Found problems: 85335
2005 AMC 10, 2
For each pair of real numbers $ a\not\equal{} b$, define the operation $ \star$ as \[(a \star b) \equal{} \frac{a \plus{} b}{a \minus{} b}.\] What is the value of $ ((1 \star 2) \star 3)$?
$ \textbf{(A)}\ \minus{}\frac{2}{3}\qquad
\textbf{(B)}\ \minus{}\frac{1}{5}\qquad
\textbf{(C)}\ 0\qquad
\textbf{(D)}\ \frac{1}{2}\qquad
\textbf{(E)}\ \text{This value is not defined.}$
2019 Poland - Second Round, 2
Determine all nonnegative integers $x, y$ satisfying the equation
\begin{align*}
\sqrt{xy}=\sqrt{x+y}+\sqrt{x}+\sqrt{y}.
\end{align*}
2023 Moldova EGMO TST, 5
Find all pairs of real numbers $(x, y)$, that satisfy the system of equations: $$\left\{\begin{matrix} 6(1-x)^2=\dfrac{1}{y} \\ \\6(1-y)^2=\dfrac{1}{x}.\end{matrix}\right.$$
2017 Taiwan TST Round 3, 1
There are $m$ real numbers $x_i \geq 0$ ($i=1,2,...,m$), $n \geq 2$, $\sum_{i=1}^{m} x_i=S$. Prove that\\
\[
\sum_{i=1}^{m} \sqrt[n]{\frac{x_i}{S-x_i}} \geq 2,
\]
The equation holds if and only if there are exactly two of $x_i$ are equal(not equal to $0$), and the rest are equal to $0$.
1999 Hungary-Israel Binational, 1
$ c$ is a positive integer. Consider the following recursive sequence: $ a_1\equal{}c, a_{n\plus{}1}\equal{}ca_{n}\plus{}\sqrt{(c^2\minus{}1)(a_n^2\minus{}1)}$, for all $ n \in N$.
Prove that all the terms of the sequence are positive integers.
2022 Yasinsky Geometry Olympiad, 5
Let $ABC$ be a right triangle with leg $CB = 2$ and hypotenuse $AB= 4$. Point $K$ is chosen on the hypotenuse $AB$, and point $L$ is chosen on the leg $AC$.
a) Describe and justify how to construct such points $K$ and $ L$ so that the sum of the distances $CK+KL$ is the smallest possible.
b) Find the smallest possible value of $CK+KL$.
(Olexii Panasenko)
2019 Korea National Olympiad, 7
For prime $p\equiv 1\pmod{7} $, prove that there exists some positive integer $m$ such that $m^3+m^2-2m-1$ is a multiple of $p$.
2023 Durer Math Competition Finals, 16
What is the remainder of $2025\wedge (2024\wedge (2022\wedge (2021\wedge (2020\wedge ...\wedge (2\wedge 1) . . .)))))$ when it is divided by $2023$?
Here $\wedge$ is the exponential symbol, for example $2\wedge (3\wedge 2) = 2\wedge 9 = 512$. The power tower contains the integers from $2025$ to $1$ exactly once, except that the number $2023$ is missing.
2007 Tournament Of Towns, 1
Black and white checkers are placed on an $8 \times 8$ chessboard, with at most one checker on each cell. What is the maximum number of checkers that can be placed such that each row and each column contains twice as many white checkers as black ones?
1975 All Soviet Union Mathematical Olympiad, 219
a) Given real numbers $a_1,a_2,b_1,b_2$ and positive $p_1,p_2,q_1,q_2$. Prove that in the table $2\times 2$
$$(a_1 + b_1)/(p_1 + q_1) , (a_1 + b_2)/(p_1 + q_2) $$
$$(a_2 + b_1)/(p_2 + q_1) , (a_2 + b_2)/(p_2 + q_2)$$
there is a number in the table, that is not less than another number in the same row and is not greater than another number in the same column (a saddle point).
b) Given real numbers $a_1, a_2, ... , a_n, b_1, b_2, ... , b_n$ and positive $p_1, p_2, ... , p_n, q_1, q_2, ... , q_n$. We construct the table $n\times n$, with the numbers ($0 < i,j \le n$)
$$(a_i + b_j)/(p_i + q_j)$$
in the intersection of the $i$-th row and $j$-th column. Prove that there is a number in the table, that is not less than arbitrary number in the same row and is not greater than arbitrary number in the same column (a saddle point).
MOAA Gunga Bowls, 2021.9
William is biking from his home to his school and back, using the same route. When he travels to school, there is an initial $20^\circ$ incline for $0.5$ kilometers, a flat area for $2$ kilometers, and a $20^\circ$ decline for $1$ kilometer. If William travels at $8$ kilometers per hour during uphill $20^\circ$ sections, $16$ kilometers per hours during flat sections, and $20$ kilometers per hour during downhill $20^\circ$ sections, find the closest integer to the number of minutes it take William to get to school and back.
[i]Proposed by William Yue[/i]
2018 Israel Olympic Revenge, 2
Is it possible to disassemble and reassemble a $4\times 4\times 4$ Rubik's Cuble in at least $577,800$ non-equivalent ways?
Notes:
1. When we reassemble the cube, a corner cube has to go to a corner cube, an edge cube must go to an edge cube and a central cube must go to a central cube.
2. Two positions of the cube are called equivalent if they can be obtained from one two another by rotating the faces or layers which are parallel to the faces.
2010 Singapore MO Open, 1
Let $CD$ be a chord of a circle $\Gamma_1$ and $AB$ a diameter of $\Gamma_1$ perpendicular to $CD$ at $N$ with $AN > NB$. A circle $\Gamma_2$ centered at $C$ with radius $CN$ intersects $\Gamma_1$ at points $P$ and $Q$. The line $PQ$ intersects $CD$ at $M$ and $AC$ at $K$; and the extension of $NK$ meets $\Gamma_2$ at $L$. Prove that $PQ$ is perpendicular to $AL$
2019 USAMO, 3
Let $K$ be the set of all positive integers that do not contain the digit $7$ in their base-$10$ representation. Find all polynomials $f$ with nonnegative integer coefficients such that $f(n)\in K$ whenever $n\in K$.
[i]Proposed by Titu Andreescu, Cosmin Pohoata, and Vlad Matei[/i]
2009 IberoAmerican, 3
Let $ C_1$ and $ C_2$ be two congruent circles centered at $ O_1$ and $ O_2$, which intersect at $ A$ and $ B$. Take a point $ P$ on the arc $ AB$ of $ C_2$ which is contained in $ C_1$. $ AP$ meets $ C_1$ at $ C$, $ CB$ meets $ C_2$ at $ D$ and the bisector of $ \angle CAD$ intersects $ C_1$ and $ C_2$ at $ E$ and $ L$, respectively. Let $ F$ be the symmetric point of $ D$ with respect to the midpoint of $ PE$. Prove that there exists a point $ X$ satisfying $ \angle XFL \equal{} \angle XDC \equal{} 30^\circ$ and $ CX \equal{} O_1O_2$.
[i]
Author: Arnoldo Aguilar (El Salvador)[/i]
1993 Romania Team Selection Test, 3
Suppose that each of the diagonals $AD,BE,CF$ divides the hexagon $ABCDEF$ into two parts of the same area and perimeter. Does the hexagon necessarily have a center of symmetry?
2007 Indonesia TST, 4
Given a collection of sets $X = \{A_1, A_2, ..., A_n\}$. A set $\{a_1, a_2, ..., a_n\}$ is called a single representation of $X$ if $a_i \in A_i$ for all i. Let $|S| = mn$, $S = A_1\cup A_2 \cup ... \cup A_n = B_1 \cup B_2 \cup ... \cup B_n$ with $|A_i| = |B_i| = m$ for all $i$. Prove that $S = C_1 \cup C_2 \cup ... \cup C_n$ where for every $i, C_i $ is a single represenation for $\{A_j\}_{j=1}^n $and $\{B_j\}_{j=1}^n$.
2000 Harvard-MIT Mathematics Tournament, 31
Given collinear points $A,B,C$ such that $AB = BC$. How can you construct a point $D$ on $AB$ such that $AD = 2DB$, using only a straightedge? (You are not allowed to measure distances)
2005 International Zhautykov Olympiad, 2
The inner point $ X$ of a quadrilateral is [i]observable[/i] from the side $ YZ$ if the perpendicular to the line $ YZ$ meet it in the colosed interval $ [YZ].$ The inner point of a quadrilateral is a $ k\minus{}$point if it is observable from the exactly $ k$ sides of the quadrilateral. Prove that if a convex quadrilateral has a 1-point then it has a $ k\minus{}$point for each $ k\equal{}2,3,4.$
2012 Irish Math Olympiad, 1
Let $S(n)$ be the sum of the decimal digits of $n$. For example. $S(2012)=2+0+1+2=5$. Prove that there is no integer $n>0$ for which $n-S(n)=9990$.
2009 Costa Rica - Final Round, 5
Suppose the polynomial $ x^{n} \plus{} a_{n \minus{} 1}x^{n \minus{} 1} \plus{} ... \plus{} a_{1} \plus{} a_{0}$ can be factorized as $ (x \plus{} r_{1})(x \plus{} r_{2})...(x \plus{} r_{n})$, with $ r_{1}, r_{2}, ..., r_{n}$ real numbers.
Show that $ (n \minus{} 1)a_{n \minus{} 1}^{2}\geq\ 2na_{n \minus{} 2}$
2015 Iran Team Selection Test, 6
If $a,b,c$ are positive real numbers such that $a+b+c=abc$ prove that
$$\frac{abc}{3\sqrt{2}}\left ( \sum_{cyc}\frac{\sqrt{a^3+b^3}}{ab+1} \right )\geq \sum_{cyc}\frac{a}{a^2+1}$$
2020 CCA Math Bonanza, T5
Find all pairs of real numbers $(x,y)$ satisfying both equations
\[
3x^2+3xy+2y^2 =2
\]
\[
x^2+2xy+2y^2 =1.
\]
[i]2020 CCA Math Bonanza Team Round #5[/i]
LMT Speed Rounds, 2014
[b]p1.[/b] What is $6\times 7 + 4 \times 7 + 6\times 3 + 4\times 3$?
[b]p2.[/b] How many integers $n$ have exactly $\sqrt{n}$ factors?
[b]p3.[/b] A triangle has distinct angles $3x+10$, $2x+20$, and $x+30$. What is the value of $x$?
[b]p4.[/b] If $4$ people of the Math Club are randomly chosen to be captains, and Henry is one of the $30$ people eligible to be chosen, what is the probability that he is not chosen to be captain?
[b]p5.[/b] $a, b, c, d$ is an arithmetic sequence with difference $x$ such that $a, c, d$ is a geometric sequence. If $b$ is $12$, what is $x$? (Note: the difference of an aritmetic sequence can be positive or negative, but not $0$)
[b]p6.[/b] What is the smallest positive integer that contains only $0$s and $5$s that is a multiple of $24$.
[b]p7.[/b] If $ABC$ is a triangle with side lengths $13$, $14$, and $15$, what is the area of the triangle made by connecting the points at the midpoints of its sides?
[b]p8.[/b] How many ways are there to order the numbers $1,2,3,4,5,6,7,8$ such that $1$ and $8$ are not adjacent?
[b]p9.[/b] Find all ordered triples of nonnegative integers $(x, y, z)$ such that $x + y + z = xyz$.
[b]p10.[/b] Noah inscribes equilateral triangle $ABC$ with area $\sqrt3$ in a cricle. If $BR$ is a diameter of the circle, then what is the arc length of Noah's $ARC$?
[b]p11.[/b] Today, $4/12/14$, is a palindromic date, because the number without slashes $41214$ is a palindrome. What is the last palindromic date before the year $3000$?
[b]p12.[/b] Every other vertex of a regular hexagon is connected to form an equilateral triangle. What is the ratio of the area of the triangle to that of the hexagon?
[b]p13.[/b] How many ways are there to pick four cards from a deck, none of which are the same suit or number as another, if order is not important?
[b]p14.[/b] Find all functions $f$ from $R \to R$ such that $f(x + y) + f(x - y) = x^2 + y^2$.
[b]p15.[/b] What are the last four digits of $1(1!) + 2(2!) + 3(3!) + ... + 2013(2013!)$/
[b]p16.[/b] In how many distinct ways can a regular octagon be divided up into $6$ non-overlapping triangles?
[b]p17.[/b] Find the sum of the solutions to the equation $\frac{1}{x-3} + \frac{1}{x-5} + \frac{1}{x-7} + \frac{1}{x-9} = 2014$ .
[b]p18.[/b] How many integers $n$ have the property that $(n+1)(n+2)(n+3)(n+4)$ is a perfect square of an integer?
[b]p19.[/b] A quadrilateral is inscribed in a unit circle, and another one is circumscribed. What is the minimum possible area in between the two quadrilaterals?
[b]p20.[/b] In blindfolded solitary tic-tac-toe, a player starts with a blank $3$-by-$3$ tic-tac-toe board. On each turn, he randomly places an "$X$" in one of the open spaces on the board. The game ends when the player gets $3$ $X$s in a row, in a column, or in a diagonal as per normal tic-tac-toe rules. (Note that only $X$s are used, not $O$s). What fraction of games will run the maximum $7$ amount of moves?
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
PEN M Problems, 27
Let $ p \ge 3$ be a prime number. The sequence $ \{a_{n}\}_{n \ge 0}$ is defined by $ a_{n}=n$ for all $ 0 \le n \le p-1$, and $ a_{n}=a_{n-1}+a_{n-p}$ for all $ n \ge p$. Compute $ a_{p^{3}}\; \pmod{p}$.