Found problems: 85335
2017 China Team Selection Test, 4
An integer $n>1$ is given . Find the smallest positive number $m$ satisfying the following conditions： for any set $\{a,b\}$ $\subset \{1,2,\cdots,2n-1\}$ ,there are non-negative integers $ x, y$ ( not all zero) such that $2n|ax+by$ and $x+y\leq m.$
2012 IFYM, Sozopol, 3
Find all pairs of positive integers $(x,y) $ for which $x^3 + y^3 = 4(x^2y + xy^2 - 5) .$
2012 Saint Petersburg Mathematical Olympiad, 1
$a,b,c$ are reals, such that every pair of equations of $x^3-ax^2+b=0,x^3-bx^2+c=0,x^3-cx^2+a=0$ has common root.
Prove $a=b=c$
2018 CCA Math Bonanza, L2.1
Let $S$ be the set of the first $2018$ positive integers, and let $T$ be the set of all distinct numbers of the form $ab$, where $a$ and $b$ are distinct members of $S$. What is the $2018$th smallest member of $T$?
[i]2018 CCA Math Bonanza Lightning Round #2.1[/i]
1953 Moscow Mathematical Olympiad, 257
Let $x_0 = 10^9$, $x_n = \frac{x^2_{n-1}+2}{2x_{n-1}}$ for $n > 0$. Prove that $0 < x_{36} - \sqrt2 < 10^{-9}$.
2016 Portugal MO, 6
The natural numbers are colored green or blue so that:
$\bullet$ The sum of a green and a blue is blue;
$\bullet$ The product of a green and a blue is green.
How many ways are there to color the natural numbers with these rules, so that $462$ are blue and $2016$ are green?
2012 AMC 8, 8
A shop advertises everything is "half price in today's sale." In addition, a coupon gives a $20\%$ discount on sale prices. Using the coupon, the price today represents what percentage off the original price?
$\textbf{(A)}\hspace{.05in}10 \qquad \textbf{(B)}\hspace{.05in}33 \qquad \textbf{(C)}\hspace{.05in}40 \qquad \textbf{(D)}\hspace{.05in}60 \qquad \textbf{(E)}\hspace{.05in}70 $
2005 Junior Tuymaada Olympiad, 1
In each cell of the table $ 3 \times 3 $ there is one of the numbers $1, 2$ and $3$. Dima counted the sum of the numbers in each row and in each column. What is the greatest number of different sums he could get?
2021 Science ON grade VI, 3
Consider positive integers $a<b$ and the set $C\subset\{a,a+1,a+2,\dots ,b-2,b-1,b\}$. Suppose $C$ has more than $\frac{b-a+1}{2}$ elements. Prove that there are two elements $x,y\in C$ that satisfy $x+y=a+b$.
[i] (From "Radu Păun" contest, Radu Miculescu)[/i]
PEN N Problems, 4
Show that if an infinite arithmetic progression of positive integers contains a square and a cube, it must contain a sixth power.
1977 AMC 12/AHSME, 25
Determine the largest positive integer $n$ such that $1005!$ is divisible by $10^n$.
$\textbf{(A) }102\qquad\textbf{(B) }112\qquad\textbf{(C) }249\qquad\textbf{(D) }502\qquad \textbf{(E) }\text{none of these}$
2013 IMO, 2
A configuration of $4027$ points in the plane is called Colombian if it consists of $2013$ red points and $2014$ blue points, and no three of the points of the configuration are collinear. By drawing some lines, the plane is divided into several regions. An arrangement of lines is good for a Colombian configuration if the following two conditions are satisfied:
i) No line passes through any point of the configuration.
ii) No region contains points of both colors.
Find the least value of $k$ such that for any Colombian configuration of $4027$ points, there is a good arrangement of $k$ lines.
Proposed by [i]Ivan Guo[/i] from [i]Australia.[/i]
2016 CHMMC (Fall), 14
For a unit circle $O$, arrange points $A,B,C,D$ and $E$ in that order evenly along $O$'s circumference. For each of those points, draw the arc centered at that point inside O from the point to its left to the point to its right. Denote the outermost intersections of these arcs as $A', B', C', D'$ and $E'$, where the prime of any point is opposite the point. The length of $AC'$ can be written as an expression $f(x)$, where $f$ is a trigonometric function. Find this expression.
2020 Online Math Open Problems, 3
Compute the number of ways to write the numbers 1, 2, 3, 4, 5, 6, 7, 8, and 9 in the cells of a 3 by 3 grid such that
[list]
[*] each cell has exactly one number,
[*] each number goes in exactly one cell,
[*] the numbers in each row are increasing from left to right,
[*] the numbers in each column are increasing from top to bottom, and
[*]the numbers in the diagonal from the upper-right corner cell to the lower-left corner cell are increasing from upper-right to lower-left.
[/list]
[i]Proposed by Ankit Bisain & Luke Robitaille[/i]
2021 Abels Math Contest (Norwegian MO) Final, 2b
If $a_1,\cdots,a_n$ and $b_1,\cdots,b_n$ are real numbers satisfying $a_1^2+\cdots+a_n^2 \le 1$ and $b_1^2+\cdots+b_n^2 \le 1$ , show that:
$$(1-(a_1^2+\cdots+a_n^2))(1-(b_1^2+\cdots+b_n^2)) \le (1-(a_1b_1+\cdots+a_nb_n))^2$$
2002 China Western Mathematical Olympiad, 1
Given a trapezoid $ ABCD$ with $ AD\parallel BC, E$ is a moving point on the side $ AB,$ let $ O_{1},O_{2}$ be the circumcenters of triangles $ AED,BEC$, respectively. Prove that the length of $ O_{1}O_{2}$ is a constant value.
2018 Vietnam Team Selection Test, 4
Let $a\in\left[ \tfrac{1}{2},\ \tfrac{3}{2}\right]$ be a real number. Sequences $(u_n),\ (v_n)$ are defined as follows:
$$u_n=\frac{3}{2^{n+1}}\cdot (-1)^{\lfloor2^{n+1}a\rfloor},\ v_n=\frac{3}{2^{n+1}}\cdot (-1)^{n+\lfloor 2^{n+1}a\rfloor}.$$
a. Prove that
$${{({{u}_{0}}+{{u}_{1}}+\cdots +{{u}_{2018}})}^{2}}+{{({{v}_{0}}+{{v}_{1}}+\cdots +{{v}_{2018}})}^{2}}\le 72{{a}^{2}}-48a+10+\frac{2}{{{4}^{2019}}}.$$
b. Find all values of $a$ in the equality case.
2006 AMC 12/AHSME, 5
Doug and Dave shared a pizza with $ 8$ equally-sized slices. Doug wanted a plain pizza, but Dave wanted anchovies on half the pizza. The cost of a plain pizza was $ \$8$, and there was an additional cost of $ \$2$ for putting anchovies on one half. Dave ate all the slices of anchovy pizza and one plain slice. Doug ate the remainder. Each paid for what he had eaten. How many more dollars did Dave pay than Doug?
$ \textbf{(A) } 1\qquad \textbf{(B) } 2\qquad \textbf{(C) } 3\qquad \textbf{(D) } 4\qquad \textbf{(E) } 5$
2011 LMT, 6
Define a sequence by $a_1=a_2=1, a_3=2,$ and
$$a_n+a_{n-3}=a_{n-1}+a_{n-2}$$
for all $n>3.$ What is the value of $a_7?$
1999 Canada National Olympiad, 3
Determine all positive integers $n$ with the property that $n = (d(n))^2$. Here $d(n)$ denotes the number of positive divisors of $n$.
Putnam 1938, B1
Do either $(1)$ or $(2)$
$(1)$ Let $A$ be matrix $(a_{ij}), 1 \leq i,j \leq 4.$ Let $d =$ det$(A),$ and let $A_{ij}$ be the cofactor of $a_{ij}$, that is, the determinant of the $3 \times 3$ matrix formed from $A$ by deleting $a_{ij}$ and other elements in the same row and column. Let $B$ be the $4 \times 4$ matrix $(A_{ij})$ and let $D$ be det $B.$ Prove $D = d^3$.
$(2)$ Let $P(x)$ be the quadratic $Ax^2 + Bx + C.$ Suppose that $P(x) = x$ has unequal real roots. Show that the roots are also roots of $P(P(x)) = x.$ Find a quadratic equation for the other two roots of this equation. Hence solve $(y^2 - 3y + 2)2 - 3(y^2 - 3y + 2) + 2 - y = 0.$
2023 Abelkonkurransen Finale, 2b
Arne and Berit are playing a game. They have chosen positive integers $m$ and $n$ with $n\geq 4$ and $m \leq 2n + 1$. Arne begins by choosing a number from the set $\{1, 2, \dots , n \}$, and writes it on a blackboard. Then Berit picks another number from the same set, and writes it on the board. They continue alternating turns, always choosing numbers that are not already on the blackboard. When the sum of all the numbers on the board exceeds or equals $m$, the game is over, and whoever wrote the last number has won. For which combinations of $m$ and $n$ does Arne have a winning strategy?
Ukraine Correspondence MO - geometry, 2019.7
Given a triangle $ABC$. Construct a point $D$ on the side $AB$ and point $E$ on the side $AC$ so that $BD = CE$ and $\angle ADC = \angle BEC$
1968 German National Olympiad, 2
Which of all planes, the one and the same body diagonal of a cube with the edge length $a$, cuts out a cut figure with the smallest area from the cube? Calculate the area of such a cut figure.
[hide=original wording]Welche von allen Ebenen, die eine und dieselbe Korperdiagonale eines Wurfels mit der Kantenlange a enthalten, schneiden aus den W¨urfel eine Schnittfigur kleinsten Flacheninhaltes heraus? Berechnen
Sie den Fl¨acheninhalt solch einer Schnittfigur![/hide]
2015 Harvard-MIT Mathematics Tournament, 8
Let $S$ be the set of [b]discs[/b] $D$ contained completely in the set $\{ (x,y) : y<0\}$ (the region below the $x$-axis) and centered (at some point) on the curve $y=x^2-\frac{3}{4}$. What is the area of the union of the elements of $S$?