Found problems: 85335
2011 China Team Selection Test, 3
Let $m$ and $n$ be positive integers. A sequence of points $(A_0,A_1,\ldots,A_n)$ on the Cartesian plane is called [i]interesting[/i] if $A_i$ are all lattice points, the slopes of $OA_0,OA_1,\cdots,OA_n$ are strictly increasing ($O$ is the origin) and the area of triangle $OA_iA_{i+1}$ is equal to $\frac{1}{2}$ for $i=0,1,\ldots,n-1$.
Let $(B_0,B_1,\cdots,B_n)$ be a sequence of points. We may insert a point $B$ between $B_i$ and $B_{i+1}$ if $\overrightarrow{OB}=\overrightarrow{OB_i}+\overrightarrow{OB_{i+1}}$, and the resulting sequence $(B_0,B_1,\ldots,B_i,B,B_{i+1},\ldots,B_n)$ is called an [i]extension[/i] of the original sequence. Given two [i]interesting[/i] sequences $(C_0,C_1,\ldots,C_n)$ and $(D_0,D_1,\ldots,D_m)$, prove that if $C_0=D_0$ and $C_n=D_m$, then we may perform finitely many [i]extensions[/i] on each sequence until the resulting two sequences become identical.
2005 National High School Mathematics League, 10
In tetrahedron $ABCD$, the volume of tetrahedron $ABCD$ is $\frac{1}{6}$, and $\angle ACB=45^{\circ},AD+BC+\frac{AC}{\sqrt2}=3$, then $CD=$________.
2019 District Olympiad, 3
Let $(a_n)_{n \in \mathbb{N}}$ be a sequence of real numbers such that $$2(a_1+a_2+…+a_n)=na_{n+1}~\forall~n \ge 1.$$
$\textbf{a)}$ Prove that the given sequence is an arithmetic progression.
$\textbf{b)}$ If $\lfloor a_1 \rfloor + \lfloor a_2 \rfloor +…+ \lfloor a_n \rfloor = \lfloor a_1+a_2+…+a_n \rfloor~\forall~ n \in \mathbb{N},$ prove that every term of the sequence is an integer.
PEN O Problems, 42
Let $N_{n}$ denote the number of ordered $n$-tuples of positive integers $(a_{1},a_{2},\ldots,a_{n})$ such that \[1/a_{1}+1/a_{2}+\ldots+1/a_{n}=1.\] Determine whether $N_{10}$ is even or odd.
2024 Indonesia TST, 3
Let $n$ be a positive integer and let $a_1, a_2, \ldots, a_n$ be positive reals. Show that $$\sum_{i=1}^{n} \frac{1}{2^i}(\frac{2}{1+a_i})^{2^i} \geq \frac{2}{1+a_1a_2\ldots a_n}-\frac{1}{2^n}.$$
2016 Sharygin Geometry Olympiad, P12
Let $BB_1$ be the symmedian of a nonisosceles acute-angled triangle $ABC$. Ray $BB_1$ meets the circumcircle of $ABC$ for the second time at point $L$. Let $AH_A, BH_B, CH_C$ be the altitudes of triangle $ABC$. Ray $BH_B$ meets the circumcircle of $ABC$ for the second time at point $T$. Prove that $H_A, H_C, T, L$ are concyclic.
2023 Rioplatense Mathematical Olympiad, 1
An integer $n\geq 3$ is [i]poli-pythagorean[/i] if there exist $n$ positive integers pairwise distinct such that we can order these numbers in the vertices of a regular $n$-gon such that the sum of the squares of consecutive vertices is also a perfect square. For instance, $3$ is poli-pythagorean, because if we write $44,117,240$ in the vertices of a triangle we notice:
$$44^2+117^2=125^2, 117^2+240^2=267^2, 240^2+44^2=244^2$$
Determine all poli-pythagorean integers.
1995 Austrian-Polish Competition, 6
The Alpine Club organizes four mountain trips for its $n$ members. Let $E_1, E_2, E_3, E_4$ be the teams participating in these trips. In how many ways can these teams be formed so as to satisfy
$E_1 \cap E_2 \ne\varnothing$, $E_2 \cap E_3 \ne\varnothing$ , $E_3 \cap E_4 \ne\varnothing$ ?
2011 Tokyo Instutute Of Technology Entrance Examination, 1
Let $f_n\ (n=1,\ 2,\ \cdots)$ be a linear transformation expressed by a matrix $\left(
\begin{array}{cc}
1-n & 1 \\
-n(n+1) & n+2
\end{array}
\right)$ on the $xy$ plane. Answer the following questions:
(1) Prove that there exists 2 lines passing through the origin $O(0,\ 0)$ such that all points of the lines are mapped to the same lines, then find the equation of the lines.
(2) Find the area $S_n$ of the figure enclosed by the lines obtained in (1) and the curve $y=x^2$.
(3) Find $\sum_{n=1}^{\infty} \frac{1}{S_n-\frac 16}.$
[i]2011 Tokyo Institute of Technlogy entrance exam, Problem 1[/i]
1967 Putnam, A1
Let $f(x)= a_1 \sin x + a_2 \sin 2x+\cdots +a_{n} \sin nx $, where $a_1 ,a_2 ,\ldots,a_n $ are real numbers and where $n$ is a positive integer. Given that $|f(x)| \leq | \sin x |$ for all real $x,$ prove that
$$|a_1 +2a_2 +\cdots +na_{n}|\leq 1.$$
2019 Durer Math Competition Finals, 4
Let $ABC$ be an acute-angled triangle having angles $\alpha,\beta,\gamma$ at vertices $A, B, C$ respectively. Let isosceles triangles $BCA_1, CAB_1, ABC_1$ be erected outwards on its sides, with apex angles $2\alpha ,2\beta ,2\gamma$ respectively. Let $A_2$ be the intersection point of lines $AA_1$ and $B_1C_1$ and let us define points $B_2$ and $C_2$ analogously. Find the exact value of the expression $$\frac{AA_1}{A_2A_1}+\frac{BB_1}{B_2B_1}+\frac{CC_1}{C_2C_1}$$
2007 Turkey Junior National Olympiad, 2
In a qualification group with $15$ volleyball teams, each team plays with all the other teams exactly once. Since there is no tie in volleyball, there is a winner in every match. After all matches played, a team would be qualified if its total number of losses is not exceeding $N$. If there are at least $7$ teams qualified, find the possible least value of $N$.
1999 USAMO, 3
Let $p > 2$ be a prime and let $a,b,c,d$ be integers not divisible by $p$, such that
\[ \left\{ \dfrac{ra}{p} \right\} + \left\{ \dfrac{rb}{p} \right\} + \left\{ \dfrac{rc}{p} \right\} + \left\{ \dfrac{rd}{p} \right\} = 2 \]
for any integer $r$ not divisible by $p$. Prove that at least two of the numbers $a+b$, $a+c$, $a+d$, $b+c$, $b+d$, $c+d$ are divisible by $p$.
(Note: $\{x\} = x - \lfloor x \rfloor$ denotes the fractional part of $x$.)
1906 Eotvos Mathematical Competition, 3
Let $a_1, a_2, ...,a_n$ represent an arbitrary arrangement of the numbers $1, 2, ...,n$. Prove that, if $n$ is odd, the product $$(a_1 - 1)(a_2 - 2) ... (a_n -n)$$ is an even number.
2014 South East Mathematical Olympiad, 2
Let $n\geq 4$ be a positive integer.Out of $n$ people,each of two individuals play table tennis game(every game has a winner).Find the minimum value of $n$,such that for any possible outcome of the game,there always exist an ordered four people group $(a_{1},a_{2},a_{3},a_{4})$,such that the person $a_{i}$ wins against $a_{j}$ for any $1\leq i<j\leq 4$
2020 DMO Stage 1, 4.
[b]Q.[/b] We paint the numbers $1,2,3,4,5$ with red or blue. Prove that the equation $x+y=z$ have a monocolor solution (that is, all the 3 unknown there are the same color . It not needed that $x, y, z$ must be different!)
[i]Proposed by TuZo[/i]
1998 May Olympiad, 1
With six rods a piece like the one in the figure is constructed. The three outer rods are equal to each other. The three inner rods are equal to each other. You want to paint each rod a single color so that at each joining point, the three arriving rods have a different color. The rods can only be painted blue, white, red or green. In how many ways can the piece be painted?
[img]https://cdn.artofproblemsolving.com/attachments/1/1/91e6b388498613486477ab6b51735055e920cc.gif[/img]
2004 Oral Moscow Geometry Olympiad, 6
The length of each side and each non-principal diagonal of a convex hexagon does not exceed $1$. Prove that this hexagon contains a principal diagonal whose length does not exceed $\frac{2}{\sqrt3}$.
2013 India Regional Mathematical Olympiad, 3
In an acute-angled triangle $ABC$ with $AB < AC$, the circle $\omega$ touches $AB$ at $B$ and passes through $C$ intersecting $AC$ again at $D$. Prove that the orthocentre of triangle $ABD$ lies on $\omega$ if and only if it lies on the perpendicular bisector of $BC$.
2012 NIMO Problems, 8
A convex 2012-gon $A_1A_2A_3 \dots A_{2012}$ has the property that for every integer $1 \le i \le 1006$, $\overline{A_iA_{i+1006}}$ partitions the polygon into two congruent regions. Show that for every pair of integers $1 \le j < k \le 1006$, quadrilateral $A_jA_kA_{j+1006}A_{k+1006}$ is a parallelogram.
[i]Proposed by Lewis Chen[/i]
2010 Austria Beginners' Competition, 4
In the right-angled triangle $ABC$ with a right angle at $C$, the side $BC$ is longer than the side $AC$. The perpendicular bisector of $AB$ intersects the line $BC$ at point $D$ and the line $AC$ at point $E$. The segments $DE$ has the same length as the side $AB$. Find the measures of the angles of the triangle $ABC$.
(R. Henner, Vienna)
1997 Moscow Mathematical Olympiad, 2
$9$ different pieces of cheese are placed on a plate. Is it always possible to cut one of them into two parts so that the $10$ pieces obtained were divisible into two portions of equal mass of $5$ pieces each?
2017 Harvard-MIT Mathematics Tournament, 3
There are $2017$ jars in a row on a table, initially empty. Each day, a nice man picks ten consecutive jars and deposits one coin in each of the ten jars. Later, Kelvin the Frog comes back to see that $N$ of the jars all contain the same positive integer number of coins (i.e. there is an integer $d>0$ such that $N$ of the jars have exactly $d$ coins). What is the maximum possible value of $N$?
2025 Kyiv City MO Round 2, Problem 3
Does there exist a sequence of positive integers \( a_1, a_2, \ldots, a_{100} \) such that every number from \( 1 \) to \( 100 \) appears exactly once, and for each \( 1 \leq i \leq 100 \), the condition
\[
a_{a_i + i} = i
\]
holds? Here it is assumed that \( a_{k+100} = a_k \) for each \( 1 \leq k \leq 100 \).
[i]Proposed by Mykhailo Shtandenko[/i]
2021 Princeton University Math Competition, A1 / B3
A circle is inscribed in a regular octagon with area $2024$. A second regular octagon is inscribed in the circle, and its area can be expressed as $a + b\sqrt{c}$, where $a, b, c$ are integers and $c$ is square-free. Compute $a + b + c$.