Found problems: 85335
2018 BMT Spring, Tie 2
An integer $a$ is a quadratic nonresidue modulo a prime $p$ if there does not exist $x \in Z$ such that $x^2 \equiv a$ (mod $p$). How many ordered pairs $(a, b)$ modulo $29$ exist such that
$$a + b\equiv 1 \,\,\, (mod \,\,\, 29)$$
where both $a$ and $b$ are quadratic nonresidues modulo $29$?
2019 Miklós Schweitzer, 5
Let $S \subset \mathbb{R}^d$ be a convex compact body with nonempty interior. Show that there is an $\alpha > 0$ such that if $S = \cap_{i \in I} H_i$, where $I$ is an index set and $(H_i)_{i \in I}$ are halfspaces, then for any $P \in \mathbb{R}^d$, there is an $i \in I$ for which $\mathrm{dist}(P, H_i) \ge \alpha \, \mathrm{dist}(P, S)$.
2015 Caucasus Mathematical Olympiad, 5
Are there natural $a, b >1000$ , such that for any $c$ that is a perfect square, the three numbers $a, b$ and $c$ are not the lengths of the sides of a triangle?
2023 Israel National Olympiad, P7
Ana and Banana are playing a game. Initially, Ana secretly picks a number $1\leq A\leq 10^6$. In each subsequent turn of the game, Banana may pick a positive integer $B$, and Ana will reveal to him the most common digit in the product $A\cdot B$ (written in decimal notation). In the case when at least two digits are tied for being the most common, Ana will reveal all of them to Banana. For example, if $A\cdot B=2022$, Ana will tell Banana that the digit $2$ is the most common, while if $A\cdot B=5783783$, Ana will reveal that $3, 7$ and $8$ are the most common. Banana's goal is to determine with certainty the number $A$ after some number of turns. Does he have a winning strategy?
2010 Harvard-MIT Mathematics Tournament, 8
How many polynomials of degree exactly $5$ with real coefficients send the set $\{1, 2, 3, 4, 5, 6\}$ to a permutation of itself?
2004 National High School Mathematics League, 1
If the equation $x^2+4x\cos\theta+\cot\theta=0$ has a repeated root, where $\theta$ is an acute angle, then the radian of $\theta$ is
$\text{(A)}\frac{\pi}{6}\qquad\text{(B)}\frac{\pi}{12}\text{ or }\frac{5\pi}{12}\qquad\text{(C)}\frac{\pi}{6}\text{ or }\frac{5\pi}{12}\qquad\text{(D)}\frac{\pi}{12}$
1990 AIME Problems, 13
Let $T = \{9^k : k \ \text{is an integer}, 0 \le k \le 4000\}$. Given that $9^{4000}$ has 3817 digits and that its first (leftmost) digit is 9, how many elements of $T$ have 9 as their leftmost digit?
2021 Taiwan APMO Preliminary First Round, 6
Find all positive integers $A,B$ satisfying the following properties:
(i) $A$ and $B$ has same digit in decimal.
(ii) $2\cdot A\cdot B=\overline{AB}$ (Here $\cdot$ denotes multiplication, $\overline{AB}$ denotes we write $A$ and $B$ in turn. For example, if $A=12,B=34$, then $\overline{AB}=1234$)
2007 Brazil National Olympiad, 3
Consider $ n$ points in a plane which are vertices of a convex polygon. Prove that the set of the lengths of the sides and the diagonals of the polygon has at least $ \lfloor n/2\rfloor$ elements.
2008 iTest Tournament of Champions, 3
Simon and Garfunkle play in a round-robin golf tournament. Each player is awarded one point for a victory, a half point for a tie, and no points for a loss. Simon beat Garfunkle in the first game by a record margin as Garfunkle sent a shot over the bridge and into troubled waters on the final hole. Garfunkle went on to score $8$ total victories, but no ties at all. Meanwhile, Simon wound up with exactly $8$ points, including the point for a victory over Garfunkle. Amazingly, every other player at the tournament scored exactly $n$. Find the sum of all possible values of $n$.
2007 Putnam, 6
For each positive integer $ n,$ let $ f(n)$ be the number of ways to make $ n!$ cents using an unordered collection of coins, each worth $ k!$ cents for some $ k,\ 1\le k\le n.$ Prove that for some constant $ C,$ independent of $ n,$
\[ n^{n^2/2\minus{}Cn}e^{\minus{}n^2/4}\le f(n)\le n^{n^2/2\plus{}Cn}e^{\minus{}n^2/4}.\]
2005 Purple Comet Problems, 9
Let $T$ be a $30-60-90$ triangle with hypotenuse of length $20$. Three circles, each externally tangent to the other two, have centers at the three vertices of $T$. The area of the union of the circles intersected with $T$ is $(m + n \sqrt{3}) \pi$ for rational numbers $m$ and $n$. Find $m + n$.
2014 Bosnia And Herzegovina - Regional Olympiad, 2
Let $a$, $b$ and $c$ be positive real numbers such that $ab+bc+ca=1$. Prove the inequality: $$\frac{1}{a}+\frac{1}{b}+\frac{1}{c} \geq 3(a+b+c)$$
2016 PUMaC Number Theory B, 4
For a positive integer $n$, let $P(n)$ be the product of the factors of $n$ (including $n$ itself).
A positive integer $n$ is called [i]deplorable [/i] if $n > 1$ and $\log_n P(n)$ is an odd integer.
How many factors of $2016$ are [i]deplorable[/i]?
2020 Brazil Team Selection Test, 7
Each of the $n^2$ cells of an $n \times n$ grid is colored either black or white. Let $a_i$ denote the number of white cells in the $i$-th row, and let $b_i$ denote the number of black cells in the $i$-th column. Determine the maximum value of $\sum_{i=1}^n a_ib_i$ over all coloring schemes of the grid.
[i]Proposed by Alex Zhai[/i]
1959 AMC 12/AHSME, 24
A chemist has $m$ ounces of salt that is $m\%$ salt. How many ounces of salt must he add to make a solution that is $2m\%$ salt?
$ \textbf{(A)}\ \frac{m}{100+m} \qquad\textbf{(B)}\ \frac{2m}{100-2m}\qquad\textbf{(C)}\ \frac{m^2}{100-2m}\qquad\textbf{(D)}\ \frac{m^2}{100+2m}\qquad\textbf{(E)}\ \frac{2m}{100+2m} $
2021 LMT Fall, 13
Find the sum of $$\frac{\sigma(n) \cdot d(n)}{ \phi (n)}$$ over all positive $n$ that divide $ 60$.
Note: The function $d(i)$ outputs the number of divisors of $i$, $\sigma (i)$ outputs the sum of the factors of $i$, and $\phi (i)$ outputs the number of positive integers less than or equal to $i$ that are relatively prime to $i$.
1978 Putnam, A5
Let $0 < x_i < \pi$ for $i=1,2,\ldots, n$ and set
$$x= \frac{ x_1 +x_2 + \ldots+ x_n }{n}.$$
Prove that
$$ \prod_{i=1}^{n} \frac{ \sin x_i }{x_i } \leq \left( \frac{ \sin x}{x}\right)^{n}.$$
1983 Iran MO (2nd round), 3
Find a matrix $A_{(2 \times 2)}$ for which
\[ \begin{bmatrix}2 &1 \\ 3 & 2\end{bmatrix} A \begin{bmatrix}3 & 2 \\ 4 & 3\end{bmatrix} = \begin{bmatrix}1 & 2 \\ 2 & 1\end{bmatrix}.\]
2017 BAMO, A
Consider the $4 \times 4$ “multiplication table” below. The numbers in the first column multiplied by the numbers in the first row give the remaining numbers in the table. For example, the $3$ in the first column times the $4$ in the first row give the $12 (= 3 \cdot 4)$ in the cell that is in the 3rd row and 4th column.
[asy]
size(3cm);
for (int x=0; x<=4; ++x)
draw((x, 0) -- (x, 4), linewidth(.5pt));
for (int y=0; y<=4; ++y)
draw((0, y) -- (4, y), linewidth(.5pt));
draw((0,0)--(4,0)--(4,4)--(0,4)--cycle);
void foo(int x, int y, string n)
{
label(n, (x+0.5, y+0.5));
}
foo(0, 3, "1");
foo(1, 3, "2");
foo(2, 3, "3");
foo(3, 3, "4");
foo(0, 2, "2");
foo(1, 2, "4");
foo(2, 2, "6");
foo(3, 2, "8");
foo(0, 1, "3");
foo(1, 1, "6");
foo(2, 1, "9");
foo(3, 1, "12");
foo(0, 0, "4");
foo(1, 0, "8");
foo(2, 0, "12");
foo(3, 0, "16");
[/asy]
We create a path from the upper-left square to the lower-right square by always moving one cell either to the right or down. For example, here is one such possible path, with all the numbers along the path circled:
[asy]
import graph;
size(3cm);
for (int x=0; x<=4; ++x)
draw((x, 0) -- (x, 4), linewidth(.5pt));
for (int y=0; y<=4; ++y)
draw((0, y) -- (4, y), linewidth(.5pt));
draw((0,0)--(4,0)--(4,4)--(0,4)--cycle);
void foo(int x, int y, string n)
{
label(n, (x+0.5, y+0.5));
}
draw(Circle((0.5,3.5),0.5));
draw(Circle((1.5,3.5),0.5));
draw(Circle((2.5,3.5),0.5));
draw(Circle((2.5,2.5),0.5));
draw(Circle((3.5,2.5),0.5));
draw(Circle((3.5,1.5),0.5));
draw(Circle((3.5,0.5),0.5));
foo(0, 3, "1");
foo(1, 3, "2");
foo(2, 3, "3");
foo(3, 3, "4");
foo(0, 2, "2");
foo(1, 2, "4");
foo(2, 2, "6");
foo(3, 2, "8");
foo(0, 1, "3");
foo(1, 1, "6");
foo(2, 1, "9");
foo(3, 1, "12");
foo(0, 0, "4");
foo(1, 0, "8");
foo(2, 0, "12");
foo(3, 0, "16");
[/asy]
If we add up the circled numbers in the example above (including the start and end squares), we get $48$. Considering all such possible paths:
(a) What is the smallest sum we can possibly get when we add up the numbers along such a path? Prove your answer is correct.
(b) What is the largest sum we can possibly get when we add up the numbers along such a path? Prove your answer is correct.
2023 AMC 10, 1
Mrs. Jones is pouring orange juice for her 4 kids into 4 identical glasses. She fills the first 3 full, but only has enough orange juice to fill one third of the last glass. What fraction of a glass of orange juice does she need to pour from the 3 full glasses into the last glass so that all glasses have an equal amount of orange juice?
$\textbf{(A) }\frac{1}{12}\qquad\textbf{(B) }\frac{1}{4}\qquad\textbf{(C) }\frac{1}{6}\qquad\textbf{(D) }\frac{1}{8}\qquad\textbf{(E) }\frac{2}{9}$
1998 Romania Team Selection Test, 1
We are given an isosceles triangle $ABC$ such that $BC=a$ and $AB=BC=b$. The variable points $M\in (AC)$ and $N\in (AB)$ satisfy $a^2\cdot AM \cdot AN = b^2 \cdot BN \cdot CM$. The straight lines $BM$ and $CN$ intersect in $P$. Find the locus of the variable point $P$.
[i]Dan Branzei[/i]
2023 BAMO, D/2
Given a positive integer $N$ (written in base $10$), define its [i]integer substrings[/i] to be integers that are equal to strings of one or more consecutive digits from $N$, including $N$ itself. For example, the integer substrings of $3208$ are $3$, $2$, $0$, $8$, $32$, $20$, $320$, $208$, $3208$. (The substring $08$ is omitted from this list because it is the same integer as the substring $8$, which is already listed.)
What is the greatest integer $N$ such that no integer substring of $N$ is a multiple of $9$? (Note: $0$ is a multiple of $9$.)
2023 ELMO Shortlist, A4
Let \(f:\mathbb R\to\mathbb R\) be a function such that for all real numbers \(x\neq1\), \[f(x-f(x))+f(x)=\frac{x^2-x+1}{x-1}.\] Find all possible values of \(f(2023)\).
[i]Proposed by Linus Tang[/i]
2022 Moldova EGMO TST, 12
On a board there are $2022$ numbers: $1,\frac{1}{2},\frac{1}{3},\frac{1}{4},\dots,\frac{1}{2022}$. During a $move$ two numbers are chosen, $a$ and $b$, they are erased and $a+b+ab$ is written in their place. The moves take place until only one number is left on the board. What are the possible values of this number?