Found problems: 85335
2017 Harvard-MIT Mathematics Tournament, 1
Find the sum of all positive integers whose largest proper divisor is $55$. (A proper divisor of $n$ is a divisor that is strictly less than $n$.)
2021 CCA Math Bonanza, I7
The image below consists of a large triangle divided into $13$ smaller triangles. Let $N$ be the number of ways to color each smaller triangle one of red, green, and blue such that if $T_1$ and $T_2$ are smaller triangles whose perimeters intersect at more than one point, $T_1$ and $T_2$ have two different colors. Compute the number of positive integer divisors of $N$.
[asy]
size(5 cm);
draw((-4,0)--(4,0)--(0,6.928)--cycle);
draw((0,0)--(2,3.464)--(-2,3.464)--cycle);
draw((-2,0)--(-1,1.732)--(-3,1.732)--cycle);
draw((2,0)--(1,1.732)--(3,1.732)--cycle);
draw((0,3.464)--(1,5.196)--(-1,5.196)--cycle);
[/asy]
[i]2021 CCA Math Bonanza Individual Round #7[/i]
1998 Bulgaria National Olympiad, 2
The polynomials $P_n(x,y), n=1,2,... $ are defined by \[P_1(x,y)=1, P_{n+1}(x,y)=(x+y-1)(y+1)P_n(x,y+2)+(y-y^2)P_n(x,y)\] Prove that $P_{n}(x,y)=P_{n}(y,x)$ for all $x,y \in \mathbb{R}$ and $n $.
2007 German National Olympiad, 4
Find all triangles such that its angles form an arithmetic sequence and the corresponding sides form a geometric sequence.
2021 Romania National Olympiad, 1
Let $f:[a,b] \rightarrow \mathbb{R}$ a function with Intermediate Value property such that $f(a) * f(b) < 0$. Show that there exist $\alpha$, $\beta$ such that $a < \alpha < \beta < b$ and $f(\alpha) + f(\beta) = f(\alpha) * f(\beta)$.
2013 Junior Balkan Team Selection Tests - Moldova, 1
Given are positive integers $a, b, c$ such that $a$ is odd, $b>c$, $a, b, c$ are coprime and $a(b-c) =2bc$. Prove that $abc$ is square
2010 Contests, 4
Find all polynomials $P(x)$ with real coefficients such that
\[(x-2010)P(x+67)=xP(x) \]
for every integer $x$.
2015 Bulgaria National Olympiad, 6
In a mathematical olympiad students received marks for any of the four areas: algebra, geometry, number theory and combinatorics. Any two of the students have distinct marks for all four areas. A group of students is called [i]nice [/i] if all students in the group can be ordered in increasing order simultaneously of at least two of the four areas. Find the least positive integer N, such that among any N students there exist a [i]nice [/i] group of ten students.
2005 AIME Problems, 14
Consider the points $A(0,12)$, $B(10,9)$, $C(8,0)$, and $D(-4,7)$. There is a unique square $S$ such that each of the four points is on a different side of $S$. Let $K$ be the area of $S$. Find the remainder when $10K$ is divided by $1000$.
2018 Greece JBMO TST, 3
$12$ friends play a tennis tournament, where each plays only one game with any of the other eleven. Winner gets one points. Loser getos zero points, and there is no draw. Final points of the participants are $B_1, B_2, ..., B_{12}$. Find the largest possible value of the sum $\Sigma_3=B_1^3+B_2^3+ ... + B_{12}^3$ .
2010 AIME Problems, 3
Let $ K$ be the product of all factors $ (b\minus{}a)$ (not necessarily distinct) where $ a$ and $ b$ are integers satisfying $ 1\le a < b \le 20$. Find the greatest positive integer $ n$ such that $ 2^n$ divides $ K$.
1998 Iran MO (3rd Round), 1
A one-player game is played on a $m \times n$ table with $m \times n$ nuts. One of the nuts' sides is black, and the other side of them is white. In the beginning of the game, there is one nut in each cell of the table and all nuts have their white side upwards except one cell in one corner of the table which has the black side upwards. In each move, we should remove a nut which has its black side upwards from the table and reverse all nuts in adjacent cells (i.e. the cells which share a common side with the removed nut's cell). Find all pairs $(m,n)$ for which we can remove all nuts from the table.
2014 Postal Coaching, 1
Two circles $\omega_1$ and $\omega_2$ touch externally at point $P$.Let $A$ be a point on $\omega_2$ not lying on the line through the centres of the two circles.Let $AB$ and $AC$ be the tangents to $\omega_1$.Lines $BP$ and $CP$ meet $\omega_2$ for the second time at points $E$ and $F$.Prove that the line $EF$,the tangent to $\omega_2$ at $A$ and the common tangent at $P$ concur.
2002 Korea Junior Math Olympiad, 3
For square $ABCD$, $M$ is a midpoint of segment $CD$ and $E$ is a point on $AD$ satisfying $\angle BEM = \angle MED$. $P$ is an intersection of $AM$, $BE$. Find the value of $\frac{PE}{BP}$
2002 Croatia National Olympiad, Problem 1
In triangle $ABC$, the angles $\alpha=\angle A$ and $\beta=\angle B$ are acute. The isosceles triangle $ACD$ and $BCD$ with the bases $AC$ and $BC$ and $\angle ADC=\beta$, $\angle BEC=\alpha$ are constructed in the exterior of the triangle $ABC$. Let $O$ be the circumcenter of $\triangle ABC$. Prove that $DO+EO$ equals the perimeter of triangle $ABC$ if and only if $\angle ACB$ is right.
1999 Croatia National Olympiad, Problem 2
For a real parameter $a$, solve the equation $x^4-2ax^2+x+a^2-a=0$. Find all $a$ for which all solutions are real.
2019 Saint Petersburg Mathematical Olympiad, 3
Let $a, b$ and $c$ be non-zero natural numbers such that $c \geq b$ . Show that
$$a^b\left(a+b\right)^c>c^b a^c.$$
1969 Polish MO Finals, 2
Given distinct real numbers $a_1,a_2,...,a_n$, find the minimum value of the function
$$y = |x-a_1|+|x-a_2|+...+|x-a_n|, \,\,\, x \in R.$$
KoMaL A Problems 2020/2021, A. 789
Let $p(x) = a_{21} x^{21} + a_{20} x^{20} + \dots + a_1 x + 1$ be a polynomial with integer coefficients and real roots such that the absolute value of all of its roots are less than $1/3$, and all the coefficients of $p(x)$ are lying in the interval $[-2019a,2019a]$ for some positive integer $a$. Prove that if this polynomial is reducible in $\mathbb{Z}[x]$, then the coefficients of one of its factors are less than $a$.
[i]Submitted by Navid Safaei, Tehran, Iran[/i]
2021 Iberoamerican, 4
Let $a,b,c,x,y,z$ be real numbers such that
\[ a^2+x^2=b^2+y^2=c^2+z^2=(a+b)^2+(x+y)^2=(b+c)^2+(y+z)^2=(c+a)^2+(z+x)^2 \]
Show that $a^2+b^2+c^2=x^2+y^2+z^2$.
2012 German National Olympiad, 4
Let $a,b$ be positive real numbers and $n\geq 2$ a positive integer. Prove that if $x^n \leq ax+b$ holds for a positive real number $x$, then it also satisfies the inequality $x < \sqrt[n-1]{2a} + \sqrt[n]{2b}.$
2011 Morocco National Olympiad, 3
Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that for all $x,y, \in \mathbb{R}$,
\[xf(x+xy)=xf(x)+f(x^{2})\cdot f(y).\]
2017 HMIC, 4
Let $G$ be a weighted bipartite graph $A \cup B$, with $|A| = |B| = n$. In other words, each edge in the graph is assigned a positive integer value, called its [i]weight.[/i] Also, define the weight of a perfect matching in $G$ to be the sum of the weights of the edges in the matching.
Let $G'$ be the graph with vertex set $A \cup B$, and (which) contains the edge $e$ if and only if $e$ is part of some minimum weight perfect matching in $G$.
Show that all perfect matchings in $G'$ have the same weight.
1985 AIME Problems, 2
When a right triangle is rotated about one leg, the volume of the cone produced is $800 \pi$ $\text{cm}^3$. When the triangle is rotated about the other leg, the volume of the cone produced is $1920 \pi$ $\text{cm}^3$. What is the length (in cm) of the hypotenuse of the triangle?
2019 Math Hour Olympiad, 8-10
[u]Round 1[/u]
[b]p1.[/b] The alphabet of the Aau-Bau language consists of two letters: A and B. Two words have the same meaning if one of them can be constructed from the other by replacing any AA with A, replacing any BB with B, or by replacing any ABA with BAB. For example, the word AABA means the same thing as ABA, and AABA also means the same thing as ABAB. In this language, is it possible to name all seven days of the week?
[b]p2.[/b] A museum has a $4\times 4$ grid of rooms. Every two rooms that share a wall are connected by a door. Each room contains some paintings. The total number of paintings along any path of $7$ rooms from the lower left to the upper right room is always the same. Furthermore, the total number of paintings along any path of $7$ rooms from the lower right to the upper left room is always the same. The guide states that the museum has exactly $500$ paintings. Show that the guide is mistaken.
[img]https://cdn.artofproblemsolving.com/attachments/7/6/0fd93a0deaa71a5bb1599d2488f8b4eac5d0eb.jpg[/img]
[b]p3.[/b] A playground has a swing-set with exactly three swings. When 3rd and 4th graders from Dr. Anna’s math class play during recess, she has a rule that if a $3^{rd}$ grader is in the middle swing there must be $4^{th}$ graders on that person’s left and right. And if there is a $4^{th}$ grader in the middle, there must be $3^{rd}$ graders on that person’s left and right. Dr. Anna calculates that there are $350$ different ways her students can arrange themselves on the three swings with no empty seats. How many students are in her class?
[img]https://cdn.artofproblemsolving.com/attachments/5/9/4c402d143646582376d09ebbe54816b8799311.jpg[/img]
[b]p4.[/b] The archipelago Artinagos has $19$ islands. Each island has toll bridges to at least $3$ other islands. An unsuspecting driver used a bad mapping app to plan a route from North Noether Island to South Noether Island, which involved crossing $12$ bridges. Show that there must be a route with fewer bridges.
[img]https://cdn.artofproblemsolving.com/attachments/e/3/4eea2c16b201ff2ac732788fe9b78025004853.jpg[/img]
[b]p5.[/b] Is it possible to place the numbers from $1$ to $121$ in an $11\times 11$ table so that numbers that differ by $1$ are in horizontally or vertically adjacent cells and all the perfect squares $(1, 4, 9, ... , 121)$ are in one column?
[u]Round 2[/u]
[b]p6.[/b] Hungry and Sneaky have opened a rectangular box of chocolates and are going to take turns eating them. The chocolates are arranged in a $2m \times 2n$ grid. Hungry can take any two chocolates that are side-by-side, but Sneaky can take only one at a time. If there are no more chocolates located side-by-side, all remaining chocolates go to Sneaky. Hungry goes first. Each player wants to eat as many chocolates as possible. What is the maximum number of chocolates Sneaky can get, no matter how Hungry picks his?
[img]https://cdn.artofproblemsolving.com/attachments/b/4/26d7156ca6248385cb46c6e8054773592b24a3.jpg[/img]
[b]p7.[/b] There is a thief hiding in the sultan’s palace. The palace contains $2019$ rooms connected by doors. One can walk from any room to any other room, possibly through other rooms, and there is only one way to do this. That is, one cannot walk in a loop in the palace. To catch the thief, a guard must be in the same room as the thief at the same time. Prove that $11$ guards can always find and catch the thief, no matter how the thief moves around during the search.
[img]https://cdn.artofproblemsolving.com/attachments/a/b/9728ac271e84c4954935553c4d58b3ff4b194d.jpg[/img]
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