Found problems: 85335
2010 Contests, 4
Let's consider the inequality $ a^3\plus{}b^3\plus{}c^3<k(a\plus{}b\plus{}c)(ab\plus{}bc\plus{}ca)$ where $ a,b,c$ are the sides of a triangle and $ k$ a real number.
[b]a)[/b] Prove the inequality for $ k\equal{}1$.
[b]b) [/b]Find the smallest value of $ k$ such that the inequality holds for all triangles.
1990 AMC 12/AHSME, 5
Which of these numbers is the largest?
$\textbf{(A)} \sqrt{\sqrt[3]{5\cdot 6}}\qquad
\textbf{(B)} \sqrt{6\sqrt[3]{5}}\qquad
\textbf{(C)} \sqrt{5\sqrt[3]{6}}\qquad
\textbf{(D)} \sqrt[3]{5\sqrt{6}}\qquad
\textbf{(E)} \sqrt[3]{6\sqrt{5}}$
2020 Iran Team Selection Test, 5
For every positive integer $k>1$ prove that there exist a real number $x$ so that for every positive integer $n<1398$:
$$\left\{x^n\right\}<\left\{x^{n-1}\right\} \Longleftrightarrow k\mid n.$$
[i]Proposed by Mohammad Amin Sharifi[/i]
2014 NZMOC Camp Selection Problems, 1
Prove that for all positive real numbers $a$ and $ b$: $$\frac{(a + b)^3}{4} \ge a^2b + ab^2$$
2012 Hanoi Open Mathematics Competitions, 11
[b]Q11.[/b] Let be given a sequense $a_1=5, \; a_2=8$ and $a_{n+1}=a_n+3a_{n-1}, \qquad n=1,2,3,...$ Calculate the greatest common divisor of $a_{2011}$ and $a_{2012}$.
2015 China Girls Math Olympiad, 6
Let $\Gamma_1$ and $\Gamma_2$ be two non-overlapping circles. $A,C$ are on $\Gamma_1$ and $B,D$ are on $\Gamma_2$ such that $AB$ is an external common tangent to the two circles, and $CD$ is an internal common tangent to the two circles. $AC$ and $BD$ meet at $E$. $F$ is a point on $\Gamma_1$, the tangent line to $\Gamma_1$ at $F$ meets the perpendicular bisector of $EF$ at $M$. $MG$ is a line tangent to $\Gamma_2$ at $G$. Prove that $MF=MG$.
2023 Brazil National Olympiad, 4
Determine the smallest integer $k$ for which there are three distinct positive integers $a$, $b$ and $c$, such that $$a^2 =bc \text{ and } k = 2b+3c-a.$$
1995 Belarus Team Selection Test, 1
Prove that the number of odd coefficients in the polynomial $(1+x)^n$ is a power of $2$ for every positive integer $N$
2004 India IMO Training Camp, 3
An integer $n$ is said to be [i]good[/i] if $|n|$ is not the square of an integer. Determine all integers $m$ with the following property: $m$ can be represented, in infinitely many ways, as a sum of three distinct good integers whose product is the square of an odd integer.
[i]Proposed by Hojoo Lee, Korea[/i]
2020 AIME Problems, 2
Let $P$ be a point chosen uniformly at random in the interior of the unit square with vertices at $(0,0), (1,0), (1,1)$, and $(0,1)$. The probability that the slope of the line determined by $P$ and the point $\left(\frac58, \frac38 \right)$ is greater than $\frac12$ can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
2024 Harvard-MIT Mathematics Tournament, 2
Suppose $a$ and $b$ are positive integers. Isabella and Vidur both fill up an $a \times b$ table. Isabella fills it up with numbers $1, 2, . . . , ab$, putting the numbers $1, 2, . . . , b$ in the first row, $b + 1, b + 2, . . . , 2b$ in the second row, and so on. Vidur fills it up like a multiplication table, putting $ij$ in the cell in row $i$ and column $j$.
(Examples are shown for a $3 \times 4$ table below.)
[img]https://cdn.artofproblemsolving.com/attachments/6/8/a0855d790069ecd2cd709fbc5e70f21f1fa423.png[/img]
Isabella sums up the numbers in her grid, and Vidur sums up the numbers in his grid; the difference between these two quantities is $1200$. Compute $a + b$.
2006 Baltic Way, 18
For a positive integer $n$ let $a_n$ denote the last digit of $n^{(n^n)}$. Prove that the sequence $(a_n)$ is periodic and determine the length of the minimal period.
2007 China Girls Math Olympiad, 7
Let $ a$, $ b$, $ c$ be integers each with absolute value less than or equal to $ 10$. The cubic polynomial $ f(x) \equal{} x^3 \plus{} ax^2 \plus{} bx \plus{} c$ satisfies the property
\[ \Big|f\left(2 \plus{} \sqrt 3\right)\Big| < 0.0001.
\]
Determine if $ 2 \plus{} \sqrt 3$ is a root of $ f$.
LMT Speed Rounds, 2018 S
[b]p1.[/b] Evaluate $6^4 +5^4 +3^4 +2^4$.
[b]p2.[/b] What digit is most frequent between $1$ and $1000$ inclusive?
[b]p3.[/b] Let $n = gcd \, (2^2 \cdot 3^3 \cdot 4^4,2^4 \cdot 3^3 \cdot 4^2)$. Find the number of positive integer factors of $n$.
[b]p4.[/b] Suppose $p$ and $q$ are prime numbers such that $13p +5q = 91$. Find $p +q$.
[b]p5.[/b] Let $x = (5^3 -5)(4^3 -4)(3^3 -3)(2^3 -2)(1^3 -1)$. Evaluate $2018^x$ .
[b]p6.[/b] Liszt the lister lists all $24$ four-digit integers that contain each of the digits $1,2,3,4$ exactly once in increasing order. What is the sum of the $20$th and $18$th numbers on Liszt’s list?
[b]p7.[/b] Square $ABCD$ has center $O$. Suppose $M$ is the midpoint of $AB$ and $OM +1 =OA$. Find the area of square $ABCD$.
[b]p8.[/b] How many positive $4$-digit integers have at most $3$ distinct digits?
[b]p9.[/b] Find the sumof all distinct integers obtained by placing $+$ and $-$ signs in the following spaces
$$2\_3\_4\_5$$
[b]p10.[/b] In triangle $ABC$, $\angle A = 2\angle B$. Let $I$ be the intersection of the angle bisectors of $B$ and $C$. Given that $AB = 12$, $BC = 14$,and $C A = 9$, find $AI$ .
[b]p11.[/b] You have a $3\times 3\times 3$ cube in front of you. You are given a knife to cut the cube and you are allowed to move the pieces after each cut before cutting it again. What is the minimumnumber of cuts you need tomake in order to cut the cube into $27$ $1\times 1\times 1$ cubes?
p12. How many ways can you choose $3$ distinct numbers fromthe set $\{1,2,3,...,20\}$ to create a geometric sequence?
[b]p13.[/b] Find the sum of all multiples of $12$ that are less than $10^4$ and contain only $0$ and $4$ as digits.
[b]p14.[/b] What is the smallest positive integer that has a different number of digits in each base from $2$ to $5$?
[b]p15.[/b] Given $3$ real numbers $(a,b,c)$ such that $$\frac{a}{b +c}=\frac{b}{3a+3c}=\frac{c}{a+3b},$$ find all possible values of $\frac{a +b}{c}$.
[b]p16.[/b] Let S be the set of lattice points $(x, y, z)$ in $R^3$ satisfying $0 \le x, y, z \le 2$. How many distinct triangles exist with all three vertices in $S$?
[b]p17.[/b] Let $\oplus$ be an operator such that for any $2$ real numbers $a$ and $b$, $a \oplus b = 20ab -4a -4b +1$. Evaluate $$\frac{1}{10} \oplus \frac19 \oplus \frac18 \oplus \frac17 \oplus \frac16 \oplus \frac15 \oplus \frac14 \oplus \frac13 \oplus \frac12 \oplus 1.$$
[b]p18.[/b] A function $f :N \to N$ satisfies $f ( f (x)) = x$ and $f (2f (2x +16)) = f \left(\frac{1}{x+8} \right)$ for all positive integers $x$. Find $f (2018)$.
[b]p19.[/b] There exists an integer divisor $d$ of $240100490001$ such that $490000 < d < 491000$. Find $d$.
[b]p20.[/b] Let $a$ and $b$ be not necessarily distinct positive integers chosen independently and uniformly at random from the set $\{1,2, 3, ... ,511,512\}$. Let $x = \frac{a}{b}$ . Find the probability that $(-1)^x$ is a real number.
[b]p21[/b]. In $\vartriangle ABC$ we have $AB = 4$, $BC = 6$, and $\angle ABC = 135^o$. $\angle ABC$ is trisected by rays $B_1$ and $B_2$. Ray $B_1$ intersects side $C A$ at point $F$, and ray $B_2$ intersects side $C A$ at point $G$. What is the area of $\vartriangle BFG$?
[b]p22.[/b] A level number is a number which can be expressed as $x \cdot \lfloor x \rfloor \cdot \lceil x \rceil$ where $x$ is a real number. Find the number of positive integers less than or equal to $1000$ which are also level numbers.
[b]p23.[/b] Triangle $\vartriangle ABC$ has sidelengths $AB = 13$, $BC = 14$, $C A = 15$ and circumcenter $O$. Let $D$ be the intersection of $AO$ and $BC$. Compute $BD/DC$.
[b]p24.[/b] Let $f (x) = x^4 -3x^3 +2x^2 +5x -4$ be a quartic polynomial with roots $a,b,c,d$. Compute
$$\left(a+1 +\frac{1}{a} \right)\left(b+1 +\frac{1}{b} \right)\left(c+1 +\frac{1}{c} \right)\left(d+1 +\frac{1}{d} \right).$$
[b]p25.[/b] Triangle $\vartriangle ABC$ has centroid $G$ and circumcenter $O$. Let $D$ be the foot of the altitude from $A$ to $BC$. If $AD = 2018$, $BD =20$, and $CD = 18$, find the area of triangle $\vartriangle DOG$.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
2024 Junior Balkan Team Selection Tests - Romania, P3
In the exterior of the acute-angles triangle $ABC$ we construct the isosceles triangles $DAB$ and $EAC$ with bases $AB{}$ and $AC{}$ respectively such that $\angle DBC=\angle ECB=90^\circ.$ Let $M$ and $N$ be the reflections of $A$ with respect to $D$ and $E$ respectively. Prove that the line $MN$ passes through the orthocentre of the triangle $ABC.$
[i]Florin Bojor[/i]
2018 Bosnia And Herzegovina - Regional Olympiad, 3
Solve equation $x \lfloor{x}\rfloor+\{x\}=2018$, where $x$ is real number
2007 Switzerland - Final Round, 10
The plane is divided into equilateral triangles of side length $1$. Consider a equilateral triangle of side length $n$ whose sides lie on the grid lines. On every grid point on the edge and inside of this triangle lies a stone. In a move, a unit triangle is selected, which has exactly $2$ corners with is covered with a stone. The two stones are removed, and the third corner is turned a new stone was laid. For which $n$ is it possible that after finitely many moves only one stone left?
1979 Swedish Mathematical Competition, 4
$f(x)$ is continuous on the interval $[0, \pi]$ and satisfies
\[
\int\limits_0^\pi f(x)dx=0, \qquad \int\limits_0^\pi f(x)\cos x dx=0
\]
Show that $f(x)$ has at least two zeros in the interval $(0, \pi)$.
2021 IMC, 7
Let $D \subseteq \mathbb{C}$ be an open set containing the closed unit disk $\{z : |z| \leq 1\}$. Let $f : D \rightarrow \mathbb{C}$ be a holomorphic function, and let $p(z)$ be a monic polynomial. Prove that
$$
|f(0)| \leq \max_{|z|=1} |f(z)p(z)|
$$
2021 Science ON grade IX, 4
$\textbf{(a)}$ On the sides of triangle $ABC$ we consider the points $M\in \overline{BC}$, $N\in \overline{AC}$ and $P\in \overline{AB}$ such that the quadrilateral $MNAP$ with right angles $\angle MNA$ and $\angle MPA$ has an inscribed circle. Prove that $MNAP$ has to be a kite.
$\textbf{(b)}$ Is it possible for an isosceles trapezoid to be orthodiagonal and circumscribed too?
[i] (Călin Udrea) [/i]
2014 Iranian Geometry Olympiad (junior), P4
In a triangle ABC we have $\angle C = \angle A + 90^o$. The point $D$ on the continuation of $BC$ is given such that $AC = AD$. A point $E$ in the side of $BC$ in which $A$ doesn’t lie is chosen such that $\angle EBC = \angle A, \angle EDC = \frac{1}{2} \angle A$ . Prove that $\angle CED = \angle ABC$.
by Morteza Saghafian
2017 AIME Problems, 8
Two real numbers $a$ and $b$ are chosen independently and uniformly at random from the interval $(0, 75)$. Let $O$ and $P$ be two points on the plane with $OP = 200$. Let $Q$ and $R$ be on the same side of line $OP$ such that the degree measures of $\angle POQ$ and $\angle POR$ are $a$ and $b$ respectively, and $\angle OQP$ and $\angle ORP$ are both right angles. The probability that $QR \leq 100$ is equal to $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
2010 Today's Calculation Of Integral, 619
Consider a function $f(x)=\frac{\sin x}{9+16\sin ^ 2 x}\ \left(0\leq x\leq \frac{\pi}{2}\right).$ Let $a$ be the value of $x$ for which $f(x)$ is maximized.
Evaluate $\int_a^{\frac{\pi}{2}} f(x)\ dx.$
[i]2010 Saitama University entrance exam/Mathematics[/i]
Last Edited
2015 ASDAN Math Tournament, 3
For a math tournament, each person is assigned an ID which consists of two uppercase letters followed by two digits. All IDs have the property that either the letters are the same, the digits are the same, or both the letters are the same and the digits are the same. Compute the number of possible IDs that the tournament can generate.
2024 CMIMC Theoretical Computer Science, 1
Mellon Game Lab has come up with a concept for a new game: Square Finder. The premise is as follows. You are given an $n\times n$ grid of squares (for integer $n\geq 2$), each of which is either blank or has an arrow pointing up, down, left, or right. You are also given a $2\times 2$ grid of squares that appears somewhere in this grid, possibly rotated. For example, see if you can find the following $2\times 2$ grid inside the larger $4\times 4$ grid.
[asy]
size(2cm);
defaultpen(fontsize(16pt));
string b = "";
string u = "$\uparrow$";
string d = "$\downarrow$";
string l = "$\leftarrow$";
string r = "$\rightarrow$";
// input should be n x n
string[][] input =
{{b,u},{r,l}};
int n = input.length;
// draw table
for (int i=0; i<=n; ++i)
{
draw((i,0)--(i,n));
draw((0,i)--(n,i));
}
// fill table
for (int i=1; i<=n; ++i)
{
for (int j=1; j<=n; ++j)
{
label(input[i-1][j-1], (j-0.5,n-i+0.5));
}
}
[/asy]
[asy]
size(4cm);
defaultpen(fontsize(16pt));
string b = "";
string u = "$\uparrow$";
string d = "$\downarrow$";
string l = "$\leftarrow$";
string r = "$\rightarrow$";
// input should be n x n
string[][] input =
{{u,b,b,r},{b,r,u,d},{d,b,u,b},{u,r,b,l}};
int n = input.length;
// draw table
for (int i=0; i<=n; ++i)
{
draw((i,0)--(i,n));
draw((0,i)--(n,i));
}
// fill table
for (int i=1; i<=n; ++i)
{
for (int j=1; j<=n; ++j)
{
label(input[i-1][j-1], (j-0.5,n-i+0.5));
}
}
[/asy]
Did you spot it? It's in the bottom left, rotated by $90^\circ$ clockwise. To make the game as interesting as possible, Mellon Game Lab would like the grid to be as large as possible and for no $2\times 2$ grid to appear more than once in the big grid. The grid above doesn't work, as the following $2\times 2$ grid appears twice, once in the top left corner (rotated $90^\circ$ counterclockwise) and once directly below it (overlapping).
[asy]
size(2cm);
defaultpen(fontsize(16pt));
string b = "";
string u = "$\uparrow$";
string d = "$\downarrow$";
string l = "$\leftarrow$";
string r = "$\rightarrow$";
// input should be n x n
string[][] input =
{{b,r},{d,b}};
int n = input.length;
// draw table
for (int i=0; i<=n; ++i)
{
draw((i,0)--(i,n));
draw((0,i)--(n,i));
}
// fill table
for (int i=1; i<=n; ++i)
{
for (int j=1; j<=n; ++j)
{
label(input[i-1][j-1], (j-0.5,n-i+0.5));
}
}
[/asy]
Let's call a grid that avoids such repeats a [i]repeat-free grid[/i]. We are interested in finding out for which $n$ constructing an $n\times n$ repeat-free grid is possible. Here's what we know so far.
[list]
[*] Any $2\times 2$ grid is repeat-free, as there is only one subgrid to worry about, and there can't possibly be any repeats.
[*] If we can construct an $n\times n$ repeat-free grid, we can also construct a $k\times k$ repeat-free grid for any $k\leq n$ by just taking the top left $k\times k$ of the original one we found.
[*] By the previous observation, if it is impossible to construct such an $n\times n$ repeat-free grid, we cannot construct a $k\times k$ repeat-free grid for any $k\geq n$, as otherwise we could take the top left $n\times n$ to get one working for $n$.
[/list]
These three observations together tell us that either we can construct an $n\times n$ repeat-free grid for all $n\geq 2$, or there exists some upper limit $N\geq 2$ such that we can construct an $n\times n$ repeat-free grid for all $n\leq N$ but cannot construct one for any $n> N$. Your goal is to determine if such an $N$ exists, and if so, place bounds on its value.
More precisely, this problem consists of two parts: a lower bound and an upper bound. For the lower bound, to show that $N\geq n$ for some $n$, you need to construct an $n\times n$ repeat-free grid (you do not need to prove your construction works). For the upper bound, to show that $N$ is at most some value $n$, you must prove that it is impossible to construct an $(n+1)\times (n+1)$ repeat-free grid.
[i]Proposed by Connor Gordon and Eric Oh[/i]