Found problems: 85335
2014 India IMO Training Camp, 2
For $j=1,2,3$ let $x_{j},y_{j}$ be non-zero real numbers, and let $v_{j}=x_{j}+y_{j}$.Suppose that the following statements hold:
$x_{1}x_{2}x_{3}=-y_{1}y_{2}y_{3}$
$x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=y_{1}^{2}+y_{2}^{2}+y_{3}^2$
$v_{1},v_{2},v_{3}$ satisfy triangle inequality
$v_{1}^{2},v_{2}^{2},v_{3}^{2}$ also satisfy triangle inequality.
Prove that exactly one of $x_{1},x_{2},x_{3},y_{1},y_{2},y_{3}$ is negative.
2016 Croatia Team Selection Test, Problem 2
Let $N$ be a positive integer. Consider a $N \times N$ array of square unit cells. Two corner cells that lie on the same longest diagonal are colored black, and the rest of the array is white. A [i]move[/i] consists of choosing a row or a column and changing the color of every cell in the chosen row or column.
What is the minimal number of additional cells that one has to color black such that, after a finite number of moves, a completely black board can be reached?
2019 Junior Balkan Team Selection Tests - Romania, 4
The numbers from $1$ through $100$ are written in some order on a circle.
We call a pair of numbers on the circle [i]good [/i] if the two numbers are not neighbors on the circle and if at least one of the two arcs they determine on the circle only contains numbers smaller then both of them. What may be the total number of good pairs on the circle.
2011 Putnam, B1
Let $h$ and $k$ be positive integers. Prove that for every $\varepsilon >0,$ there are positive integers $m$ and $n$ such that \[\varepsilon < \left|h\sqrt{m}-k\sqrt{n}\right|<2\varepsilon.\]
1990 Baltic Way, 5
Let $*$ be an operation, assigning a real number $a * b$ to each pair of real numbers $(a, b)$. Find an equation which is true (for all possible values of variables) provided the operation $*$ is commutative or associative and which can be false otherwise.
2006 AMC 10, 21
How many four-digit positive integers have at least one digit that is a 2 or a 3?
$ \textbf{(A) } 2439 \qquad \textbf{(B) } 4096 \qquad \textbf{(C) } 4903 \qquad \textbf{(D) } 4904 \qquad \textbf{(E) } 5416$
1947 Moscow Mathematical Olympiad, 131
Calculate (without calculators, tables, etc.) with accuracy to $0.00001$ the product $\left(1-\frac{1}{10}\right)\left(1-\frac{1}{10^2}\right)...\left(1-\frac{1}{10^{99}}\right)$
2020 CCA Math Bonanza, T6
A cat can see $1$ mile in any direction. The cat walks around the $13$ mile perimeter of a triangle. Over the course of its walk, it sees every point inside of this triangle. What is the largest possible area, in square miles, of the total region it sees?
[i]2020 CCA Math Bonanza Team Round #6[/i]
2022 Czech and Slovak Olympiad III A, 6
Consider any graph with $50$ vertices and $225$ edges. We say that a triplet of its (mutually distinct) vertices is [i]connected[/i] if the three vertices determine at least two edges. Determine the smallest and the largest possible number of connected triples.
[i](Jan Mazak, Josef Tkadlec)[/i]
2022 AMC 8 -, 19
Mr. Ramos gave a test to his class of $20$ students. The dot plot below shows the distribution of test scores.
[asy]
//diagram by pog . give me 1,000,000,000 dollars for this diagram
size(5cm);
defaultpen(0.7);
dot((0.5,1));
dot((0.5,1.5));
dot((1.5,1));
dot((1.5,1.5));
dot((2.5,1));
dot((2.5,1.5));
dot((2.5,2));
dot((2.5,2.5));
dot((3.5,1));
dot((3.5,1.5));
dot((3.5,2));
dot((3.5,2.5));
dot((3.5,3));
dot((4.5,1));
dot((4.5,1.5));
dot((5.5,1));
dot((5.5,1.5));
dot((5.5,2));
dot((6.5,1));
dot((7.5,1));
draw((0,0.5)--(8,0.5),linewidth(0.7));
defaultpen(fontsize(10.5pt));
label("$65$", (0.5,-0.1));
label("$70$", (1.5,-0.1));
label("$75$", (2.5,-0.1));
label("$80$", (3.5,-0.1));
label("$85$", (4.5,-0.1));
label("$90$", (5.5,-0.1));
label("$95$", (6.5,-0.1));
label("$100$", (7.5,-0.1));
[/asy]
Later Mr. Ramos discovered that there was a scoring error on one of the questions. He regraded the tests, awarding some of the students $5$ extra points, which increased the median test score to $85$. What is the minimum number of students who received extra points?
(Note that the [i]median[/i] test score equals the average of the $2$ scores in the middle if the $20$ test scores are arranged in increasing order.)
$\textbf{(A)} ~2\qquad\textbf{(B)} ~3\qquad\textbf{(C)} ~4\qquad\textbf{(D)} ~5\qquad\textbf{(E)} ~6\qquad$
2013 BMT Spring, 1
A rectangle with sides $a$ and $b$ has an area of $24$ and a diagonal of length $11$. Find the perimeter of this rectangle.
1962 Poland - Second Round, 6
Find a three-digit number with the property that the number represented by these digits and in the same order, but with a numbering base different than $ 10 $, is twice as large as the given number.
LMT Speed Rounds, 2016.10
There are sixteen buildings all on the same side of a street. How many ways can we choose a nonempty subset of the buildings such that there is an odd number of buildings between each pair of buildings in the subset?
[i]Proposed by Yiming Zheng
2024 Brazil Team Selection Test, 3
Let \( n \) be a positive integer. A function \( f : \{0, 1, \dots, n\} \to \{0, 1, \dots, n\} \) is called \( n \)-Bolivian if it satisfies the following conditions:
• \( f(0) = 0 \);
• \( f(t) \in \{ t-1, f(t-1), f(f(t-1)), \dots \} \) for all \( t = 1, 2, \dots, n \).
For example, if \( n = 3 \), then the function defined by \( f(0) = f(1) = 0 \), \( f(2) = f(3) = 1 \) is 3-Bolivian, but the function defined by \( f(0) = f(1) = f(2) = 0 \), \( f(3) = 1 \) is not 3-Bolivian.
For a fixed positive integer \( n \), Gollum selects an \( n \)-Bolivian function. Smeagol, knowing that \( f \) is \( n \)-Bolivian, tries to figure out which function was chosen by asking questions of the type:
\[
\text{How many integers } a \text{ are there such that } f(a) = b?
\]
given a \( b \) of his choice. Show that if Gollum always answers correctly, Smeagol can determine \( f \) and find the minimum number of questions he needs to ask, considering all possible choices of \( f \).
2016 Dutch IMO TST, 1
Prove that for all positive reals $a, b,c$ we have: $a +\sqrt{ab}+ \sqrt[3]{abc}\le \frac43 (a + b + c)$
2002 AMC 12/AHSME, 1
Compute the sum of all the roots of $ (2x \plus{} 3)(x \minus{} 4) \plus{} (2x \plus{} 3)(x \minus{} 6) \equal{} 0$.
$ \textbf{(A)}\ 7/2 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 7 \qquad \textbf{(E)}\ 13$
2004 Tournament Of Towns, 7
Let A and B be two rectangles such that it is possible to get rectangle similar to A by putting together rectangles equal to B. Show that it is possible to get rectangle similar to B by putting together rectangles equal to A.
2016 Harvard-MIT Mathematics Tournament, 10
Quadrilateral $ABCD$ satisfies $AB = 8, BC = 5, CD = 17, DA = 10$. Let $E$ be the intersection of $AC$ and $BD$. Suppose $BE : ED = 1 : 2$. Find the area of $ABCD$.
2007 Hanoi Open Mathematics Competitions, 13
Let ABC be an acute-angle triangle with BC > CA. Let O, H and F be the circumcenter, orthocentre and the foot of its altitude CH, respectively. Suppose that the perpendicular to OF at F meet the side CA at P. Prove FHP = BAC.
1947 Moscow Mathematical Olympiad, 126
Given a convex pentagon $ABCDE$, prove that if an arbitrary point $M$ inside the pentagon is connected by lines with all the pentagon’s vertices, then either one or three or five of these lines cross the sides of the pentagon opposite the vertices they pass.
Note: In reality, we need to exclude the points of the diagonals, because that in this case the drawn lines can pass not through the internal points of the sides, but through the vertices. But if the drawn diagonals are not considered or counted twice (because they are drawn from two vertices), then the statement remains true.
2003 BAMO, 3
A lattice point is a point $(x, y)$ with both $x$ and $y$ integers. Find, with proof, the smallest $n$ such that every set of $n$ lattice points contains three points that are the vertices of a triangle with integer area. (The triangle may be degenerate, in other words, the three points may lie on a straight line and hence form a triangle with area zero.)
MMPC Part II 1958 - 95, 1992
[b]p1.[/b] The English alphabet consists of $21$ consonants and $5$ vowels. (We count $y$ as a consonant.)
(a) Suppose that all the letters are listed in an arbitrary order. Prove that there must be $4$ consecutive consonants.
(b) Give a list to show that there need not be $5$ consecutive consonants.
(c) Suppose that all the letters are arranged in a circle. Prove that there must be $5$ consecutive consonants.
[b]p2.[/b] From the set $\{1,2,3,... , n\}$, $k$ distinct integers are selected at random and arranged in numerical order (lowest to highest). Let $P(i, r, k, n)$ denote the probability that integer $i$ is in position $r$. For example, observe that $P(1, 2, k, n) = 0$.
(a) Compute $P(2, 1,6,10)$.
(b) Find a general formula for $P(i, r, k, n)$.
[b]p3.[/b] (a) Write down a fourth degree polynomial $P(x)$ such that $P(1) = P(-1)$ but $P(2) \ne P(-2)$
(b) Write down a fifth degree polynomial $Q(x)$ such that $Q(1) = Q(-1)$ and $Q(2) = Q(-2)$ but $Q(3) \ne Q(-3)$.
(c) Prove that, if a sixth degree polynomial $R(x)$ satisfies $R(1) = R(-1)$, $R(2) = R(-2)$, and $R(3) = R(-3)$, then $R(x) = R(-x)$ for all $x$.
[b]p4.[/b] Given five distinct real numbers, one can compute the sums of any two, any three, any four, and all five numbers and then count the number $N$ of distinct values among these sums.
(a) Give an example of five numbers yielding the smallest possible value of $N$. What is this value?
(b) Give an example of five numbers yielding the largest possible value of $N$. What is this value?
(c) Prove that the values of $N$ you obtained in (a) and (b) are the smallest and largest possible ones.
[b]p5.[/b] Let $A_1A_2A_3$ be a triangle which is not a right triangle. Prove that there exist circles $C_1$, $C_2$, and $C_3$ such that $C_2$ is tangent to $C_3$ at $A_1$, $C_3$ is tangent to $C_1$ at $A_2$, and $C_1$ is tangent to $C_2$ at $A_3$.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
1988 IMO Longlists, 50
Prove that the numbers $A,B$ and $C$ are equal, where:
- $A=$ number of ways that we can cover a $2 \times n$ rectangle with $2 \times 1$ retangles.
- $B=$ number of sequences of ones and twos that add up to $n$
- $C= \sum^m_{k=0} \binom{m + k}{2 \cdot k}$ if $n = 2 \cdot m,$ and
- $C= \sum^m_{k=0} \binom{m + k + 1}{2 \cdot k + 1}$ if $n = 2 \cdot m + 1.$
2023 ELMO Shortlist, A6
Let \(\mathbb R_{>0}\) denote the set of positive real numbers and \(\mathbb R_{\ge0}\) the set of nonnegative real numbers. Find all functions \(f:\mathbb R\times \mathbb R_{>0}\to \mathbb R_{\ge0}\) such that for all real numbers \(a\), \(b\), \(x\), \(y\) with \(x,y>0\), we have \[f(a,x)+f(b,y)=f(a+b,x+y)+f(ay-bx,xy(x+y)).\]
[i]Proposed by Luke Robitaille[/i]
2015 CCA Math Bonanza, L2.1
What is the sum of the first $10$ primes?
[i]2015 CCA Math Bonanza Lightning Round #2.1[/i]