Found problems: 85335
2007 AMC 12/AHSME, 23
How many non-congruent right triangles with positive integer leg lengths have areas that are numerically equal to $ 3$ times their perimeters?
$ \textbf{(A)}\ 6 \qquad \textbf{(B)}\ 7 \qquad \textbf{(C)}\ 8 \qquad \textbf{(D)}\ 10 \qquad \textbf{(E)}\ 12$
2014 Harvard-MIT Mathematics Tournament, 15
Given a regular pentagon of area $1$, a pivot line is a line not passing through any of the pentagon's vertices such that there are $3$ vertices of the pentagon on one side of the line and $2$ on the other. A pivot point is a point inside the pentagon with only finitely many non-pivot lines passing through it. Find the area of the region of pivot points.
Brazil L2 Finals (OBM) - geometry, 2020.5
Let $ABC$ be a triangle and $M$ the midpoint of $AB$. Let circumcircles of triangles $CMO$ and $ABC$ intersect at $K$ where $O$ is the circumcenter of $ABC$. Let $P$ be the intersection of lines $OM$ and $CK$. Prove that $\angle{PAK} = \angle{MCB}$.
2020/2021 Tournament of Towns, P1
Is it possible that a product of 9 consecutive positive integers is equal to a sum of 9 consecutive (not necessarily the same) positive integers?
[i]Boris Frenkin[/i]
2006 AMC 8, 19
Triangle $ ABC$ is an isosceles triangle with $ \overline{AB} \equal{}\overline{BC}$. Point $ D$ is the midpoint of both $ \overline{BC}$ and $ \overline{AE}$, and $ \overline{CE}$ is 11 units long. Triangle $ ABD$ is congruent to triangle $ ECD$. What is the length of $ \overline{BD}$?
[asy]size(100);
draw((0,0)--(2,4)--(4,0)--(6,4)--cycle--(4,0),linewidth(1));
label("$A$", (0,0), SW);
label("$B$", (2,4), N);
label("$C$", (4,0), SE);
label("$D$", shift(0.2,0.1)*intersectionpoint((0,0)--(6,4),(2,4)--(4,0)), N);
label("$E$", (6,4), NE);[/asy]
$ \textbf{(A)}\ 4 \qquad
\textbf{(B)}\ 4.5 \qquad
\textbf{(C)}\ 5 \qquad
\textbf{(D)}\ 5.5 \qquad
\textbf{(E)}\ 6$
1994 Romania TST for IMO, 3:
Let $a_1, a_2, . . ., a_n$ be a finite sequence of $0$ and $1$. Under any two consecutive terms of this sequence $0$ is written if the digits are equal and $1$ is written otherwise. This way a new sequence of length $n -1$ is obtained.
By repeating this procedure $n - 1$ times one obtains a triangular table of $0$ and $1$. Find the maximum possible number of ones that can appear on this table
2018 AIME Problems, 2
Let $a_0 = 2$, $a_1 = 5$, and $a_2 = 8$, and for $n>2$ define $a_n$ recursively to be the remainder when $4(a_{n-1} + a_{n-2} + a_{n-3})$ is divided by $11$. Find $a_{2018}\cdot a_{2020}\cdot a_{2022}$.
1988 IMO Shortlist, 11
The lock of a safe consists of 3 wheels, each of which may be set in 8 different ways positions. Due to a defect in the safe mechanism the door will open if any two of the three wheels are in the correct position. What is the smallest number of combinations which must be tried if one is to guarantee being able to open the safe (assuming the "right combination" is not known)?
2017 Online Math Open Problems, 5
Henry starts with a list of the first 1000 positive integers, and performs a series of steps on the list. At each step, he erases any nonpositive integers or any integers that have a repeated digit, and then decreases everything in the list by 1. How many steps does it take for Henry's list to be empty?
[i]Proposed by Michael Ren[/i]
2023 Austrian MO Beginners' Competition, 1
Let $x, y, z$ be nonzero real numbers with $$\frac{x + y}{z}=\frac{y + z}{x}=\frac{z + x}{y}.$$
Determine all possible values of $$\frac{(x + y)(y + z)(z + x)}{xyz}.$$
[i](Walther Janous)[/i]
1972 Poland - Second Round, 3
The coordinates of the triangle's vertices in the Cartesian system $XOY$ are integers. Prove that the diameter of the circle circumscribed by this triangle is not greater than the product of the lengths of the triangle's sides.
1993 China National Olympiad, 1
Given an odd $n$, prove that there exist $2n$ integers $a_1,a_2,\cdots ,a_n$; $b_1,b_2,\cdots ,b_n$, such that for any integer $k$ ($0<k<n$), the following $3n$ integers:
$a_i+a_{i+1}, a_i+b_i, b_i+b_{i+k}$ ($i=1,2,\cdots ,n; a_{n+1}=a_1, b_{n+j}=b_j, 0<j<n$) are of different remainders on division by $3n$.
2007 Sharygin Geometry Olympiad, 11
A boy and his father are standing on a seashore. If the boy stands on his tiptoes, his eyes are at a height of $1$ m above sea-level, and if he seats on father’s shoulders, they are at a height of $2$ m. What is the ratio of distances visible for him in two eases?
(Find the answer to $0,1$, assuming that the radius of Earth equals $6000$ km.)
2016 Estonia Team Selection Test, 7
On the sides $AB, BC$ and $CA$ of triangle $ABC$, points $L, M$ and $N$ are chosen, respectively, such that the lines $CL, AM$ and $BN$ intersect at a common point O inside the triangle and the quadrilaterals $ALON, BMOL$ and $CNOM$ have incircles. Prove that
$$\frac{1}{AL\cdot BM} +\frac{1}{BM\cdot CN} +\frac{1}{CN \cdot AL} =\frac{1}{AN\cdot BL} +\frac{1}{BL\cdot CM} +\frac{1}{CM\cdot AN} $$
2022 Portugal MO, 2
Let $P$ be a point on a circle $C_1$ and let $C_2$ be a circle with center $P$ that intersects $C_1$ at two points Q and R. The circle $C_3$, with center $Q$ and which passes through $R$, intersects $C_2$ at another point S, as in figure. Shows that $QS$ is tangent to $C_1$.
[img]https://cdn.artofproblemsolving.com/attachments/7/5/f48d414c68c33c4efaf4d6c8bebcf6f1fad4ba.png[/img]
1984 Brazil National Olympiad, 1
Find all solutions in positive integers to $(n+1)^k -1 = n!$
2007 Princeton University Math Competition, 7
Positive reals $p$ and $q$ are such that the graph of $y = x^2 - 2px + q$ does not intersect the $x$-axis. Find $q$ if there is a unique pair of points $A, B$ on the graph with $AB$ parallel to the $x$-axis and $\angle AOB = \frac{\pi}{2}$, where $O$ is the origin.
2019 CMIMC, 4
Determine the sum of all positive integers $n$ between $1$ and $100$ inclusive such that \[\gcd(n,2^n - 1) = 3.\]
2016 Azerbaijan IMO TST First Round, 4
Find the solution of the functional equation $P(x)+P(1-x)=1$ with power $2015$
P.S: $P(y)=y^{2015}$ is also a function with power $2015$
1990 Turkey Team Selection Test, 4
Let $ABCD$ be a convex quadrilateral such that \[\begin{array}{rl} E,F \in [AB],& AE = EF = FB \\ G,H \in [BC],& BG = GH = HC \\ K,L \in [CD],& CK = KL = LD \\ M,N \in [DA],& DM = MN = NA \end{array}\] Let \[[NG] \cap [LE] = \{P\}, [NG]\cap [KF] = \{Q\},\] \[{[}MH] \cap [KF] = \{R\}, [MH]\cap [LE]=\{S\}\]
Prove that [list=a][*]$Area(ABCD) = 9 \cdot Area(PQRS)$ [*] $NP=PQ=QG$ [/list]
2017 Germany, Landesrunde - Grade 11/12, 1
Solve the equation \[ x^5+x^4+x^3+x^2=x+1 \] in $\mathbb{R}$.
2023 OMpD, 1
Determine all functions $f : \mathbb{R} \rightarrow \mathbb{R}$ such that, for all real numbers $x$ and $y$, $$f(x)(x+f(f(y))) = f(x^2)+xf(y)$$
2016 Online Math Open Problems, 12
For each positive integer $n\ge 2$, define $k\left(n\right)$ to be the largest integer $m$ such that $\left(n!\right)^m$ divides $2016!$. What is the minimum possible value of $n+k\left(n\right)$?
[i]Proposed by Tristan Shin[/i]
2005 Serbia Team Selection Test, 3
problem 3:
(a) Show that there exists a multiple of 2005 whose sum of (decimal) digits equals 2.
(b) Let $x_n$ denote the number obtained by writing natural numbers from $1$ to $n$ one after another (for example, $x_1 = 1, x_2 = 12,...,x_{13} = 12345678910111213$).
Prove that the sequence $x_1,x_2,...$ contains infinitely many terms that are divisiblenby 2005.
2019 Belarus Team Selection Test, 2.4
Cells of $11\times 11$ table are colored with $n$ colors (each cell is colored with exactly one color). For each color, the total amount of the cells of this color is not less than $7$ and not greater than $13$.
Prove that there exists at least one row or column which contains cells of at least four different colors.
[i](N. Sedrakyan)[/i]