Found problems: 85335
2017 ASDAN Math Tournament, 21
In trapezoid $ABCD$, we have $\overline{AD}\parallel\overline{BC}$, $BC=3$, and $CD=4$. In addition, $\cos\angle ADC=\tfrac{1}{3}$ and $\angle ABC=2\angle ADC$. Compute $AC$.
2021 Saudi Arabia IMO TST, 8
The Fibonacci numbers $F_0, F_1, F_2, . . .$ are defined inductively by $F_0=0, F_1=1$, and $F_{n+1}=F_n+F_{n-1}$ for $n \ge 1$. Given an integer $n \ge 2$, determine the smallest size of a set $S$ of integers such that for every $k=2, 3, . . . , n$ there exist some $x, y \in S$ such that $x-y=F_k$.
[i]Proposed by Croatia[/i]
1970 All Soviet Union Mathematical Olympiad, 134
Given five segments. It is possible to construct a triangle of every subset of three of them. Prove that at least one of those triangles is acute-angled.
2002 ITAMO, 4
Find all values of $n$ for which all solutions of the equation $x^3-3x+n=0$ are integers.
2022 CCA Math Bonanza, I11
A river is bounded by the lines $x=0$ and $x=25$, with a current of 2 units/s in the positive y-direction. At $t=0$, a mallard is at $(0, 0)$, and a wigeon is at $(25, 0)$. They start swimming with a constant speed such that they meet at $(x,22)$. The mallard has a speed of 4 units/s relative to the water, and the wigeon has a speed of 3 units/s relative to the water. Find the value of $x$.
[i]2022 CCA Math Bonanza Individual Round #11[/i]
2024 Iran MO (3rd Round), 2
Two intelligent people playing a game on the $1403 \times 1403$ table with $1403^2$ cells. The first one in each turn chooses a cell that didn't select before and draws a vertical line segment from the top to the bottom of the cell. The second person in each turn chooses a cell that didn't select before and draws a horizontal line segment from the left to the right of the cell. After $1403^2$ steps the game will be over. The first person gets points equal to the longest verticals line segment and analogously the second person gets point equal to the longest horizonal line segment. At the end the person who gets the more point will win the game. What will be the result of the game?
2016 District Olympiad, 3
[b]a)[/b] Prove that, for any integer $ k, $ the equation $ x^3-24x+k=0 $ has at most an integer solution.
[b]b)[/b] Show that the equation $ x^3+24x-2016=0 $ has exactly one integer solution.
2009 Federal Competition For Advanced Students, P2, 2
(i) For positive integers $a<b$, let $M(a,b)=\frac{\Sigma^{b}_{k=a}\sqrt{k^2+3k+3}}{b-a+1}$.
Calculate $[M(a,b)]$
(ii) Calculate $N(a,b)=\frac{\Sigma^{b}_{k=a}[\sqrt{k^2+3k+3}]}{b-a+1}$.
2017 Canadian Mathematical Olympiad Qualification, 3
Determine all functions $f : \mathbb{R} \rightarrow \mathbb{R}$ that satisfy the following equation for all $x, y \in \mathbb{R}$.
$$(x+y)f(x-y) = f(x^2-y^2).$$
1976 Canada National Olympiad, 6
If $ A,B,C,D$ are four points in space, such that
\[ \angle ABC\equal{}\angle BCD\equal{}\angle CDA\equal{}\angle DAB\equal{}\pi/2,
\]
prove that $ A,B,C,D$ lie in a plane.
2014 Peru IMO TST, 10
Let $ABCDEF$ be a convex hexagon that does not have two parallel sides, such that $\angle AF B = \angle F DE, \angle DF E = \angle BDC$ and $\angle BFC = \angle ADF.$ Prove that the lines $ AB, FC$ and $DE$ are concurrent if and only if the lines $ AF, BE$ and $CD$ are concurrent.
2021 Purple Comet Problems, 10
A semicircle has diameter $AB$ with $AB = 100$. Points $C$ and $D$ lie on the semicircle such that $AC = 28$ and $BD = 60$. Find $CD$.
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$?