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Find $x$ such that $\log_x 81=\log_2 16$. | open2math |
How many non-negative integers can be formed using all the digits of the number 123456789 exactly once, such that the resulting number is divisible by 27? | open2math |
The product of two positive integers is 24. The positive difference of these two integers is 2. What is the sum of the two integers? | open2math |
The red parabola shown is the graph of the equation $x = ay^2 + by + c$. Find $a+b+c$.
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pen axispen=black+1.3bp;
real vectorarrowsize=0.2cm;
real tickdown=-0.5;
real tickdownlength=-0.15inch;
real tickdownbase=0.3;
re... | open2math |
What is the smallest positive integer that is guaranteed to be a factor of the sum of any three consecutive positive integers? | open2math |
Let $\mathbf{a},$ $\mathbf{b},$ and $\mathbf{c}$ be three points in 3-dimensional space. Determine the largest possible value of the sum of the squares of the areas of the three triangles formed by connecting any two of the three points with the origin. | open2math |
$\triangle ABC$ is inscribed in a semicircle of radius $r$ so that its base $AB$ coincides with diameter $AB$. Point $C$ does not coincide with either $A$ or $B$. Let $s=AC+BC$. Then, for all permissible positions of $C$:
$\textbf{(A)}\ s^2\le8r^2\qquad \textbf{(B)}\ s^2=8r^2 \qquad \textbf{(C)}\ s^2 \ge 8r^2 \qquad\\ ... | open2math |
The function $f(x)$ satisfies
\[f(xy) = \frac{f(x)}{y}\]for all positive real numbers $x$ and $y.$ If $f(30) = 20,$ find $f(40).$ | open2math |
Find the sum of all possible positive integer values of $c$ such that the quadratic equation $3x^2 + 4x + c = 0$ has rational roots. | open2math |
Solve the equation $\sqrt{2x+1}=\sqrt{x+2}$, where $x$ is a real number. | open2math |
The sum of the squares of two consecutive integers is 85. Determine the absolute value of their difference. | open2math |
In a rectangular coordinate system, the vertices of a triangle are located at (0, 0), (a, 0), and (0, b), where a and b are positive real numbers. The area of the triangle is equal to 16 square units. Find the value of a + b. | open2math |
Let $a$ and $b$ be two positive integers such that their sum and product are given. Determine the value of $a^2 + b^2$ if the product of $a$ and $b$ is 16900 and their sum is 260. | open2math |
Compute the largest integer $n$ such that $2401^n$ divides $2401!$. | open2math |
How many different positive integers can be represented as the sum of two distinct members of the set $\{1, 2, 3, 4, 5, 6\}$? | open2math |
A square pyramid has a base with a side length of $s$ units and four triangular faces that meet at the apex. The sum of the $x$-coordinates of the vertices of the base is $s+1$. Find the $x$-coordinate of the centroid of the pyramid. | open2math |
In a regular dodecagon, a diagonal is drawn, dividing the dodecagon into two regions. What is the smallest number of diagonals that can be drawn in the dodecagon such that each of the resulting regions has an equal number of vertices? | open2math |
Let $a$ and $b$ be positive real numbers such that $a + 2b = 1.$ Find the minimum value of
\[\frac{1}{a} + \frac{1}{b}.\] | open2math |
Let $q(x)$ be a polynomial with integer coefficients such that $q(50) = 50.$ Find the maximum number of integer roots $r$ of the equation $q(r) = r^2.$ | open2math |
Let $a, b, c,$ and $d$ be positive integers such that $\gcd(a, b)=24$, $\gcd(b, c)=36$, $\gcd(c, d)=54$, and $70<\gcd(d, a)<100$. Which of the following must be a divisor of $a$?
$\textbf{(A)} \text{ 5} \qquad \textbf{(B)} \text{ 7} \qquad \textbf{(C)} \text{ 11} \qquad \textbf{(D)} \text{ 13} \qquad \textbf{(E)} \text... | open2math |
In the base-3 system, let $S$ be the set of all positive integers whose cube, when written in base 3, contains exactly three digits.
Determine the smallest integer $n$ such that all integers less than $n$ are in $S$. | open2math |
Problem:
Let $S$ be the set of all complex numbers of the form $r+si$, where $r$ and $s$ are integers, that can be expressed as a finite sum of terms of the form $a_n(2+i)^n$, where $a_n$ is a non-negative integer less than or equal to $2^n$. Find the maximum value of $|z|$ for all $z \in S$ that can be expressed as a... | open2math |
Given three non-overlapping cubes of unit length, with centers at $(3, 4, 1)$, $(1, 2, 3)$, and $(5, 6, 7)$, find the equation of the plane passing through $(1, 2, 3)$ such that the volume of the parts of the cubes on one side of the plane is equal to the volume of the parts of the cubes on the other side. | open2math |
Determine the value of $x$ such that $x\ast x = x\ast(x+1)$. | open2math |
Three positive numbers are in an arithmetic progression. If the sum of the squares of the two smaller numbers is 65, and the product of the two larger numbers is 105, find the largest number. | open2math |
Determine the monic polynomial in $x$ with real coefficients that has the following roots: $2$, $1-i$, and $3+\sqrt{2}$. | open2math |
Let's assume that the angle measures are $4x$ and $5x$. Since the two angles are complementary, $4x+5x=90$.
Therefore, $9x=90$, and the measures are $\ds\frac{4}{9}\cdot 90=40^\circ$ and $\ds\frac{5}{9}\cdot 90=50^\circ$. Therefore, the ratio of the measures is $\ds\frac{40}{50}=\frac{4}{5}$.
The smaller angle's measur... | open2math |
Find the sum of all numbers $x$ satisfying $x^2+10/x = 20$. | open2math |
A circle is inscribed in a right triangle with legs 8 and 15. The sum of the areas of the two smaller triangles formed by the circle and the legs of the triangle is equal to half the area of the triangle. Find the radius of the inscribed circle. | open2math |
How many four-digit whole numbers have at least two digits that are perfect squares? | open2math |
Find the number of integer values of x that satisfy the condition:
$$2x < 5$$
$$3^x < 10$$ | open2math |
Find the smallest positive integer $n$ such that the sum of its prime factors is equal to the number of its divisors. | open2math |
Two reals $a$ and $b$ are such that $a+b=7$ and $a^3+b^3=91$. Compute $ab$. | open2math |
Find the value of $a$ for which the equation $ax^2+3x-2=0$ has a solution $x=1$ and the sum of the roots is equal to $-\frac{7}{2}$. | open2math |
Given that $n$ is an integer and $0 < 4n <30$, what is the sum of all possible integer values of $n$? | open2math |
Given that $a$ and $b$ are positive integers such that $\gcd(a,b) = 18$, find the smallest possible value of $\gcd(a^2,b^3)$. | open2math |
A positive number is called $n$-primable if it is divisible by $n$ and each of its digits is a one-digit prime number. How many 3-primable positive integers are there that are less than 1000? | open2math |
Determine the number of distinct subsets of the set {A, L, B, M, O, R, T} such that each subset contains exactly three elements. | open2math |
What is the smallest positive integer $n$ such that $2^n-1$ is divisible by $5^3$? | open2math |
The sequence $a, ar, ar^2, \ldots$ is a geometric sequence, and the sum of all the terms is $18$. The sum of the terms at even places is $3$ times the sum of the terms at odd places. Find the common ratio $r$. | open2math |
The probability that both $a_1$ divides $a_2$ and $a_2$ divides $a_3$ can be written as $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers. Determine the value of $b$. | open2math |
Find the smallest positive integer $k$ such that $
z^{10} + z^9 + z^6+z^5+z^4+z+1
$ divides $z^k-1$. | open2math |
Find the common ratio of the infinite geometric series:
$$\frac{3}{4}+\frac{3}{4}x+\frac{3}{4}x^2+\dots$$
where $x$ is a real number such that $|x|<1$.
Given that the sum of the series is equal to $2$, find the value of $x$. | open2math |
At the 2007 Math Olympics, Team Canada won $17$ out of a possible $100$ medals. Which one of the following is closest to the fraction of medals that they won? $$
\frac{1}{4} \qquad \frac{1}{5} \qquad \frac{1}{6} \qquad \frac{1}{7} \qquad \frac{1}{8}
$$ | open2math |
Find the number of 5-digit numbers that have exactly three identical digits and start with a digit other than the repeated one. | open2math |
Find the sum of the elements in the 10th row of Pascal's triangle that are not divisible by 3. | open2math |
Count the number of positive integers $m$ such that there exists an integer $n$, for which $m$ is the smallest positive integer satisfying $2^n + m = 3^k$, where $k$ is a positive integer. | open2math |
A sequence of integers $b_1, b_2, b_3, \ldots$ is chosen so that $b_n = b_{n - 1} + b_{n - 2}$ for each $n \ge 3.$ The sum of the first $m$ terms of this sequence is $m$ for every positive integer $m.$ Determine the sum of the squares of the first $m$ terms of this sequence for every positive integer $m.$ | open2math |
A palindrome is a number that reads the same forwards and backwards, such as 3003. How many positive four-digit integers are palindromes? | open2math |
Find the smallest four-digit integer that is divisible by the sum of its distinct, non-zero digits. | open2math |
How many 5-digit numbers can be formed using the digits 0-9, such that the sum of the digits is even and the number contains at least one 3 or 5? | open2math |
Let $a,$ $b,$ and $c$ be three positive integers whose sum is 120. If $a,$ $b,$ and $c$ are consecutive terms of an arithmetic progression, then find the maximum value of the sum of their squares $a^2+b^2+c^2.$ | open2math |
Let A denote a set of positive integers such that any element in A is divisible by at least one element less than or equal to its square root. Determine the smallest possible value of the largest element in A. | open2math |
Determine the number of positive integers $b$ less than $20$ such that the congruence $bx \equiv 2 \pmod{20}$ has a solution in $x$. | open2math |
Given that the point $(4,7)$ is on the graph of $y=f(x)$, there is one point that must be on the graph of $2y=3f(4x)+5$. What is the sum of the coordinates of that point? | open2math |
Let $g : \mathbb{R} \to \mathbb{R}$ be a function such that $g(0) = 0$ and
\[g(x^2 + y^2) = (x + y) (g(x) - g(y))\]for all real numbers $x$ and $y.$
Let $m$ be the number of possible values of $g(1),$ and let $t$ be the sum of all possible values of $g(1).$ Find $m \times t.$ | open2math |
A math book has a large number of pages, each having a unique number from 1 to n. However, one page is missing. Given that the sum of all the page numbers is 141, what is the number of the missing page? | open2math |
Let $a_1,a_2,\ldots,a_n$ be real numbers greater than 1 such that $a_1\cdot a_2\cdots a_n>2^n$. Find the minimum value of
\[\frac{a_1+1}{2}+\frac{a_2+1}{2}+\cdots+\frac{a_n+1}{2}.\] | open2math |
There are values $C$ and $D$ such that
\[\frac{Dx+15}{x^2+8x+12}=\frac{2}{x+2}+\frac{C}{x+6}.\]
Find $C+D$. | open2math |
Find the sum of all positive divisors of $70$ that are also divisors of $21$. | open2math |
Find the number of permutations of the letters in the word "LISTEN" such that no two vowels are adjacent to each other. | open2math |
The product of three consecutive positive integers is equal to the sum of the cubes of the same integers. What is the largest integer? | open2math |
The equation $x^3-4x^2+x+6=0$ has two roots whose sum is $-1$.
Find the product of the three roots of this equation. | open2math |
Let $f(x)$ be a real-valued function such that $f(xy)=f(x)+f(y)$ and $f(x^2)+f(x)=0$ for all $x\neq 0$. Determine the value of $f(123)$. | open2math |
Find the sum of the first 10 positive integers that are one more than a multiple of 5. | open2math |
Given that $\gcd(2a,3b) = 56$ and $\gcd(3a,4c) = 168$, where $a, b, c$ are positive integers, determine the smallest possible value of $\gcd(2b,3c)$ | open2math |
Find the number of distinct positive integers that divide 360 and are multiples of 3. | open2math |
There are three consecutive numbers $n$, $n+1$, and $n+2$. The square of the largest number is equal to the sum of the squares of the other two numbers. Determine the middle number. | open2math |
How many positive integers less than 1000 have the property that the sum of their digits is 9 and the product of their digits is divisible by 7? | open2math |
The domain of the function $f(x) = \arccos(\log_{m}(nx))$ is a closed interval of length $\frac{1}{2024}$, where $m$ and $n$ are positive integers and $m>1$. Find the the smallest possible value of $m+n.$ | open2math |
In the equation $5x^2-kx+1=0$ determine $k$ such that the difference of the roots be equal to unity.
New Problem:
The roots $x_1$ and $x_2$ of the equation $x^2-3ax+a^2=0$ are such that $x_1^2+x_2^2=1.75$. Determine $a$. | open2math |
Find the smallest positive integer that is a multiple of 7 and has exactly three digits that are the same when its digits are reversed. | open2math |
Given that $\gcd(3m+1,13m-4)=1$, find the largest possible value of $\gcd(7m-11,2m+27)$. | open2math |
A positive integer $k$ can be represented as the product of three distinct factors $a$, $b$, and $a^2+b^2$, where $a$ and $b$ are each one-digit numbers. What is the largest possible value of $k$ that is a three-digit number? | open2math |
Find the maximum value of $\sqrt{x} - x\sqrt{x},$ over all real numbers $x > 0.$ | open2math |
A circle has three chords of the following lengths: $11$, $15$, and $x^3$. What are all of the positive integer values of $x$ such that the circle exists and the three chords can form a triangle inside the circle? Separate your answers using commas and express them in increasing order. | open2math |
Compute the value of $p$ such that the sum of the squares of the roots of the equation $x^3 - 12x^2 + px - 45 = 0$ is equal to 116. | open2math |
What is the least positive integer that can be multiplied by 27 to yield a number that ends with the digit 9? | open2math |
Let $p$ be a prime number such that $p^2-p+1$ is divisible by a perfect square greater than 1. Prove that $p^3+1$ is divisible by the same perfect square. | open2math |
A fair six-sided die is rolled twice, and the numbers shown on the top face of the die are recorded each time. What is the probability that the sum of the two numbers is a prime number? | open2math |
Consider a sequence $b_n$ defined as follows: $b_1=2$ and for each positive integer $n>1$, $b_n=2^nb_{n-1}+n^2$. Determine the smallest positive integer $n$ such that $b_n$ is divisible by $3$. | open2math |
Find the number of four-digit numbers such that the sum of the first and last digits is equal to the sum of the second and third digits, and the sum of the first and second digits is equal to the sum of the third and last digits. | open2math |
Let $n$ be the number of positive divisors of $x$ and $m$ be the number of positive divisors of $y$. If $\lceil{\frac{n}{m}}\rceil=2$, how many possible ordered pairs of integers $(x,y)$ are there? | open2math |
How many six-digit positive integers have at least two consecutive digits that are the same? | open2math |
Find the minimum value of
\[x^2 + xy + y^2\]over all real numbers $x$ and $y.$ | open2math |
Let $x$ be a three-digit number. Define a function $f(x)$ such that $f(x)$ is the integer obtained by rotating the digits of $x$ one position to the left (e.g., $f(123)=231$). Find the least positive integer $n$ such that for any three-digit number $x$, if $x+f(x)+f(f(x))$ is divisible by $n$, then so is $f(f(f(x)))$. | open2math |
Let $x$ be a real number such that the expression $x + \frac{x}{\sqrt{1+x^2}} + \frac{x}{\sqrt{1+\frac{x^2}{1+x^2}}} + ...$ is finite. Find all possible values of x. | open2math |
The parabola $y=ax^2+bx+c$ has vertex $(2, -1)$. Given that $a$ and $c$ are integers and that the parabola passes through the point $(0, 5)$, find the value of $a$. | open2math |
In triangle $ABC$, the bisectors of $\angle A$ and $\angle C$ intersect at $I$, and $AB = 24$. Extend $BI$ to intersect the circumcircle of $ABC$ at $D$. The length of arc $BCD$ is $m\sqrt{n}$, where $m$ and $n$ are positive integers and $n$ is not divisible by the square of any prime. Find $m+n$. | open2math |
Let $x$ and $y$ be the side lengths of two rectangles with integral dimensions. Given that the ratio of their areas is $\frac{441}{400}$ and that the ratio of their perimeters is $\frac{21}{20}$, find the value of $xy$. | open2math |
A circle with a radius of 5 units is centered at the point (3,4). Find the length of the segment that connects the point (2,3) with the center of the circle. | open2math |
Determine the number of distinct, natural-number factors of $n$ that are also multiples of $k$, given that $n=2^3 \cdot 3^2 \cdot 5$ and $k=2^2 \cdot 3$. | open2math |
Let $a,$ $b,$ and $c$ be positive real numbers such that $a + b + c = 6.$ Find the minimum value of
\[\frac{1}{a} + \frac{4}{b} + \frac{9}{c}.\] | open2math |
Let $\mathcal{S}$ be the set of real numbers that can be represented as repeating decimals of the form $0.\overline{abc}$ where $a, b, c$ are distinct digits. Find the sum of the elements of $\mathcal{S}.$
| open2math |
Determine the value of $k$ such that the sum of the digits of the number $4k27$ is a multiple of $3$.
Let $S$ denote the sum of the digits of the number $4k27$, then
$S=4+k+2+7=k+13$. In order for $S$ to be a multiple of $3$, the digit $k$ should be $2$, $5$ or $8$. | open2math |
In the $xyz$-coordinate system, there are eight points with coordinates $(0,0,0), (3,0,0), (0,3,0), (0,0,3), (3,3,0), (3,0,3), (0,3,3)$, and $(3,3,3)$. Find the point that can be removed so that the remaining points have a greater average distance between them. | open2math |
Find the largest three-digit positive integer that is divisible by 53. | open2math |
Find $w$, such that $5^65^w=25$. | open2math |
Find all real numbers $a$ such that the roots of the polynomial
$$x^3 - 6x^2 + 21x + a$$form an arithmetic progression and are not all real. | open2math |
Let $d$ be a positive number such that when $109$ is divided by $d$, the remainder is $4.$ Compute the sum of all possible two-digit values of $d$. | open2math |
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