Found problems: 85335
DMM Individual Rounds, 2002 Tie
[b]p1.[/b] Suppose $a$, $b$ and $c$ are integers such that $c$ divides $a^n + b^n$ for all integers, $n \ge 1$. If the greatest common divisor of $a$ and $b$ is $7$, what is the largest possible value of $c$?
[b]p2.[/b] Consider a sequence of points $\{P_1, P_2,...\}$ on a circle w with the property that $\overline{P_{i+1}P_{i+2}}$ is parallel to the tangent line through $P_i$ for each $i \ge 1$. If $P_5 = P_1$, what is the largest possible angle formed by $P_1P_3P_2$?
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
2011 Dutch BxMO TST, 5
A trapezoid $ABCD$ is given with $BC // AD$. Assume that the bisectors of the angles $BAD$ and $CDA$ intersect on the perpendicular bisector of the line segment $BC$. Prove that $|AB|= |CD|$ or $|AB| +|CD| =|AD|$.
1976 Dutch Mathematical Olympiad, 4
For $a,b, x \in R$ holds: $x^2 - (2a^2 + 4)x + a^2 + 2a + b = 0$. For which $b$ does this equation have at least one root between $0$ and $1$ for all $a$?
1958 Kurschak Competition, 1
Given any six points in the plane, no three collinear, show that we can always find three which form an obtuse-angled triangle with one angle at least $120^o$.
2014 Belarus Team Selection Test, 1
Find all functions$ f : R_+ \to R_+$ such that $f(f(x)+y)=x+f(y)$ , for all $x, y \in R_+$
(Folklore)
[hide=PS]Using search terms [color=#f00]+ ''f(x+f(y))'' + ''f(x)+y[/color]''
I found the same problem [url=https://artofproblemsolving.com/community/c6h1122140p5167983]in Q[/url], [url=https://artofproblemsolving.com/community/c6h1597644p9926878]continuous in R[/url], [url=https://artofproblemsolving.com/community/c6h1065586p4628238]strictly monotone in R[/url] , [url=https://artofproblemsolving.com/community/c6h583742p3451211 ]without extra conditions in R[/url]
[/hide]
1991 Romania Team Selection Test, 6
Let $n \ge 3$ be an integer. A finite number of disjoint arcs with the total sum of length $1 -\frac{1}{n}$ are given on a circle of perimeter $1$. Prove that there is a regular $n$-gon whose all vertices lie on the considered arcs
2021 Korea - Final Round, P6
Find all functions $f,g: \mathbb{R} \to \mathbb{R}$ such that satisfies
$$f(x^2-g(y))=g(x)^2-y$$
for all $x,y \in \mathbb{R}$
LMT Guts Rounds, 7
A team of four students goes to LMT, and each student brings a lunch. However, on the bus, the students’ lunches get mixed up, and during lunch time, each student chooses a random lunch to eat (no two students may eat the same lunch). What is the probability that each student chooses his or her own lunch correctly?
2022 LMT Spring, Tie
Let $L$ be the number of times the letter $L$ appeared on the Speed Round, $M$ be the number of times the letter $M$ appeared on the Speed Round, and $T$ be the number of times the letter $T$ appeared on the Speed Round. Find the value of $LMT$.
2023 Turkey Team Selection Test, 1
Let $ABCD$ be a trapezoid with $AB \parallel CD$. A point $T$ which is inside the trapezoid satisfies $ \angle ATD = \angle CTB$. Let line $AT$ intersects circumcircle of $ACD$ at $K$ and line $BT$ intersects circumcircle of $BCD$ at $L$.($K \neq A$ , $L \neq B$) Prove that $KL \parallel AB$.
2024 Alborz Mathematical Olympiad, P3
A person is locked in a room with a password-protected computer. If they enter the correct password, the door opens and they are freed. However, the password changes every time it is entered incorrectly. The person knows that the password is always a 10-digit number, and they also know that the password change follows a fixed pattern. This means that if the current password is \( b \) and \( a \) is entered, the new password is \( c \), which is determined by \( b \) and \( a \) (naturally, the person does not know \( c \) or \( b \)). Prove that regardless of the characteristics of this computer, the prisoner can free themselves.
Proposed by Reza Tahernejad Karizi
2019 India PRMO, 16
Let $N$ denote the number of all natural numbers $n$ such that $n$ is divisible by a prime $p> \sqrt{n}$ and $p<20$. What is the value of $N$ ?
1989 IMO Longlists, 6
Let $ E$ be the set of all triangles whose only points with integer coordinates (in the Cartesian coordinate system in space), in its interior or on its sides, are its three vertices, and let $ f$ be the function of area of a triangle. Determine the set of values $ f(E)$ of $ f.$
2003 All-Russian Olympiad Regional Round, 11.4
Points $ A_1,A_2,...,A_n$ and $ B_1,B_2,...,B_n$ are given on a plane. Show that the points $ B_i$ can be renumbered in such a way that the angle between vectors $ A_iA_j^{\longrightarrow}$ and $ B_iB_j^{\longrightarrow}$ is acute or right whenever $ i\neq j$.
1992 Putnam, B3
For any pair $(x,y)$ of real numbers, a sequence $(a_{n}(x,y))$ is defined as follows:
$$a_{0}(x,y)=x, \;\;\;\; a_{n+1}(x,y) =\frac{a_{n}(x,y)^{2} +y^2 }{2} \;\, \text{for}\, n\geq 0$$
Find the area of the region $\{(x,y)\in \mathbb{R}^{2} \, |\, (a_{n}(x,y)) \,\, \text{converges} \}$.
1958 February Putnam, A3
Real numbers are chosen at random from the interval $[0,1].$ If after choosing the $n$-th number the sum of the numbers so chosen first exceeds $1$, show that the expected value for $n$ is $e$.
2004 Regional Olympiad - Republic of Srpska, 2
The positive real numbers $x,y,z$ satisfy $x+y+z=1$. Show that
\[\sqrt{3xyz}\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}+\frac{1}{1-x}+\frac{1}{1-y}+\frac{1}{1-z}\right)\geq4+
\frac{4xyz}{(1-x)(1-y)(1-z)}.\]
1997 Brazil Team Selection Test, Problem 2
We say that a subset $A$ of $\mathbb N$ is good if for some positive integer $n$, the equation $x-y=n$ admits infinitely many solutions with $x,y\in A$. If $A_1,A_2,\ldots,A_{100}$ are sets whose union is $\mathbb N$, prove that at least one of the $A_i$s is good.
1999 National High School Mathematics League, 1
Give a geometric series $(a_n)$ with common ratio of $q$, let $b_1=a_1+a_2+a_3,b_2=a_4+a_5+a_6,\cdots,b_n=a_{3n}+a_{3n+1}+a_{3n+2}$, then sequence $(b_n)$
$\text{(A)}$ is an arithmetic sequence
$\text{(B)}$ is a geometric series with common ratio of $q$
$\text{(C)}$ is a geometric series with common ratio of $q^3$
$\text{(D)}$ is neither an arithmetic sequence nor a geometric series
2022 Taiwan TST Round 2, G
Let $I$, $O$, $H$, and $\Omega$ be the incenter, circumcenter, orthocenter, and the circumcircle of the triangle $ABC$, respectively. Assume that line $AI$ intersects with $\Omega$ again at point $M\neq A$, line $IH$ and $BC$ meets at point $D$, and line $MD$ intersects with $\Omega$ again at point $E\neq M$. Prove that line $OI$ is tangent to the circumcircle of triangle $IHE$.
[i]Proposed by Li4 and Leo Chang.[/i]
2015 Baltic Way, 3
Let $n>1$ be an integer. Find all non-constant real polynomials $P(x)$ satisfying , for any real $x$ , the identy \[P(x)P(x^2)P(x^3)\cdots P(x^n)=P(x^{\frac{n(n+1)}{2}})\]
1980 Bundeswettbewerb Mathematik, 3
Given 2n+3 points in the plane, no three on a line and no four on a circle, prove that it is always possible to find a circle C that goes through three of the given points and splits the other 2n in half, that is, has n on the inside and n on the outside.
2011 Croatia Team Selection Test, 3
Let $K$ and $L$ be the points on the semicircle with diameter $AB$. Denote intersection of $AK$ and $AL$ as $T$ and let $N$ be the point such that $N$ is on segment $AB$ and line $TN$ is perpendicular to $AB$. If $U$ is the intersection of perpendicular bisector of $AB$ an $KL$ and $V$ is a point on $KL$ such that angles $UAV$ and $UBV$ are equal. Prove that $NV$ is perpendicular to $KL$.
2016 India IMO Training Camp, 3
An equilateral triangle with side length $3$ is divided into $9$ congruent triangular cells as shown in the figure below. Initially all the cells contain $0$. A [i]move[/i] consists of selecting two adjacent cells (i.e., cells sharing a common boundary) and either increasing or decreasing the numbers in both the cells by $1$ simultaneously. Determine all positive integers $n$ such that after performing several such moves one can obtain $9$ consecutive numbers $n,(n+1),\cdots ,(n+8)$ in some order.
[asy] size(3cm);
pair A=(0,0),D=(1,0),B,C,E,F,G,H,I;
G=rotate(60,A)*D;
B=(1/3)*D; C=2*B;I=(1/3)*G;H=2*I;E=C+I-A;F=H+B-A;
draw(A--D--G--A^^B--F--H--C--E--I--B,black);[/asy]
2024 HMNT, 8
Let $$f(x) = \left|\left|\cdots\left|\left|\left|\left|x\right|-1\right|-2\right|-3\right|-\cdots \right|-10\right|.$$ Compute $f(1)+f(2)+\cdots+f(54)+f(55).$