Found problems: 85335
2021 Taiwan TST Round 3, C
There are $2020$ points on the coordinate plane {$A_i = (x_i, y_i):i = 1, ..., 2020$}, satisfying
$$0=x_1<x_2<...<x_{2020}$$
$$0=y_{2020}<y_{2019}<...<y_1$$
Let $O=(0, 0)$ be the origin, $OA_1A_2...A_{2020}$ forms a polygon $C$.
Now, you want to blacken the polygon $C$. Every time you can choose a point $(x,y)$ with $x, y > 0$, and blacken the area {$(x', y'): 0\leq x' \leq x, 0\leq y' \leq y$}. However, you have to pay $xy$ dollars for doing so.
Prove that you could blacken the whole polygon $C$ by using $4|C|$ dollars. Here, $|C|$ stands for the area of the polygon $C$.
[i]Proposed by me[/i]
2014 Regional Olympiad of Mexico Center Zone, 1
Find the smallest positive integer $n$ that satisfies that for any $n$ different integers, the product of all the positive differences of these numbers is divisible by $2014$.
2021 CMIMC, 1.8
Let $ABC$ be a triangle with $AB < AC$ and $\omega$ be a circle through $A$ tangent to both the $B$-excircle and the $C$-excircle. Let $\omega$ intersect lines $AB, AC$ at $X,Y$ respectively and $X,Y$ lie outside of segments $AB, AC$. Let $O$ be the center of $\omega$ and let $OI_C, OI_B$ intersect line $BC$ at $J,K$ respectively. Suppose $KJ = 4$, $KO = 16$ and $OJ = 13$. Find $\frac{[KI_BI_C]}{[JI_BI_C]}$.
[i]Proposed by Grant Yu[/i]
2011 Today's Calculation Of Integral, 728
Evaluate
\[\int_{\frac {\pi}{12}}^{\frac{\pi}{6}} \frac{\sin x-\cos x-x(\sin x+\cos x)+1}{x^2-x(\sin x+\cos x)+\sin x\cos x}\ dx.\]
1996 IMO Shortlist, 6
Let the sides of two rectangles be $ \{a,b\}$ and $ \{c,d\},$ respectively, with $ a < c \leq d < b$ and $ ab < cd.$ Prove that the first rectangle can be placed within the second one if and only if
\[ \left(b^2 \minus{} a^2\right)^2 \leq \left(bc \minus{} ad \right)^2 \plus{} \left(bd \minus{} ac \right)^2.\]
2016 CMIMC, 8
Brice is eating bowls of rice. He takes a random amount of time $t_1 \in (0,1)$ minutes to consume his first bowl, and every bowl thereafter takes $t_n = t_{n-1} + r_n$ minutes, where $t_{n-1}$ is the time it took him to eat his previous bowl and $r_n \in (0,1)$ is chosen uniformly and randomly. The probability that it takes Brice at least 12 minutes to eat 5 bowls of rice can be expressed as simplified fraction $\tfrac{m}{n}$. Compute $m+n$.
2008 Putnam, B6
Let $ n$ and $ k$ be positive integers. Say that a permutation $ \sigma$ of $ \{1,2,\dots n\}$ is $ k$-[i]limited[/i] if $ |\sigma(i)\minus{}i|\le k$ for all $ i.$ Prove that the number of $ k$-limited permutations of $ \{1,2,\dots n\}$ is odd if and only if $ n\equiv 0$ or $ 1\pmod{2k\plus{}1}.$
2012-2013 SDML (High School), 4
For what digit $A$ is the numeral $1AA$ a perfect square in base-$5$ and a perfect cube in base-$6$?
$\text{(A) }0\qquad\text{(B) }1\qquad\text{(C) }2\qquad\text{(D) }3\qquad\text{(E) }4$
2024 Korea Summer Program Practice Test, 2
Find all integer sequences $a_1 , a_2 , \ldots , a_{2024}$ such that $1\le a_i \le 2024$ for $1\le i\le 2024$ and
$$i+j|ia_i-ja_j$$
for each pair $1\le i,j \le 2024$.
2020 BMT Fall, Tie 3
Three distinct integers $a_1$, $a_2$, $a_3$ between $1$ and $21$, inclusive, are selected uniformly at random. The probability that the greatest common factor of $a_i-a_j$ and $21$ is $7$ for some positive integers $i $ and $j$, where $1 \le i \ne j \le3 $, can be written in the form $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Compute $m + n$.
2023 VIASM Summer Challenge, Problem 1
Find the largest positive real number $k$ such that the inequality$$a^3+b^3+c^3-3\ge k(3-ab-bc-ca)$$holds for all positive real triples $(a;b;c)$ satisfying $a+b+c=3.$
2018 MIG, 19
Rectangle $ABCD$, with integer side lengths, has equal area and perimeter. What is the positive difference between the two possible areas of $ABCD$?
$\textbf{(A) } 0\qquad\textbf{(B) } 2\qquad\textbf{(C) } 4\qquad\textbf{(D) } 5\qquad\textbf{(E) } 6$
2009 Croatia Team Selection Test, 2
In each field of 2009*2009 table you can write either 1 or -1.
Denote Ak multiple of all numbers in k-th row and Bj the multiple of all numbers in j-th column.
Is it possible to write the numbers in such a way that
$ \sum_{i\equal{}1}^{2009}{Ai}\plus{} \sum_{i\equal{}1}^{2009}{Bi}\equal{}0$?
2011 Mathcenter Contest + Longlist, 3 sl3
We will call the sequence of positive real numbers. $a_1,a_2,\dots ,a_n$ of [i]length [/i] $n$ when $$a_1\geq \frac{a_1+a_2}{2}\geq \dots \geq \frac{a_1+a_2+\cdots +a_n}{n}.$$
Let $x_1,x_2,\dots ,x_n$ and $y_1,y_2,\dots ,y_n$ be sequences of length $n.$ Prove that
$$\sum_{i = 1}^{n}x_iy_i\geq\frac{1}{n}\left(\sum_{i = 1}^{n}x_i\right)\left(\sum_{i = 1}^{n}y_i\right).$$
[i](tatari/nightmare)[/i]
2002 AMC 12/AHSME, -1
This test and the matching AMC 10P were developed for the use of a group of Taiwan schools, in early January of 2002. When Taiwan had taken the contests, the AMC released the questions here as a set of practice questions for the 2002 AMC 10 and AMC 12 contests.
2020 Centroamerican and Caribbean Math Olympiad, 3
Find all the functions $f: \mathbb{Z}\to \mathbb{Z}$ satisfying the following property: if $a$, $b$ and $c$ are integers such that $a+b+c=0$, then
$$f(a)+f(b)+f(c)=a^2+b^2+c^2.$$
2022 Thailand TST, 1
For each integer $n\ge 1,$ compute the smallest possible value of \[\sum_{k=1}^{n}\left\lfloor\frac{a_k}{k}\right\rfloor\] over all permutations $(a_1,\dots,a_n)$ of $\{1,\dots,n\}.$
[i]Proposed by Shahjalal Shohag, Bangladesh[/i]
2013 National Chemistry Olympiad, 60
Which vitamin is the most soluble in water?
${ \textbf{(A)}\ \text{A} \qquad\textbf{(B)}\ \text{K} \qquad\textbf{(C)}\ \text{C} \qquad\textbf{(D)}}\ \text{D} \qquad $
2015 Irish Math Olympiad, 7
Let $n > 1$ be an integer and $\Omega=\{1,2,...,2n-1,2n\}$ the set of all positive integers that are not larger than $2n$.
A nonempty subset $S$ of $\Omega$ is called [i]sum-free[/i] if, for all elements $x, y$ belonging to $S, x + y$ does not belong to $S$. We allow $x = y$ in this condition.
Prove that $\Omega$ has more than $2^n$ distinct [i]sum-free[/i] subsets.
2014 All-Russian Olympiad, 2
Let $M$ be the midpoint of the side $AC$ of $ \triangle ABC$. Let $P\in AM$ and $Q\in CM$ be such that $PQ=\frac{AC}{2}$. Let $(ABQ)$ intersect with $BC$ at $X\not= B$ and $(BCP)$ intersect with $BA$ at $Y\not= B$. Prove that the quadrilateral $BXMY$ is cyclic.
[i]F. Ivlev, F. Nilov[/i]
2023 Estonia Team Selection Test, 2
For any natural number $n{}$ and positive integer $k{}$, we say that $n{}$ is $k-good$ if there exist non-negative integers $a_1,\ldots, a_k$ such that $$n=a_1^2+a_2^4+a_3^8+\ldots+a_k^{2^k}.$$ Is there a positive integer $k{}$ for which every natural number is $k-good$?
2014 Sharygin Geometry Olympiad, 23
Let $A, B, C$ and $D$ be a triharmonic quadruple of points, i.e $AB\cdot CD = AC \cdot BD = AD \cdot BC.$ Let $A_1$ be a point distinct from $A$ such that the quadruple $A_1, B, C$ and $D$ is triharmonic. Points $B_1, C_1$ and $D_1$ are defined similarly. Prove that
a) $A, B, C_1, D_1$ are concyclic;
b) the quadruple $A_1, B_1, C_1, D_1$ is triharmonic.
2017 Taiwan TST Round 1, 2
Let $ABC$ be a triangle with $AB = AC \neq BC$ and let $I$ be its incentre. The line $BI$ meets $AC$ at $D$, and the line through $D$ perpendicular to $AC$ meets $AI$ at $E$. Prove that the reflection of $I$ in $AC$ lies on the circumcircle of triangle $BDE$.
2007 German National Olympiad, 1
Determine all real numbers $x$ such that for all positive integers $n$ the inequality $(1+x)^n \leq 1+(2^n -1)x$ is true.
Durer Math Competition CD Finals - geometry, 2012.D5
The points of a circle of unit radius are colored in two colors. Prove that $3$ points of the same color can be chosen such that the area of the triangle they define is at least $\frac{9}{10}$.