Problems

For each integer $n\geq 2$ , let $S_n$ be the sum of all products $jk$ , where $j$ and $k$ are integers and $1\leq j<k\leq n$ . What is the sum of the 10 least values of $n$ such that $S_n$ is divisible by $3$ ?

Assuming $a\neq3$ , $b\neq4$ , and $c\neq5$ , what is the value in simplest form of the following expression? $$\frac{a-3}{5-c} \cdot \frac{b-4}{3-a} \cdot \frac{c-5}{4-b}$$

What is the sum of all real numbers $x$ for which $|x^2-12x+34|=2?$

Real numbers $x$ and $y$ satisfy $x + y = 4$ and $x \cdot y = -2$ . What is the value of $$x + \frac{x^3}{y^2} + \frac{y^3}{x^2} + y?$$

There exists a unique strictly increasing sequence of nonnegative integers $a_1 < a_2 < … < a_k$ such that $$\frac{2^{289}+1}{2^{17}+1} = 2^{a_1} + 2^{a_2} + … + 2^{a_k}.$$ What is $k?$

How many ordered pairs of integers $(x,y)$ satisfy the equation $$x^{2020} + y^2 = 2y?$$

What is the remainder when $2^{202} +202$ is divided by $2^{101}+2^{51}+1$ ?

Ana and Bonita are born on the same date in different years, $n$ years apart. Last year Ana was $5$ times as old as Bonita. This year Ana's age is the square of Bonita's age. What is $n?$

What is the greatest three-digit positive integer $n$ for which the sum of the first $n$ positive integers is $\underline{\text{not}}$ a divisor of the product of the first $n$ positive integers?

Let $p$ , $q$ , and $r$ be the distinct roots of the polynomial $x^3 - 22x^2 + 80x - 67$ . It is given that there exist real numbers $A$ , $B$ , and $C$ such that $$\dfrac{1}{s^3 - 22s^2 + 80s - 67} = \dfrac{A}{s-p} + \dfrac{B}{s-q} + \frac{C}{s-r}$$ for all $s\not\in\{p,q,r\}$ . What is $\tfrac1A+\tfrac1B+\tfrac1C$ ?

There is a real $n$ such that $(n+1)! + (n+2)! = n! \cdot 440$ . What is the sum of the digits of $n$ ?

Right triangles $T_1$ and $T_2$ have areas 1 and 2, respectively. A side of $T_1$ is congruent to a side of $T_2$ , and a different side of $T_1$ is congruent to a different side of $T_2$ . What is the square of the product of the other (third) sides of $T_1$ and $T_2$ ?

Suppose that real number $x$ satisfies $$\sqrt{49-x^2}-\sqrt{25-x^2}=3.$$ What is the value of $\sqrt{49-x^2}+\sqrt{25-x^2}$ ?

For a positive integer $n$ and nonzero digits $a$ , $b$ , and $c$ , let $A_n$ be the $n$ -digit integer each of whose digits is equal to $a$ ; let $B_n$ be the $n$ -digit integer each of whose digits is equal to $b$ , and let $C_n$ be the $2n$ -digit (not $n$ -digit) integer each of whose digits is equal to $c$ . What …

A three-dimensional rectangular box with dimensions $X$ , $Y$ , and $Z$ has faces whose surface areas are $24, 24, 48, 48, 72,$ and $72$ square units. What is $X+Y+Z$ ?

Let $a_1,a_2,\dots,a_{2018}$ be a strictly increasing sequence of positive integers such that $$a_1+a_2+\cdots+a_{2018}=2018^{2018}.$$ What is the remainder when $a_1^3+a_2^3+\cdots+a_{2018}^3$ is divided by $6$ ?

The sum of two nonzero real numbers is $4$ times their product. What is the sum of the reciprocals of the two numbers?

A quadrilateral is inscribed in a circle of radius $200\sqrt{2}$ . Three of the sides of this quadrilateral have length $200$ . What is the length of the fourth side?

All the numbers $2, 3, 4, 5, 6, 7$ are assigned to the six faces of a cube, one number to each face. For each of the eight vertices of the cube, a product of three numbers is computed, where the three numbers are the numbers assigned to the three faces that include that vertex. What is the greatest possible value of …

In how many ways can $345$ be written as the sum of an increasing sequence of two or more consecutive positive integers?

What is the area of the region enclosed by the graph of the equation $x^2+y^2=|x|+|y|?$

Points $(\sqrt{\pi}, a)$ and $(\sqrt{\pi}, b)$ are distinct points on the graph of $y^2+x^4=2x^2y+1$ . What is $|a-b|$ ?

If $y+4 = (x-2)^2, x+4 = (y-2)^2$ , and $x \neq y$ , what is the value of $x^2+y^2$ ?

A rectangle with positive integer side lengths in $\text{cm}$ has area $A$ $\text{cm}^2$ and perimeter $P$ $\text{cm}$ . Which of the following numbers cannot equal $A+P$ ?

The zeroes of the function $f(x)=x^2-ax+2a$ are integers. What is the sum of the possible values of $a$ ?