Found problems: 85335
2019 IMO Shortlist, G8
Let $\mathcal L$ be the set of all lines in the plane and let $f$ be a function that assigns to each line $\ell\in\mathcal L$ a point $f(\ell)$ on $\ell$. Suppose that for any point $X$, and for any three lines $\ell_1,\ell_2,\ell_3$ passing through $X$, the points $f(\ell_1),f(\ell_2),f(\ell_3)$, and $X$ lie on a circle.
Prove that there is a unique point $P$ such that $f(\ell)=P$ for any line $\ell$ passing through $P$.
[i]Australia[/i]
2020 Malaysia IMONST 1, 16
Find the number of positive integer solutions $(a,b,c,d)$ to the equation \[(a^2+b^2)(c^2-d^2)=2020.\]
Note: The solutions $(10,1,6,4)$ and $(1,10,6,4)$ are considered different.
2012 Israel National Olympiad, 6
Let $A,B,C,O$ be points in the plane such that angles $\angle AOB,\angle BOC, \angle COA$ are obtuse. On $OA,OB,OC$ points $X,Y,Z$ respectively are chosen, such that $OX=OY=OZ$. On segments $OX,OY,OZ$ points $K,L,M$ respectively are chosen.
The lines $AL$ and $BK$ intersect at point $R$, which isn't on $XY$. The segment $XY$ intersects $AL,BK$ at points $R_1,R_2$.
The lines $BM$ and $CL$ intersect at point $P$, which isn't on $YZ$. The segment $YZ$ intersects $BM,CL$ at points $P_1,P_2$.
The lines $CK$ and $AM$ intersect at point $Q$, which isn't on $ZX$. The segment $ZX$ intersects $CK,AM$ at points $Q_1,Q_2$.
Suppose that $PP_1=PP_2$ and $QQ_1=QQ_2$. Prove that $RR_1=RR_2$.
1952 Polish MO Finals, 3
Construct the quadrilateral $ ABCD $ given the lengths of the sides $ AB $ and $ CD $ and the angles of the quadrilateral.
1991 Federal Competition For Advanced Students, P2, 3
$ (a)$ Prove that $ 91$ divides $ n^{37}\minus{}n$ for all integers $ n$.
$ (b)$ Find the largest $ k$ that divides $ n^{37}\minus{}n$ for all integers $ n$.
2022 Vietnam TST, 6
Given a set $A=\{1;2;...;4044\}$. They color $2022$ numbers of them by white and the rest of them by black. With each $i\in A$, called the [b][i]important number[/i][/b] of $i$ be the number of all white numbers smaller than $i$ and black numbers larger than $i$. With every natural number $m$, find all positive integers $k$ that exist a way to color the numbers that can get $k$ important numbers equal to $m$.
2001 AIME Problems, 2
Each of the 2001 students at a high school studies either Spanish or French, and some study both. The number who study Spanish is between 80 percent and 85 percent of the school population, and the number who study French is between 30 percent and 40 percent. Let $m$ be the smallest number of students who could study both languages, and let $M$ be the largest number of students who could study both languages. Find $M-m$.
2020 Yasinsky Geometry Olympiad, 3
Point $M$ is the midpoint of the side $CD$ of the trapezoid $ABCD$, point $K$ is the foot of the perpendicular drawn from point $M$ to the side $AB$. Give that $3BK \le AK$. Prove that $BC + AD\ge 2BM$.
Russian TST 2018, P1
Let $x,y,z \in\mathbb{Q}$,such that $(x+y+z)^3=9(x^2y+y^2z+z^2x).$ Prove that $x=y=z$
2023 Germany Team Selection Test, 3
Two triangles $ABC, A’B’C’$ have the same orthocenter $H$ and the same circumcircle with center $O$. Letting $PQR$ be the triangle formed by $AA’, BB’, CC’$, prove that the circumcenter of $PQR$ lies on $OH$.
2004 VJIMC, Problem 4
Let $f:\mathbb R\to\mathbb R$ be an infinitely differentiable function. Assume that for every $x\in\mathbb R$ there is an $n\in\mathbb N$ (depending on $x$) such that
$$f^{(n)}(x)=0.$$Prove that $f$ is a polynomial.
1986 IMO Longlists, 30
Prove that a convex polyhedron all of whose faces are equilateral triangles has at most $30$ edges.
2023 Korea Junior Math Olympiad, 6
Find the maximum value of real number $A$ such that $$3x^2 + y^2 + 1 \geq A(x^2 + xy + x)$$ for all positive integers $x, y.$
Mid-Michigan MO, Grades 10-12, 2022
[b]p1.[/b] Consider a triangular grid: nodes of the grid are painted black and white. At a single step you are allowed to change colors of all nodes situated on any straight line (with the slope $0^o$ ,$60^o$, or $120^o$ ) going through the nodes of the grid. Can you transform the combination in the left picture into the one in the right picture in a finite number of steps?
[img]https://cdn.artofproblemsolving.com/attachments/3/a/957b199149269ce1d0f66b91a1ac0737cf3f89.png[/img]
[b]p2.[/b] Find $x$ satisfying $\sqrt{x\sqrt{x \sqrt{x ...}}} = \sqrt{2022}$ where it is an infinite expression on the left side.
[b]p3.[/b] $179$ glasses are placed upside down on a table. You are allowed to do the following moves. An integer number $k$ is fixed. In one move you are allowed to turn any $k$ glasses .
(a) Is it possible in a finite number of moves to turn all $179$ glasses into “bottom-down” positions if $k=3$?
(b) Is it possible to do it if $k=4$?
[b]p4.[/b] An interval of length $1$ is drawn on a paper. Using a compass and a simple ruler construct an interval of length $\sqrt{93}$.
[b]p5.[/b] Show that $5^{2n+1} + 3^{n+2} 2^{n-1} $ is divisible by $19$ for any positive integer $n$.
[b]p6.[/b] Solve the system $$\begin{cases} \dfrac{xy}{x+y}=1-z \\ \dfrac{yz}{y+z}=2-x \\ \dfrac{xz}{x+z}=2-y \end{cases}$$
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
2018 Iran Team Selection Test, 6
Consider quadrilateral $ABCD $ inscribed in circle $\omega $. $P\equiv AC\cap BD$. $E$, $F$ lie on sides $AB$, $CD$ respectively such that $\hat {APE}=\hat {DPF} $. Circles $\omega_1$, $\omega_2$ are tangent to $\omega$ at $X $, $Y $ respectively and also both tangent to the circumcircle of $\triangle PEF $ at $P $. Prove that: $$\frac {EX}{EY}=\frac {FX}{FY} $$
[i]Proposed by Ali Zamani [/i]
2000 India Regional Mathematical Olympiad, 4
All the $7$ digit numbers containing each of the digits $1,2,3,4,5,6,7$ exactly once , and not divisible by $5$ are arranged in increasing order. Find the $200th$ number in the list.
2024 Malaysian IMO Team Selection Test, 5
Let $n$ be an odd integer and $m=\phi(n)$ be the Euler's totient function. Call a set of residues $T=\{a_1, \cdots, a_k\} \pmod n$ to be [i]good[/i] if $\gcd(a_i, n) > 1$ $\forall i$, and $\gcd(a_i, a_j) = 1, \forall i \neq j$. Define the set $S_n$ consisting of the residues $$\sum_{i=1}^k a_i ^m\pmod{n}$$ over all possible residue sets $T=\{a_1,\cdots,a_k\}$ that is good. Determine $|S_n|$.
[i]Proposed by Anzo Teh Zhao Yang[/i]
2021 South East Mathematical Olympiad, 3
Let $a,b,c\geq 0$ and $a^2+b^2+c^2\leq 1.$ Prove that$$\frac{a}{a^2+bc+1}+\frac{b}{b^2+ca+1}+\frac{c}{c^2+ab+1}+3abc<\sqrt 3$$
2017 Kürschák Competition, 3
An $n$ by $n$ table has an integer in each cell, such that no two cells within a row share the same number. Prove that it is possible to permute the elements within each row to obtain a table that has $n$ distinct numbers in each column.
2011 China Team Selection Test, 1
Let $AA',BB',CC'$ be three diameters of the circumcircle of an acute triangle $ABC$. Let $P$ be an arbitrary point in the interior of $\triangle ABC$, and let $D,E,F$ be the orthogonal projection of $P$ on $BC,CA,AB$, respectively. Let $X$ be the point such that $D$ is the midpoint of $A'X$, let $Y$ be the point such that $E$ is the midpoint of $B'Y$, and similarly let $Z$ be the point such that $F$ is the midpoint of $C'Z$. Prove that triangle $XYZ$ is similar to triangle $ABC$.
2024 Australian Mathematical Olympiad, P8
Let $r=0.d_0d_1d_2\ldots$ be a real number. Let $e_n$ denote the number formed by the digits $d_n, d_{n-1}, \ldots, d_0$ written from left to right (leading zeroes are permitted). Given that $d_0=6$ and for each $n \geq 0$, $e_n$ is equal to the number formed by the $n+1$ rightmost digits of $e_n^2$. Show that $r$ is irrational.
2016 Oral Moscow Geometry Olympiad, 2
A regular heptagon $A_1A_2A_3A_4A_5A_6A_7$ is given. Straight $A_2A_3$ and $A_5A_6$ intersect at point $X$, and straight lines $A_3A_5$ and $A_1A_6$ intersect at point $Y$. Prove that lines $A_1A_2$ and $XY$ are parallel.
2003 IMO Shortlist, 7
The sequence $a_0$, $a_1$, $a_2,$ $\ldots$ is defined as follows: \[a_0=2, \qquad a_{k+1}=2a_k^2-1 \quad\text{for }k \geq 0.\] Prove that if an odd prime $p$ divides $a_n$, then $2^{n+3}$ divides $p^2-1$.
[hide="comment"]
Hi guys ,
Here is a nice problem:
Let be given a sequence $a_n$ such that $a_0=2$ and $a_{n+1}=2a_n^2-1$ . Show that if $p$ is an odd prime such that $p|a_n$ then we have $p^2\equiv 1\pmod{2^{n+3}}$
Here are some futher question proposed by me :Prove or disprove that :
1) $gcd(n,a_n)=1$
2) for every odd prime number $p$ we have $a_m\equiv \pm 1\pmod{p}$ where $m=\frac{p^2-1}{2^k}$ where $k=1$ or $2$
Thanks kiu si u
[i]Edited by Orl.[/i]
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2024 SG Originals, Q1
Find all permutations $(a_1, a_2, \cdots, a_{2024})$ of $(1, 2, \cdots, 2024)$ such that there exists a polynomial $P$ with integer coefficients satisfying $P(i) = a_i$ for each $i = 1, 2, \cdots, 2024$.
2011 AIME Problems, 5
The vertices of a regular nonagon (9-sided polygon) are to be labeled with the digits $1$ through $9$ in such a way that the sum of the numbers on every three consecutive vertices is a multiple of $3$. Two acceptable arrangements are considered to be indistinguishable if one can be obtained from the other by rotating the nonagon in the plane. Find the number of distinguishable acceptable arrangements.