Found problems: 85335
2012 NIMO Problems, 4
The degree measures of the angles of nondegenerate hexagon $ABCDEF$ are integers that form a non-constant arithmetic sequence in some order, and $\angle A$ is the smallest angle of the (not necessarily convex) hexagon. Compute the sum of all possible degree measures of $\angle A$.
[i]Proposed by Lewis Chen[/i]
1969 Putnam, A1
Let $f(x,y)$ be a polynomial with real coefficients in the real variables $x$ and $y$ defined over the entire $xy$-plane. What are the possibilities for the range of $f(x,y)?$
2005 Today's Calculation Of Integral, 7
Calculate the following indefinite integrals.
[1] $\int \sqrt{x}(\sqrt{x}+1)^2 dx$
[2] $\int (e^x+2e^{x+1}-3e^{x+2})dx$
[3] $\int (\sin ^2 x+\cos x)\sin x dx$
[4] $\int x\sqrt{2-x} dx$
[5] $\int x\ln x dx$
2017 All-Russian Olympiad, 4
Are there infinite increasing sequence of natural numbers, such that sum of every 2 different numbers are relatively prime with sum of every 3 different numbers?
1984 Kurschak Competition, 2
$A_1B_1A_2$, $B_1A_2B_2$, $A_2B_2A_3$,...,$B_{13}A_{14}B_{14}$, $A_{14}B_{14}A_1$ and $B_{14}A_1B_1$ are equilateral rigid plates that can be folded along the edges $A_1B_1$,$B_1A_2$, ..., $A_{14}B_{14}$ and $B_{14}A_1$ respectively. Can they be folded so that all $28$ plates lie in the same plane?
2010 ITAMO, 2
Every non-negative integer is coloured white or red, so that:
• there are at least a white number and a red number;
• the sum of a white number and a red number is white;
• the product of a white number and a red number is red.
Prove that the product of two red numbers is always a red number, and the sum of two red numbers is always a red number.
2016 Kyrgyzstan National Olympiad, 2
The number $N$ consists only $2's$ and $1's$ in its [b]decimal representation[/b].We know that,after deleting digits from N,we can get any number consisting $9999$- $1's$ and $one$ - $2's$ in its [b]decimal representation[/b].[b][u]Find the least number of digits in the decimal representation of N[/u][/b]
2021 Belarusian National Olympiad, 8.1
Prove that there exists a $2021$-digit positive integer $\overline{a_1a_2\ldots a_{2021}}$, with all its digits being non-zero, such that for every $1 \leq n \leq 2020$ the following equality holds
$$\overline{a_1a_2\ldots a_n} \cdot \overline{a_{n+1}a_{n+2}\ldots a_{2021}}=\overline{a_1a_2\ldots a_{2021-n}} \cdot \overline{a_{2022-n}a_{2023-n}\ldots a_{2021}}$$
and all four numbers in the equality are pairwise different.
1955 Miklós Schweitzer, 5
[b]5.[/b] Show that a ring $R$ is commutative if for every $x \in R$ the element $x^{2}-x$ belongs to the centre of $R$. [b](A. 18)[/b]
2011 BMO TST, 4
Find all prime numbers p such that $2^p+p^2 $ is also a prime number.
2017 IMO Shortlist, A7
Let $a_0,a_1,a_2,\ldots$ be a sequence of integers and $b_0,b_1,b_2,\ldots$ be a sequence of [i]positive[/i] integers such that $a_0=0,a_1=1$, and
\[
a_{n+1} =
\begin{cases}
a_nb_n+a_{n-1} & \text{if $b_{n-1}=1$} \\
a_nb_n-a_{n-1} & \text{if $b_{n-1}>1$}
\end{cases}\qquad\text{for }n=1,2,\ldots.
\]
for $n=1,2,\ldots.$ Prove that at least one of the two numbers $a_{2017}$ and $a_{2018}$ must be greater than or equal to $2017$.
1995 Bundeswettbewerb Mathematik, 3
Each diagonal of a convex pentagon is parallel to one side of the pentagon. Prove that the ratio of the length of a diagonal to that of its corresponding side is the same for all five diagonals, and compute this ratio.
2005 National High School Mathematics League, 3
$\triangle ABC$ is inscribed to unit circle. Bisector of $\angle A,\angle B,\angle C$ intersect the circle at $A_1,B_1,C_1$ respectively. The value of $\frac{\displaystyle AA_1\cdot\cos\frac{A}{2}+BB_1\cdot\cos\frac{B}{2}+CC_1\cdot\cos\frac{C}{2}}{\sin A+\sin B+\sin C}$ is
$\text{(A)}2\qquad\text{(B)}4\qquad\text{(C)}6\qquad\text{(D)}8$
1998 Iran MO (3rd Round), 1
Define the sequence $(x_n)$ by $x_0 = 0$ and for all $n \in \mathbb N,$
\[x_n=\begin{cases} x_{n-1} + (3^r - 1)/2,&\mbox{ if } n = 3^{r-1}(3k + 1);\\ x_{n-1} - (3^r + 1)/2, & \mbox{ if } n = 3^{r-1}(3k + 2).\end{cases}\]
where $k \in \mathbb N_0, r \in \mathbb N$. Prove that every integer occurs in this sequence exactly once.
Kyiv City MO Juniors 2003+ geometry, 2011.8.41
The medians $AL, BM$, and $CN$ are drawn in the triangle $ABC$. Prove that $\angle ANC = \angle ALB$ if and only if $\angle ABM =\angle LAC$.
(Veklich Bogdan)
2006 Chile National Olympiad, 6
Let $ \vartriangle ABC $ be an acute triangle and scalene, with $ BC $ its smallest side. Let $ P, Q $ points on $ AB, AC $ respectively, such that $ BQ = CP = BC $. Let $ O_1, O_2 $ be the centers of the circles circumscribed to $ \vartriangle AQB, \vartriangle APC $, respectively. Sean $ H, O $ the orthocenter and circumcenter of $ \vartriangle ABC $
a) Show that $ O_1O_2 = BC $.
b) Show that $ BO_2, CO_1 $ and $ HO $ are concurrent
2012 NIMO Summer Contest, 11
Let $a$ and $b$ be two positive integers satisfying the equation
\[
20\sqrt{12} = a\sqrt{b}.
\]
Compute the sum of all possible distinct products $ab$.
[i]Proposed by Lewis Chen[/i]
2020 USMCA, 1
If $U, S, M, C, A$ are distinct (not necessarily positive) integers such that $U \cdot S \cdot M \cdot C \cdot A = 2020$, what is the greatest possible value of $U + S + M + C + A$?
2023 All-Russian Olympiad Regional Round, 10.10
Prove that for all positive reals $x, y, z$, the inequality $(x-y)\sqrt{3x^2+y^2}+(y-z)\sqrt{3y^2+z^2}+(z-x)\sqrt{3z^2+x^2} \geq 0$ is satisfied.
2010 Contests, 1
Let $ABC$ be a triangle with $\angle BAC \neq 90^{\circ}.$ Let $O$ be the circumcenter of the triangle $ABC$ and $\Gamma$ be the circumcircle of the triangle $BOC.$ Suppose that $\Gamma$ intersects the line segment $AB$ at $P$ different from $B$, and the line segment $AC$ at $Q$ different from $C.$ Let $ON$ be the diameter of the circle $\Gamma.$ Prove that the quadrilateral $APNQ$ is a parallelogram.
2015 Singapore MO Open, 3
Find all functions $f : \mathbb{R} \rightarrow \mathbb{R}$, where $\mathbb{R}$ is the set of real numbers, such that
$f(x)f(yf(x) - 1) = x^2 f(y) - f(x) \quad\forall x,y \in \mathbb{R}$
2007 Mathematics for Its Sake, 3
Prove that there exists only one pair $ (p,q) $ of odd primes satisfying the properties that $ p^2\equiv 4\pmod q $ and $
q^2\equiv 1\pmod p. $
[i]Ana Maria Acu[/i]
2018 Tajikistan Team Selection Test, 7
Problem 7. On the board, Sabir writes 10 consecutive numbers. For each number, Salim writes the sum of its digits on his paper, and Sabrina writes the number of its divisors on her paper. Is it possible for Sabrina’s 10 numbers to be exactly the same as Salim’s 10 numbers in some order? (the repetitions of the numbers should also be the same)
1997 Croatia National Olympiad, Problem 4
An infinite sheet of paper is divided into equal squares, some of which are colored red. In each $2\times3$ rectangle, there are exactly two red squares. Now consider an arbitrary $9\times11$ rectangle. How many red squares does it contain? (The sides of all considered rectangles go along the grid lines.)
2021 BMT, 7
For a given positive integer $n$, you may perform a series of steps. At each step, you may apply an operation: you may increase your number by one, or if your number is divisible by 2, you may divide your number by 2. Let $\ell(n)$ be the minimum number of operations needed to transform the number $n$ to 1 (for example, $\ell(1) = 0$ and $\ell(7) = 4$). How many positive integers $n$ are there such that $\ell(n) \leq 12$?