Found problems: 85335
1990 Federal Competition For Advanced Students, P2, 3
In a convex quadrilateral $ ABCD$, let $ E$ be the intersection point of the diagonals, and let $ F_1,F_2,$ and $ F$ be the areas of $ ABE,CDE,$ and $ ABCD,$ respectively. Prove that:
$ \sqrt {F_1}\plus{}\sqrt {F_2} \le \sqrt {F}.$
2011 Kosovo National Mathematical Olympiad, 1
It is given the function $f:\mathbb{R} \to \mathbb{R}$ such that it holds $f(\sin x)=\sin (2011x)$. Find the value of $f(\cos x)$.
1955 AMC 12/AHSME, 4
The equality $ \frac{1}{x\minus{}1}\equal{}\frac{2}{x\minus{}2}$ is satisfied by:
$ \textbf{(A)}\ \text{no real values of }x \qquad
\textbf{(B)}\ \text{either }x\equal{}1 \text{ or }x\equal{}2 \qquad
\textbf{(C)}\ \text{only }x\equal{}1 \\
\textbf{(D)}\ \text{only }x\equal{}2 \qquad
\textbf{(E)}\ \text{only }x\equal{}0$
2014 Contests, 4
What is the probability of having $2$ adjacent white balls or $2$ adjacent blue balls in a random arrangement of $3$ red, $2$ white and $2$ blue balls?
$
\textbf{(A)}\ \dfrac{2}{5}
\qquad\textbf{(B)}\ \dfrac{3}{7}
\qquad\textbf{(C)}\ \dfrac{16}{35}
\qquad\textbf{(D)}\ \dfrac{10}{21}
\qquad\textbf{(E)}\ \dfrac{5}{14}
$
2011 Today's Calculation Of Integral, 749
Let $m$ be a positive integer. A tangent line at the point $P$ on the parabola $C_1 : y=x^2+m^2$ intersects with the parabola $C_2 : y=x^2$ at the points $A,\ B$. For the point $Q$ between $A$ and $B$ on $C_2$, denote by $S$ the sum of the areas of the region bounded by the line $AQ$,$C_2$ and the region bounded by the line $QB$, $C_2$. When $Q$ move between $A$ and $B$ on $C_2$, prove that the minimum value of $S$ doesn't depend on how we would take $P$, then find the value in terms of $m$.
2019 Bulgaria EGMO TST, 3
$A$ and $B$ play a game, given an integer $N$, $A$ writes down $1$ first, then every player sees the last number written and if it is $n$ then in his turn he writes $n+1$ or $2n$, but his number cannot be bigger than $N$. The player who writes $N$ wins. For which values of $N$ does $B$ win?
[i]Proposed by A. Slinko & S. Marshall, New Zealand[/i]
2005 Today's Calculation Of Integral, 71
Find the minimum value of $\int_{-1}^1 \sqrt{|t-x|}\ dt$
2013 Kazakhstan National Olympiad, 1
On the board written numbers from 1 to 25 . Bob can pick any three of them say $a,b,c$ and replace by $a^3+b^3+c^3$ . Prove that last number on the board can not be $2013^3$.
2005 Miklós Schweitzer, 3
Let $\alpha\leq 22$ be non-negative integer. For which $\alpha$ does the equation $$8x^{23}-5^{\alpha}y^{23}=1$$ have the most integer solutions (x,y)? What can we say about $\alpha\geq 23$?
[hide=Note]I believe the eqn has solutions only when $\alpha=0$. taking modulo 47, $\alpha\equiv 9,17\pmod{23}$ or ($23|\alpha$ and $47|x$). taking modulo 139 and 277 eliminates the $\alpha\equiv 9,17\pmod{23}$ cases. 139=23*6+1 , 277=23*12+1[/hide]
Cono Sur Shortlist - geometry, 2003.G7.3
Let $ABC$ be an acute triangle such that $\angle{B}=60$. The circle with diameter $AC$ intersects the internal angle bisectors of $A$ and $C$ at the points $M$ and $N$, respectively $(M\neq{A},$ $N\neq{C})$. The internal bisector of $\angle{B}$ intersects $MN$ and $AC$ at the points $R$ and $S$, respectively. Prove that $BR\leq{RS}$.
2010 Today's Calculation Of Integral, 599
Evaluate $\int_0^{\frac{\pi}{6}} \frac{e^x(\sin x+\cos x+\cos 3x)}{\cos^ 2 {2x}}\ dx$.
created by kunny
2023 AIME, 4
Let $x$, $y$, and $z$ be real numbers satisfying the system of equations
\begin{align*}
xy+4z&=60\\
yz+4x&=60\\
zx+4y&=60.
\end{align*}
Let $S$ be the set of possible values of $x$. Find the sum of the squares of the elements of $S$.
2023 Mexican Girls' Contest, 6
Alka finds a number $n$ written on a board that ends in $5.$ She performs a sequence of operations with the number on the board. At each step, she decides to carry out one of the following two operations:
$1.$ Erase the written number $m$ and write it´s cube $m^3$.
$2.$ Erase the written number $m$ and write the product $2023m$.
Alka performs each operation an even number of times in some order and at least once, she finally obtains the number $r$. If the tens digit of $r$ is an odd number, find all possible values that the tens digit of $n^3$ could have had.
1998 Irish Math Olympiad, 3
Show that no integer of the form $ xyxy$ in base $ 10$ can be a perfect cube. Find the smallest base $ b>1$ for which there is a perfect cube of the form $ xyxy$ in base $ b$.
2023 Stanford Mathematics Tournament, 2
Every cell in a $5\times5$ grid of paper is to be painted either red or white with equal probability. An edge of the paper is said to have a "tree" if the set of cells depicted in the diagram below are all painted red when the paper is rotated so that the edge lies at the bottom. Given that at least one edge of the paper has a tree, what is the expected number of edges that have a tree?
[center][img]https://cdn.artofproblemsolving.com/attachments/1/2/f81d8da53d7bc6819fc1dfe4acb9567d545856.png[/img][/center]
2019 Regional Olympiad of Mexico Center Zone, 6
Find all positive integers $m$ with the next property:
If $d$ is a positive integer less or equal to $m$ and it isn't coprime to $m$ , then there exist positive integers $a_{1}, a_{2}$,. . ., $a_{2019}$ (where all of them are coprimes to $m$) such that
$m+a_{1}d+a_{2}d^{2}+\cdot \cdot \cdot+a_{2019}d^{2019}$
is a perfect power.
2022 Bulgaria JBMO TST, 2
Let $ABC$ ($AB < AC$) be a triangle with circumcircle $k$. The tangent to $k$ at $A$ intersects the line $BC$ at $D$ and the point $E\neq A$ on $k$ is such that $DE$ is tangent to $k$. The point $X$ on line $BE$ is such that $B$ is between $E$ and $X$ and $DX = DA$ and the point $Y$ on the line $CX$ is such that $Y$ is between $C$ and $X$ and $DY = DA$. Prove that the lines $BC$ and $YE$ are perpendicular.
2005 China Team Selection Test, 3
Let $n$ be a positive integer, and $a_j$, for $j=1,2,\ldots,n$ are complex numbers. Suppose $I$ is an arbitrary nonempty subset of $\{1,2,\ldots,n\}$, the inequality $\left|-1+ \prod_{j\in I} (1+a_j) \right| \leq \frac 12$ always holds.
Prove that $\sum_{j=1}^n |a_j| \leq 3$.
2024/2025 TOURNAMENT OF TOWNS, P2
Peter and Basil take turns drawing roads on a plane, Peter starts. The road is either horizontal or a vertical line along which one can drive in only one direction (that direction is determined when the road is drawn). Can Basil always act in such a way that after each of his moves one could drive according to the rules between any two constructed crossroads, regardless of Peter's actions?
Alexandr Perepechko
2009 National Olympiad First Round, 26
For every $ 0 \le i \le 17$, $ a_i \equal{} \{ \minus{} 1, 0, 1\}$.
How many $ (a_0,a_1, \dots , a_{17})$ $ 18 \minus{}$tuples are there satisfying :
$ a_0 \plus{} 2a_1 \plus{} 2^2a_2 \plus{} \cdots \plus{} 2^{17}a_{17} \equal{} 2^{10}$
$\textbf{(A)}\ 9 \qquad\textbf{(B)}\ 8 \qquad\textbf{(C)}\ 7 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ 1$
2003 Baltic Way, 3
Let $x$, $y$ and $z$ be positive real numbers such that $xyz = 1$. Prove that
$$\left(1+x\right)\left(1+y\right)\left(1+z\right)\geq 2\left(1+\sqrt[3]{\frac{x}{z}}+\sqrt[3]{\frac{y}{x}}+\sqrt[3]{\frac{z}{y}}\right).$$
2025 Kyiv City MO Round 1, Problem 5
Find all quadruples of positive integers \( (a, p, q, r) \), where \( p, q, r \) are prime numbers, such that the following equation holds:
\[
p^2q^2 + q^2r^2 + r^2p^2 + 3 = 4 \cdot 13^a.
\]
[i]Proposed by Oleksii Masalitin[/i]
1995 Miklós Schweitzer, 12
Let F(x) be a known distribution function, the random variables $\eta_1 , \eta_2 ...$ be independent of the common distribution function $F( x - \vartheta)$, where $\vartheta$ is the shift parameter. Let us call the shift parameter "well estimated" if there exists a positive constant c, so that any of $\varepsilon> 0$ there exist a Lebesgue measure $\varepsilon$ Borel set E ("confidence set") and a Borel-measurable function $t_n( x_1 ,. .., x_n )$ ( n = 1,2, ...) such that for any $\vartheta$ we have
$$P ( \vartheta- t_n ( \eta_1 , ..., \eta_n ) \in E )> 1-e^{-cn} \qquad( n > n_0 ( \varepsilon, F ) )$$
Prove that
a) if F is not absolutely continuous, then the shift parameter is "well estimated",
b) if F is absolutely continuous and F' is continuous, then it is not "well estimated".
1997 Singapore Team Selection Test, 3
Suppose the numbers $a_0, a_1, a_2, ... , a_n$ satisfy the following conditions:
$a_0 =\frac12$, $a_{k+1} = a_k +\frac{1}{n}a_k^2$ for $k = 0, 1, ... , n - 1$.
Prove that $1 - \frac{1}{n}< a_n < 1$
2021 AIME Problems, 9
Let $ABCD$ be an isosceles trapezoid with $AD=BC$ and $AB<CD.$ Suppose that the distances from $A$ to the lines $BC,CD,$ and $BD$ are $15,18,$ and $10,$ respectively. Let $K$ be the area of $ABCD.$ Find $\sqrt2 \cdot K.$