Found problems: 85335
Kvant 2024, M2797
For real numbers $0 \leq a_1 \leq a_2 \leq ... \leq a_n$ and $0 \leq b_1 \leq b_2 \leq ... \leq b_n$ prove that \[ \left( \frac{a_1}{1 \cdot 2}+\frac{a_2}{2 \cdot 3}+...+\frac{a_n}{n(n+1)} \right) \times \left( \frac{b_1}{1 \cdot 2}+\frac{b_2}{2 \cdot 3}+...+\frac{b_n}{n(n+1)} \right) \leq \frac{a_1b_1}{1 \cdot 2}+\frac{a_2b_2}{2 \cdot 3}+...+\frac{a_nb_n}{n(n+1)}.\]
[i]Proposed by A. Antropov[/i]
2002 AMC 10, 11
Let $P(x)=kx^3+2k^2x^2+k^3$. Find the sum of all real numbers $k$ for which $x-2$ is a factor of $P(x)$.
$\textbf{(A) }-8\qquad\textbf{(B) }-4\qquad\textbf{(C) }0\qquad\textbf{(D) }5\qquad\textbf{(E) }8$
2008 iTest Tournament of Champions, 3
A regular $2008$-gon is located in the Cartesian plane such that $(x_1,y_1)=(p,0)$ and $(x_{1005},y_{1005})=(p+2,0)$, where $p$ is prime and the vertices, \[(x_1,y_1),(x_2,y_2),(x_3,y_3),\cdots,(x_{2008},y_{2008}),\]
are arranged in counterclockwise order. Let \begin{align*}S&=(x_1+y_1i)(x_3+y_3i)(x_5+y_5i)\cdots(x_{2007}+y_{2007}i),\\T&=(y_2+x_2i)(y_4+x_4i)(y_6+x_6i)\cdots(y_{2008}+x_{2008}i).\end{align*} Find the minimum possible value of $|S-T|$.
2003 Italy TST, 1
Find all triples of positive integers $(a,b,p)$ with $a,b$ positive integers and $p$ a prime number such that $2^a+p^b=19^a$
1958 AMC 12/AHSME, 11
The number of roots satisfying the equation $ \sqrt{5 \minus{} x} \equal{} x\sqrt{5 \minus{} x}$ is:
$ \textbf{(A)}\ \text{unlimited}\qquad
\textbf{(B)}\ 3\qquad
\textbf{(C)}\ 2\qquad
\textbf{(D)}\ 1\qquad
\textbf{(E)}\ 0$
2008 National Olympiad First Round, 23
If $a^2+b^2+c^2+d^2-ab-bc-cd-d+\frac 25 = 0$ where $a,b,c,d$ are real numbers, what is $a$?
$
\textbf{(A)}\ \frac 23
\qquad\textbf{(B)}\ \frac {\sqrt 2} 3
\qquad\textbf{(C)}\ \frac {\sqrt 3} 2
\qquad\textbf{(D)}\ \frac 15
\qquad\textbf{(E)}\ \text{None of the above}
$
2000 Stanford Mathematics Tournament, 10
Bob has a $12$ foot by $20$ foot garden. He wants to put fencing around it to keep out the neighbor’s dog. Normal fence posts cost $\$2$ each while strong ones cost $\$3$ each. If he needs one fence post for every $2$ feet and has $\$70$ to spend on the fence posts, what is the largest number of strong fence posts he can buy?
2017 Iranian Geometry Olympiad, 1
In triangle $ABC$, the incircle, with center $I$, touches the sides $BC$ at point $D$. Line $DI$ meets $AC$ at $X$. The tangent line from $X$ to the incircle (different from $AC$) intersects $AB$ at $Y$. If $YI$ and $BC$ intersect at point $Z$, prove that $AB=BZ$.
[i]Proposed by Hooman Fattahimoghaddam[/i]
1999 Portugal MO, 4
Given a number, we calculate its square and add $1$ to the sum of the digits in this square, obtaining a new number. If we start with the number $7$ we will obtain, in the first step, the number $1+(4+9)=14$, since $7^2 = 49$. What number will we obtain in the $1999$th step?
2006 Stanford Mathematics Tournament, 23
Consider two mirrors placed at a right angle to each other and two points A at $ (x,y)$ and B at $ (a,b)$. Suppose a person standing at point A shines a laser pointer so that it hits both mirrors and then hits a person standing at point B (as shown in the picture). What is the total distance that the light ray travels, in terms of $ a$, $ b$, $ x$, and $ y$? Assume that $ x$, $ y$, $ a$, and $ b$ are positive.
[asy]draw((0,4)--(0,0)--(4,0),linewidth(1));
draw((1,3)--(0,2),MidArrow);
draw((0,2)--(2,0),MidArrow);
draw((2,0)--(3,1),MidArrow);
dot((1,3));
dot((3,1));
label("$A (x,y)$", (1,3),NE);
label("$B (a,b)$", (3,1),NE);[/asy]
2016 Harvard-MIT Mathematics Tournament, 14
Let $ABC$ be a triangle such that $AB = 13$, $BC = 14$, $CA = 15$ and let $E$, $F$ be the feet of the altitudes from $B$ and $C$, respectively.
Let the circumcircle of triangle $AEF$ be $\omega$.
We draw three lines, tangent to the circumcircle of triangle $AEF$ at $A$, $E$, and $F$.
Compute the area of the triangle these three lines determine.
1976 Euclid, 3
Source: 1976 Euclid Part B Problem 3
-----
$I$ is the centre of the inscribed circle of $\triangle{ABC}$. $AI$ meets the circumcircle of $\triangle{ABC}$ at $D$. Prove that $D$ is equidistant from $I$, $B$, and $C$.
2015 ASDAN Math Tournament, 2
Jonah recently harvested a large number of lychees and wants to split them into groups. Unfortunately, for all $n$ where $3\leq n\leq8$, when the lychees are distributed evenly into $n$ groups, $n-1$ lychees remain. What is the smallest possible number of lychees that Jonah could have?
1985 All Soviet Union Mathematical Olympiad, 397
What maximal number of the men in checkers game can be put on the chess-board $8\times 8$ so, that every man can be taken by at least one other man ?
1974 Dutch Mathematical Olympiad, 3
Proove that in every five positive numbers there is a pair, say $a,b$, for which $$\left| \frac{1}{a+25}- \frac{1}{b+25}\right| <\frac{1}{100}.$$
2024 Romania National Olympiad, 4
Let $f,g:\mathbb{R}\to\mathbb{R}$ be functions with $g(x)=2f(x)+f(x^2),$ for all $x \in \mathbb{R}.$
a) Prove that, if $f$ is bounded in a neighbourhood of the origin and $g$ is continuous in the origin, then $f$ is continuous in the origin.
b) Provide an example of function $f$, discontinuous in the origin, for which the function $g$ is continuous in the origin.
2002 India IMO Training Camp, 13
Let $ABC$ and $PQR$ be two triangles such that
[list]
[b](a)[/b] $P$ is the mid-point of $BC$ and $A$ is the midpoint of $QR$.
[b](b)[/b] $QR$ bisects $\angle BAC$ and $BC$ bisects $\angle QPR$
[/list]
Prove that $AB+AC=PQ+PR$.
2019 Middle European Mathematical Olympiad, 4
Determine the smallest positive integer $n$ for which the following statement holds true: From any $n$ consecutive integers one can select a non-empty set of consecutive integers such that their sum is divisible by $2019$.
[i]Proposed by Kartal Nagy, Hungary[/i]
2003 Polish MO Finals, 6
Let $n$ be an even positive integer. Show that there exists a permutation $(x_1, x_2, \ldots, x_n)$ of the set $\{1, 2, \ldots, n\}$, such that for each $i \in \{1, 2, \ldots, n\}, x_{i+1}$ is one of the numbers $2x_i, 2x_{i}-1, 2x_i - n, 2x_i - n - 1$, where $x_{n+1} = x_1.$
2016 239 Open Mathematical Olympiad, 8
There are $n$ triangles inscribed in a circle and all $3n$ of their vertices are different. Prove that it is possible to put a boy in one of the vertices in each triangle, and a girl in the other, so that boys and girls alternate on a circle.
2018 Ramnicean Hope, 2
Let be the points $ M,N,P, $ on the sides $ BC,AC,AB $ (not on their endpoints), respectively, of a triangle $ ABC, $ such that $ \frac{BM}{MC} =\frac{CN}{NA} =\frac{AP}{PB} . $ Denote $ G_1,G_2,G_3 $ the centroids of $ APN,BMP,CNM, $ respectively. Show that the $ MNP $ has the same centroid as $ G_1G_2G_3. $
[i]Ovidiu Țâțan[/i]
2023 Cono Sur Olympiad, 4
Consider a sequence $\{a_n\}$ of integers, satisfying $a_1=1, a_2=2$ and $a_{n+1}$ is the largest prime divisor of $a_1+a_2+\ldots+a_n$. Find $a_{100}$.
2022 Swedish Mathematical Competition, 6
Bengt wants to put out crosses and rings in the squares of an $n \times n$-square, so that it is exactly one ring and exactly one cross in each row and in each column, and no more than one symbol in each box. Mona wants to stop him by setting a number in advance ban on crosses and a number of bans on rings, maximum one ban in each square. She want to use as few bans as possible of each variety. To succeed in preventing Bengt, how many prohibitions she needs to use the least of the kind of prohibitions she uses the most of?
2012 Indonesia TST, 4
Let $\mathbb{N}$ be the set of positive integers. For every $n \in \mathbb{N}$, define $d(n)$ as the number of positive divisors of $n$. Find all functions $f : \mathbb{N} \rightarrow \mathbb{N}$ such that:
a) $d(f(x)) = x$ for all $x \in \mathbb{N}$
b) $f(xy)$ divides $(x-1)y^{xy-1}f(x)$ for all $x,y \in \mathbb{N}$
2009 IMO Shortlist, 1
Let $ n$ be a positive integer and let $ a_1,a_2,a_3,\ldots,a_k$ $ ( k\ge 2)$ be distinct integers in the set $ { 1,2,\ldots,n}$ such that $ n$ divides $ a_i(a_{i + 1} - 1)$ for $ i = 1,2,\ldots,k - 1$. Prove that $ n$ does not divide $ a_k(a_1 - 1).$
[i]Proposed by Ross Atkins, Australia [/i]