Found problems: 85335
2008 Romania National Olympiad, 3
Let $ n$ be a positive integer and let $ a_i$ be real numbers, $ i \equal{} 1,2,\ldots,n$ such that $ |a_i|\leq 1$ and $ \sum_{i\equal{}1}^n a_i \equal{} 0$.
Show that $ \sum_{i\equal{}1}^n |x \minus{} a_i|\leq n$, for every $ x\in \mathbb{R}$ with $ |x|\le 1$.
1985 AMC 8, 13
If you walk for $ 45$ minutes at a rate of $ 4$ mph and then run for $ 30$ minutes at a rate of $ 10$ mph, how many miles have you gone at the end of one hour and $ 15$ minutes?
\[ \textbf{(A)}\ 3.5 \text{ miles} \qquad
\textbf{(B)}\ 8 \text{ miles} \qquad
\textbf{(C)}\ 9 \text{ miles} \qquad
\textbf{(D)}\ 25 \frac{1}{3} \text{ miles} \qquad
\textbf{(E)}\ 480 \text{ miles}
\]
2011 China Team Selection Test, 2
Let $\ell$ be a positive integer, and let $m,n$ be positive integers with $m\geq n$, such that $A_1,A_2,\cdots,A_m,B_1,\cdots,B_m$ are $m+n$ pairwise distinct subsets of the set $\{1,2,\cdots,\ell\}$. It is known that $A_i\Delta B_j$ are pairwise distinct, $1\leq i\leq m, 1\leq j\leq n$, and runs over all nonempty subsets of $\{1,2,\cdots,\ell\}$. Find all possible values of $m,n$.
2001 Putnam, 2
For each $k$, $\mathcal{C}_k$ is biased so that, when tossed, it has probability $\tfrac{1}{(2k+1)}$ of falling heads. If the $n$ coins are tossed, what is the probability that the number of heads is odd? Express the answer as a rational function $n$.
2021 Kyiv City MO Round 1, 9.5
Let $BM$ be the median of triangle $ABC$ in which $AB > BC$. The point $P$ is chosen so that $AB\parallel PC$ and $PM \perp BM$. On the line $BP$, point $Q$ is chosen so that $\angle AQC = 90^\circ$, and points $B$ and $Q$ are on opposite sides of the line $AC$. Prove that $AB = BQ$.
[i]Proposed by Mykhailo Shtandenko[/i]
2018 Online Math Open Problems, 8
Compute the number of ordered quadruples $(a,b,c,d)$ of distinct positive integers such that $\displaystyle \binom{\binom{a}{b}}{\binom{c}{d}}=21$.
[i]Proposed by Luke Robitaille[/i]
2010 China Team Selection Test, 2
Prove that there exists a sequence of unbounded positive integers $a_1\leq a_2\leq a_3\leq\cdots$, such that there exists a positive integer $M$ with the following property: for any integer $n\geq M$, if $n+1$ is not prime, then any prime divisor of $n!+1$ is greater than $n+a_n$.
2004 Tuymaada Olympiad, 2
In the plane are given 100 lines such that no 2 are parallel and no 3 meet in a point. The intersection points are marked. Then all the lines and k of the marked points are erased. Given the remained points of intersection for what max k one can reconstruct the lines?
[i]Proposed by A. Golovanov[/i]
2000 Romania National Olympiad, 3
Let be a tetahedron $ ABCD, $ and $ E $ be the projection of $ D $ on the plane formed by $ ABC. $ If $ \mathcal{A}_{\mathcal{R}} $ denotes the area of the region $ \mathcal{R}, $ show that the following affirmations are equivalent:
[b]a)[/b] $ C=E\vee CE\parallel AB $
[b]b)[/b] $ M\in\overline{CD}\implies\mathcal{A}_{ABM}^2=\frac{CM^2}{CD^2}\cdot\mathcal{A}_{ABD}^2 +\left( 1-\frac{CM^2}{CD^2}\right)\cdot\mathcal{A}_{ABC}^2 $
2009 Sharygin Geometry Olympiad, 5
Let $n$ points lie on the circle. Exactly half of triangles formed by these points are acute-angled. Find all possible $n$.
(B.Frenkin)
2010 AMC 12/AHSME, 24
The set of real numbers $ x$ for which
\[ \frac{1}{x\minus{}2009}\plus{}\frac{1}{x\minus{}2010}\plus{}\frac{1}{x\minus{}2011}\ge 1\]
is the union of intervals of the form $ a<x\le b$. What is the sum of the lengths of these intervals?
$ \textbf{(A)}\ \frac{1003}{335} \qquad \textbf{(B)}\ \frac{1004}{335} \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ \frac{403}{134} \qquad \textbf{(E)}\ \frac{202}{67}$
2023 VN Math Olympiad For High School Students, Problem 5
Given a triangle $ABC$ with [i]Lemoine[/i] point $L.$ Let $a=BC, b=CA,c=AB.$
Prove that: ${a^2}\overrightarrow {LA} + {b^2}\overrightarrow {LB} + {c^2}\overrightarrow {LC} = \overrightarrow 0 .$
2021 Canadian Junior Mathematical Olympiad, 2
How many ways are there to permute the first $n$ positive integers such that in the permutation, for each value of $k \le n$, the first $k$ elements of the permutation have distinct remainder mod $k$?
2015 India IMO Training Camp, 3
Let $G$ be a simple graph on the infinite vertex set $V=\{v_1, v_2, v_3,\ldots\}$. Suppose every subgraph of $G$ on a finite vertex subset is $10$-colorable, Prove that $G$ itself is $10$-colorable.
2011 Today's Calculation Of Integral, 702
$f(x)$ is a continuous function defined in $x>0$. For all $a,\ b\ (a>0,\ b>0)$, if $\int_a^b f(x)\ dx$ is determined by only $\frac{b}{a}$, then prove that $f(x)=\frac{c}{x}\ (c: constant).$
2022 All-Russian Olympiad, 4
There are $18$ children in the class. Parents decided to give children from this class a cake. To do this, they first learned from each child the area of the piece he wants to get. After that, they showed a square-shaped cake, the area of which is exactly equal to the sum of $18$ named numbers. However, when they saw the cake, the children wanted their pieces to be squares too. The parents cut the cake with lines parallel to the sides of the cake (cuts do not have to start or end on the side of the cake). For what maximum k the parents are guaranteed to cut out $k$ square pieces from the cake, which you can give to $k$ children so that each of them gets what they want?
2024/2025 TOURNAMENT OF TOWNS, P2
Two polynomials with real coefficients have the leading coefficients equal to 1 . Each polynomial has an odd degree that is equal to the number of its distinct real roots. The product of the values of the first polynomial at the roots of the second polynomial is equal to 2024. Find the product of the values of the second polynomial at the roots of the first one.
Sergey Yanzhinov
2020 Dutch IMO TST, 4
Let $a, b \ge 2$ be positive integers with $gcd (a, b) = 1$. Let $r$ be the smallest positive value that $\frac{a}{b}- \frac{c}{d}$ can take, where $c$ and $d$ are positive integers satisfying $c \le a$ and $d \le b$. Prove that $\frac{1}{r}$ is an integer.
1983 Iran MO (2nd round), 3
Find a matrix $A_{(2 \times 2)}$ for which
\[ \begin{bmatrix}2 &1 \\ 3 & 2\end{bmatrix} A \begin{bmatrix}3 & 2 \\ 4 & 3\end{bmatrix} = \begin{bmatrix}1 & 2 \\ 2 & 1\end{bmatrix}.\]
2006 Petru Moroșan-Trident, 1
Let be a natural number $ n\ge 3. $ Solve the equation $ \lfloor x/n \rfloor =\lfloor x-n \rfloor $ in $ \mathbb{R} . $
[i]Constantin Nicolau[/i]
1999 National High School Mathematics League, 9
In $\triangle ABC$, if $9a^2+9b^2-19c^2=0$, then $\frac{\cot C}{\cot A+\cot B}=$________.
PEN K Problems, 11
Find all functions $f: \mathbb{N}_{0}\to \mathbb{N}_{0}$ such that for all $m,n\in \mathbb{N}_{0}$: \[mf(n)+nf(m)=(m+n)f(m^{2}+n^{2}).\]
2001 All-Russian Olympiad Regional Round, 9.6
Is there such a natural number that the product of all its natural divisors (including $1$ and the number itself) ends exactly in $2001$ zeros?
2018 Yasinsky Geometry Olympiad, 3
Construct triangle $ABC$, given the altitude and the angle bisector both from $A$, if it is known for the sides of the triangle $ABC$ that $2BC = AB + AC$.
(Alexey Karlyuchenko)
2016 Online Math Open Problems, 2
Let $x$, $y$, and $z$ be real numbers such that $x+y+z=20$ and $x+2y+3z=16$. What is the value of $x+3y+5z$?
[i]Proposed by James Lin[/i]