles (without overlapping)?
1989 Irish Math Olympiad, 2
A 3x3 magic square, with magic number $m$, is a $3\times 3$ matrix such that the entries on each row, each column and each diagonal sum to $m$. Show that if the square has positive integer entries, then $m$ is divisible by $3$, and each entry of the square is at most $2n-1$, where $m=3n$. An example of a magic square with $m=6$ is
\[\left( \begin{array}{ccccc}
2 & 1 & 3\\
3 & 2 & 1\\
1 & 3 & 2
\end{array} \right)\]
2018 ABMC, Team
[u]Round 5[/u]
[b]5.1.[/b] A triangle has lengths such that one side is $12$ less than the sum of the other two sides, the semi-perimeter of the triangle is $21$, and the largest and smallest sides have a difference of $2$. Find the area of this triangle.
[b]5.2.[/b] A rhombus has side length $85$ and diagonals of integer lengths. What is the sum of all possible areas of the rhombus?
[b]5.3.[/b] A drink from YAKSHAY’S SHAKE SHOP is served in a container that consists of a cup, shaped like an upside-down truncated cone, and a semi-spherical lid. The ratio of the radius of the bottom of the cup to the radius of the lid is $\frac23$ , the volume of the combined cup and lid is $296\pi$, and the height of the cup is half of the height of the entire drink container. What is the volume of the liquid in the cup if it is filled up to half of the height of the entire drink container?
[u]Round 6[/u]
[i]Each answer in the next set of three problems is required to solve a different problem within the same set. There is one correct solution to all three problems; however, you will receive points for any correct answer regardless whether other answers are correct.[/i]
[b]6.1.[/b] Let the answer to problem $2$ be $b$. There are b people in a room, each of which is either a truth-teller or a liar. Person $1$ claims “Person $2$ is a liar,” Person $2$ claims “Person $3$ is a liar,” and so on until Person $b$ claims “Person $1$ is a liar.” How many people are truth-tellers?
[b]6.2.[/b] Let the answer to problem $3$ be $c$. What is twice the area of a triangle with coordinates $(0, 0)$, $(c, 3)$ and $(7, c)$ ?
[b]6.3.[/b] Let the answer to problem $ 1$ be $a$. Compute the smaller zero to the polynomial $x^2 - ax + 189$ which has $2$ integer roots.
[u]Round 7[/u]
[b]7.1. [/b]Sir Isaac Neeton is sitting under a kiwi tree when a kiwi falls on his head. He then discovers Neeton’s First Law of Kiwi Motion, which states:
[i]Every minute, either $\left\lfloor \frac{1000}{d} \right\rfloor$ or $\left\lceil \frac{1000}{d} \right\rceil$ kiwis fall on Neeton’s head, where d is Neeton’s distance from the tree in centimeters.[/i]
Over the next minute, $n$ kiwis fall on Neeton’s head. Let $S$ be the set of all possible values of Neeton’s distance from the tree. Let m and M be numbers such that $m < x < M$ for all elements $x$ in $S$. If the least possible value of $M - m$ is $\frac{2000}{16899}$ centimeters, what is the value of $n$?
Note that $\lfloor x \rfloor$ is the greatest integer less than or equal to $x$, and $\lceil x \rceil$ is the least integer greater than or equal to $x$.
[b]7.2.[/b] Nithin is playing chess. If one queen is randomly placed on an $ 8 \times 8$ chessboard, what is the expected number of squares that will be attacked including the square that the queen is placed on? (A square is under attack if the queen can legally move there in one move, and a queen can legally move any number of squares diagonally, horizontally or vertically.)
[b]7.3.[/b] Nithin is writing binary strings, where each character is either a $0$ or a $1$. How many binary strings of length $12$ can he write down such that $0000$ and $1111$ do not appear?
[u]Round 8[/u]
[b]8.[/b] What is the period of the fraction $1/2018$? (The period of a fraction is the length of the repeated portion of its decimal representation.) Your answer will be scored according to the following formula, where $X$ is the correct answer and $I$ is your input.
$$max \left\{ 0, \left\lceil min \left\{13 - \frac{|I-X|}{0.1 |I|}, 13 - \frac{|I-X|}{0.1 |I-2X|} \right\} \right\rceil \right\}$$
PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h2765571p24215461]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
2019 Moldova Team Selection Test, 4
Quadrilateral $ABCD$ is inscribed in circle $\Gamma$ with center $O$. Point $I$ is the incenter of triangle $ABC$, and point $J$ is the incenter of the triangle $ABD$. Line $IJ$ intersects segments $AD, AC, BD, BC$ at points $P, M, N$ and, respectively $Q$. The perpendicular from $M$ to line $AC$ intersects the perpendicular from $N$ to line $BD$ at point $X$. The perpendicular from $P$ to line $AD$ intersects the perpendicular from $Q$ to line $BC$ at point $Y$. Prove that $X, O, Y$ are colinear.
1992 India National Olympiad, 4
Find the number of permutations $( p_1, p_2, p_3 , p_4 , p_5 , p_6)$ of $1, 2 ,3,4,5,6$ such that for any $k, 1 \leq k \leq 5$, $(p_1, \ldots, p_k)$ does not form a permutation of $1 , 2, \ldots, k$.
2009 Tournament Of Towns, 7
At the entrance to a cave is a rotating round table. On top of the table are $n$ identical barrels, evenly spaced along its circumference. Inside each barrel is a herring either with its head up or its head down. In a move, Ali Baba chooses from $1$ to $n$ of the barrels and turns them upside down. Then the table spins around. When it stops, it is impossible to tell which barrels have been turned over. The cave will open if the heads of the herrings in all $n$ barrels are up or are all down. Determine all values of $n$ for which Ali Baba can open the cave in a fi nite number of moves.
[i](11 points)[/i]
2012 Today's Calculation Of Integral, 853
Let $0<a<\frac {\pi}2.$ Find $\lim_{a\rightarrow +0} \frac{1}{a^3}\int_0^a \ln\ (1+\tan a\tan x)\ dx.$