Found problems: 85335
2001 IMO Shortlist, 3
Let $x_1,x_2,\ldots,x_n$ be arbitrary real numbers. Prove the inequality
\[
\frac{x_1}{1+x_1^2} + \frac{x_2}{1+x_1^2 + x_2^2} + \cdots +
\frac{x_n}{1 + x_1^2 + \cdots + x_n^2} < \sqrt{n}.
\]
2021 IMO Shortlist, N3
Find all positive integers $n$ with the following property: the $k$ positive divisors of $n$ have a permutation $(d_1,d_2,\ldots,d_k)$ such that for $i=1,2,\ldots,k$, the number $d_1+d_2+\cdots+d_i$ is a perfect square.
2012 Indonesia TST, 3
The incircle of a triangle $ABC$ is tangent to the sides $AB,AC$ at $M,N$ respectively. Suppose $P$ is the intersection between $MN$ and the bisector of $\angle ABC$. Prove that $BP$ and $CP$ are perpendicular.
2024 Auckland Mathematical Olympiad, 2
In how many ways can $8$ people be divided into pairs?
2010 May Olympiad, 4
Let $n$ be a integer $1<n<2010$, where we have a polygon with $2010$ sides and $n$ coins, we have to paint the vertices of this polygon with $n$ colors and we've to put the $n$ coins in $n$ vertices of the polygon. In each second the coins will go to the neighbour vertex, going in the clockwise.
Determine the values of $n$ such that is possible paint and choose the initial position of the coins where in each second the $n$ coins are in vertices of distinct colors
1988 AMC 8, 11
$ \sqrt{164} $ is
$ \text{(A)}\ 42\qquad\text{(B)}\ \text{less than }10\qquad\text{(C)}\ \text{between }10\text{ and }11\qquad\text{(D)}\ \text{between }11\text{ and }12\qquad\text{(E)}\ \text{between }12\text{ and }13 $
2021 Purple Comet Problems, 3
The diagram shows a semicircle with diameter $20$ and the circle with greatest diameter that fits inside the semicircle. The area of the shaded region is $N\pi$, where $N$ is a positive integer. Find $N$.
2022 District Olympiad, P4
We call a set of $6$ points in the plane [i]splittable[/i] if we if can denote its elements by $A,B,C,D,E$ and $F$ in such a way that $\triangle ABC$ and $\triangle DEF$ have the same centroid.
[list=a]
[*]Construct a splittable set.
[*]Show that any set of $7$ points has a subset of $6$ points which is [i]not[/i] splittable.
[/list]
2023 Thailand Mathematical Olympiad, 8
Let $ABC$ be an acute triangle. The tangent at $A,B$ of the circumcircle of $ABC$ intersect at $T$. Line $CT$ meets side $AB$ at $D$. Denote by $\Gamma_1,\Gamma_2$ the circumcircle of triangle $CAD$, and the circumcircle of triangle $CBD$, respectively. Let line $TA$ meet $\Gamma_1$ again at $E$ and line $TB$ meet $\Gamma_2$ again at $F$. Line $EF$ intersects sides $AC,BC$ at $P,Q$, respectively. Prove that $EF=PQ+AB$.
IV Soros Olympiad 1997 - 98 (Russia), grade8
[b]p1.[/b] What is the maximum amount of a $12\%$ acid solution that can be obtained from $1$ liter of $5\%$, $10\%$ and $15\%$ solutions?
[b]p2.[/b] Which number is greater: $199,719,971,997^2$ or $199,719,971,996 * 19,9719,971,998$ ?
[b]p3.[/b] Is there a convex $1998$-gon whose angles are all integer degrees?
[b]p4.[/b] Is there a ten-digit number divisible by $11$ that uses all the digits from$ 0$ to $9$?
[b]p5.[/b] There are $20$ numbers written in a circle, each of which is equal to the sum of its two neighbors. Prove that the sum of all numbers is $0$.
[b]p6.[/b] Is there a convex polygon that has neither an axis of symmetry nor a center of symmetry, but which transforms into itself when rotated around some point through some angle less than $180$ degrees?
[b]p7.[/b] In a convex heptagon, draw as many diagonals as possible so that no three of them are sides of the same triangle, the vertices of which are at the vertices of the original heptagon.
[b]p8.[/b] Give an example of a natural number that is divisible by $30$ and has exactly $105$ different natural factors, including $1$ and the number itself.
[b]p9.[/b] In the writing of the antipodes, numbers are also written with the digits $0, ..., 9$, but each of the numbers has different meanings for them and for us. It turned out that the equalities are also true for the antipodes
$5 * 8 + 7 + 1 = 48$
$2 * 2 * 6 = 24$
$5* 6 = 30$
a) How will the equality $2^3 = ...$ in the writing of the antipodes be continued?
b) What does the number$ 9$ mean among the Antipodes?
Clarifications:
a) It asks to convert $2^3$ in antipodes language, and write with what number it is equal and find a valid equality in both numerical systems.
b) What does the digit $9$ mean among the antipodes, i.e. with which digit is it equal in our number system?
[b]p10.[/b] Is there a convex quadrilateral that can be cut along a straight line into two parts of the same size and shape, but neither the diagonal nor the straight line passing through the midpoints of opposite sides divides it into two equal parts?
PS.1. There was typo in problem $9$, it asks for $2^3$ and not $23$.
PS.2. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics]here.[/url]
2007 Stanford Mathematics Tournament, 19
Arrange the following four numbers from smallest to largest $ a \equal{} (10^{100})^{10}$, $ b \equal{} 10^{(10^{10})}$, $ c \equal{} 1000000!$, $ d \equal{} (100!)^{10}$
2007 IMO Shortlist, 5
Find all surjective functions $ f: \mathbb{N} \to \mathbb{N}$ such that for every $ m,n \in \mathbb{N}$ and every prime $ p,$ the number $ f(m + n)$ is divisible by $ p$ if and only if $ f(m) + f(n)$ is divisible by $ p$.
[i]Author: Mohsen Jamaali and Nima Ahmadi Pour Anari, Iran[/i]
2002 Belarusian National Olympiad, 4
This requires some imagination and creative thinking:
Prove or disprove:
There exists a solid such that, for all positive integers $n$ with $n \geq 3$, there exists a "parallel projection" (I hope the terminology is clear) such that the image of the solid under this projection is a convex $n$-gon.
2020 Israel National Olympiad, 7
Let $P$ be a point inside a triangle $ABC$, $d_a$, $d_b$ and $d_c$ be distances from $P$ to the lines $BC$, $AC$ and $AB$ respectively, $R$ be a radius of the circumcircle and $r$ be a radius of the inscribed circle for $\Delta ABC.$ Prove that:
$$\sqrt{d_a}+\sqrt{d_b}+\sqrt{d_c}\leq\sqrt{2R+5r}.$$
2010 Saudi Arabia IMO TST, 2
Find all functions $f,g : N \to N$ such that for all $m ,n \in N$ the following relation holds: $$f(m ) - f(n) = (m - n)(g(m) + g(n))$$.
Note: $N = \{0,1,2,...\}$
2024 India Regional Mathematical Olympiad, 5
Let $ABCD$ be a cyclic quadrilateral such that $AB \parallel CD$. Let $O$ be the circumcenter of $ABCD$ and $L$ be the point on $AD$ such that $OL$ is perpendicular to $AD$. Prove that
\[ OB\cdot(AB+CD) = OL\cdot(AC + BD).\]
[i]Proposed by Rijul Saini[/i]
1996 Chile National Olympiad, 1
A shoe brand proposes: Buy a pair of shoes without paying.
It's about this: you go to the factory and pay $20,000 \$ $ for a pair of shoes, get the shoes and ten stamps, with a unit cost of each stamp $2000 \$ $. By selling these stamps you will get your money back. The ones who buy these stamps go to the factory, delivers them and for $18,000 \$ $ they receive their pair of shoes and the ten stamps, thus continuing the cycle.
$\bullet$ How much does the factory receive for each pair of shoes?
$\bullet$ Can this operation be repeated a hundred times, assuming that no one repeats itself?
[hide=original wording]Una marca de zapatos propone: Compre un par de zapatos sin pagar. Se trata de lo siguiente: usted va a la fabrica y paga \$ 20000 por un par de zapatos; recibe los zapatos y diez estampillas, con un costo unitario de ]\$ 2000. Al vender estas estampillas recuperara su dinero. Quienes compren estas estampillas van a la fabrica, la entregan y por \$ 18000 reciben su par de zapatos y las diez estampillas, continuando as el ciclo.
- Cuanto recibe la fabrica por cada par de zapatos?
- Se puede repetir esta operacion cien veces, suponiendo que nadie se repite?[/hide]
2024 Yasinsky Geometry Olympiad, 3
Let \( H \) be the orthocenter of an acute triangle \( ABC \), and let \( AT \) be the diameter of the circumcircle of this triangle. Points \( X \) and \( Y \) are chosen on sides \( AC \) and \( AB \), respectively, such that \( TX = TY \) and \( \angle XTY + \angle XAY = 90^\circ \). Prove that \( \angle XHY = 90^\circ \).
[i]
Proposed by Matthew Kurskyi[/i]
2023 IRN-SGP-TWN Friendly Math Competition, 3
Let $N$ and $d$ be two positive integers with $N\geq d+2$. There are $N$ countries connected via two-way direct flights, where each country is connected to exactly $d$ other countries. It is known that for any two different countries, it is possible to go from one to another via several flights. A country is \emph{important} if after removing it and all the $d$ countries it is connected to, there exist two other countries that are no longer connected via several flights.
Show that if every country is important, then one can choose two countries so that more than $2d/3$ countries are connected to both of them via direct flights.
[i]Proposed by usjl[/i]
2014 Argentina Cono Sur TST, 2
The numbers $1$ through $9$ are written on a $3 \times 3$ board, without repetitions. A valid operation is to choose a row or a column of the board, and replace its three numbers $a, b, c$ (in order, i.e., the first number of the row/column is $a$, the second number of the row/column is $b$, the third number of the row/column is $c$) with either the three non-negative numbers $a-x, b-x, c+x$ (in order) or with the three non-negative numbers $a+x, b-x, c-x$ (in order), where $x$ is a real positive number which may vary in each operation .
a) Determine if there is a way of getting all $9$ numbers on the board to be the same, starting with the following board:
$\begin{array}{|c|c|c|c|c|c|c|c|} \hline 1 & 2 & 3 \\ \hline 4 & 5 & 6 \\ \hline 7 & 8 & 9 \\ \hline \end{array}$
b) For all posible configurations such that it is possible to get all $9$ numbers to be equal to a number $m$ using the valid operations, determine the maximum value of $m$.
2007 Junior Macedonian Mathematical Olympiad, 4
The numbers $a_{1}, a_{2}, ..., a_{20}$ satisfy the following conditions:
$a_{1} \ge a_{2} \ge ... \ge a_{20} \ge 0$
$a_{1} + a_{2} = 20$
$a_{3} + a_{4} + ... + a_{20} \le 20$ .
What is maximum value of the expression:
$a_{1}^2 + a_{2}^2 + ... + a_{20}^2$ ?
For which values of $a_{1}, a_{2}, ..., a_{20}$ is the maximum value achieved?
2009 IMO Shortlist, 8
For any integer $n\geq 2$, we compute the integer $h(n)$ by applying the following procedure to its decimal representation. Let $r$ be the rightmost digit of $n$.
[list][*]If $r=0$, then the decimal representation of $h(n)$ results from the decimal representation of $n$ by removing this rightmost digit $0$.
[*]If $1\leq r \leq 9$ we split the decimal representation of $n$ into a maximal right part $R$ that solely consists of digits not less than $r$ and into a left part $L$ that either is empty or ends with a digit strictly smaller than $r$. Then the decimal representation of $h(n)$ consists of the decimal representation of $L$, followed by two copies of the decimal representation of $R-1$. For instance, for the number $17,151,345,543$, we will have $L=17,151$, $R=345,543$ and $h(n)=17,151,345,542,345,542$.[/list]
Prove that, starting with an arbitrary integer $n\geq 2$, iterated application of $h$ produces the integer $1$ after finitely many steps.
[i]Proposed by Gerhard Woeginger, Austria[/i]
2024 Abelkonkurransen Finale, 1b
Find all functions $f:\mathbb{Z} \to \mathbb{Z}$ such that the numbers
\[n, f(n),f(f(n)),\dots,f^{m-1}(n)\]
are distinct modulo $m$ for all integers $n,m$ with $m>1$.
(Here $f^k$ is defined by $f^0(n)=n$ and $f^{k+1}(n)=f(f^{k}(n))$ for $k \ge 0$.)
2014 Contests, 2
Solve the following equation in $\mathbb{Z}$:
\[3^{2a + 1}b^2 + 1 = 2^c\]
2010 Stanford Mathematics Tournament, 6
Consider the sequence $1, 2, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 1, ...$ Find $n$ such that the first $n$ terms sum up to $2010.$