What is the value of $$ \frac{2 a^{-1}+\frac{a^{-1}}{2}}{a} $$ when $a=\frac{1}{2}$ ?
If $n \circlearrowleft m=n^{3} m^{2}$, what is $\frac{294}{49^{2}}$ ?
Let $x=-2016$. What is the value of $|||x|-x|-|x||-x$ ?
Zoey read 15 books, one at a time. The first book took her 1 day to read, the second book took her 2 days to read, the third book took her 3 days to read, and so on, with each book taking her 1 more day to read than the previous book. Zoey finished the first book on a Monday and the second on a Wednesday. On what day of the week did she finish her 15th book?
The mean age of Amanda's 4 cousins is 8 , and their median age is 5 . What is the sum of the ages of Amanda's youngest and oldest cousins?
Laura added two three-digit positive integers. All six digits in these numbers are different. Laura's sum is a three-digit number $S$. What is the smallest possible value for the sum of the digits of $S$ ?
The ratio of the measures of two acute angles is $5: 4$, and the complement of one of these two angles is twice as large as the complement of the other. What is the sum of the degree measures of the two angles?
What is the tens digit of $2015^{2016}-2017$ ?
All three vertices of $\triangle A B C$ lie on the parabola defined by $y=x^{2}$, with $A$ at the origin and $\overline{B C}$ parallel to the $x$-axis. The area of the triangle is 64 . What is the length $B C$ ?
A thin piece of wood of uniform density in the shape of an equilateral triangle with side length 3 inches weighs 12 ounces. A second piece of the same type of wood, with the same thickness, also in the shape of an equilateral triangle, has side length 5 inches. Which of the following is closest to the weight, in ounces, of the second piece?
Carl decided to fence in his rectangular garden. He bought 20 fence posts, placed one on each of the four corners, and spaced out the rest evenly along the edges of the garden, leaving exactly 4 yards between neighboring posts. The longer side of his garden, including the corners, has twice as many posts as the shorter side, including the corners. What is the area, in square yards, of Carl's garden?
Two different numbers are selected at random from $\{1,2,3,4,5\}$ and multiplied together. What is the probability that the product is even?
At Megapolis Hospital one year, multiple-birth statistics were as follows: Sets of twins, triplets, and quadruplets accounted for 1000 of the babies born. There were four times as many sets of triplets as sets of quadruplets, and three times as many sets of twins as sets of triplets. How many of these 1000 babies were in sets of quadruplets?
How many squares whose sides are parallel to the axes and whose vertices have coordinates that are integers lie entirely within the region bounded by the line $y=\pi x$, the line $y=-0.1$, and the line $x=5.1$ ?
All the numbers $1,2,3,4,5,6,7,8,9$ are written in a $3 \times 3$ array of squares, one number in each square, in such a way that if two numbers are consecutive then they occupy squares that share an edge. The numbers in the four corners add up to 18 . What number is in the center?
The sum of an infinite geometric series is a positive number $S$, and the second term in the series is 1 . What is the smallest possible value of $S$ ?
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 the sum of these eight products?
In how many ways can 345 be written as the sum of an increasing sequence of two or more consecutive positive integers?
Rectangle $A B C D$ has $A B=5$ and $B C=4$. Point $E$ lies on $\overline{A B}$ so that $E B=1$, point $G$ lies on $\overline{B C}$ so that $C G=1$, and point $F$ lies on $\overline{C D}$ so that $D F=2$. Segments $\overline{A G}$ and $\overline{A C}$ intersect $\overline{E F}$ at $Q$ and $P$, respectively. What is the value of $\frac{P Q}{E F}$ ?
A dilation of the plane-that is, a size transformation with a positive scale factor-sends the circle of radius 2 centered at $A(2,2)$ to the circle of radius 3 centered at $A^{\prime}(5,6)$. What distance does the origin $O(0,0)$ move under this transformation?
What is the area of the region enclosed by the graph of the equation $x^{2}+y^{2}= |x|+|y| ?$
A set of teams held a round-robin tournament in which every team played every other team exactly once. Every team won 10 games and lost 10 games; there were no ties. How many sets of three teams $\{A, B, C\}$ were there in which $A$ beat $B, B$ beat $C$, and $C$ beat $A$ ?
In regular hexagon $A B C D E F$, points $W, X, Y$, and $Z$ are chosen on sides $\overline{B C}$, $\overline{C D}, \overline{E F}$, and $\overline{F A}$, respectively, so that lines $A B, Z W, Y X$, and $E D$ are parallel and equally spaced. What is the ratio of the area of hexagon $W C X Y F Z$ to the area of hexagon $A B C D E F$ ?
How many four-digit positive integers $a b c d$, with $a \neq 0$, have the property that the three two-digit integers $a b<b c<c d$ form an increasing arithmetic sequence? One such number is 4692 , where $a=4, b=6, c=9$, and $d=2$.
Let $f(x)=\sum_{k=2}^{10}(\lfloor k x\rfloor-k\lfloor x\rfloor)$, where $\lfloor r\rfloor$ denotes the greatest integer less than or equal to $r$. How many distinct values does $f(x)$ assume for $x \geq 0$ ?
Publishing original scientific research in a peer-reviewed journal is an achievement many scholars work toward for years. Achieving this feat early in one's academic journey requires exceptional intellectual rigor, creativity, and persistence. Aaditya Punatar and Muskan Sadana have reached this remarkable milestone, successfully publishing their collaborative research paper in the prestigious Journal of Emerging Investigators […]
Her project on quantum-accelerated Steiner trees for 15-minute cities earned a place among the finalists at the world's largest pre-college STEM competition. Prishaa Shrimali of Marymount School of New York, was selected as a Grand Finalist at the 2026 Regeneron International Science and Engineering Fair (ISEF) in Phoenix, Arizona. Her project was Optimizing Urban Accessibility: […]
In 2026, the following Cheenta students have been successful for Indian Statistical Institute's M.Stat Entrance. They ranked within the first 50 in the entire country in these entrances. I.S.I. M.Stat Entrance
In 2026, the following Cheenta students have been successful for Indian Statistical Institute's B.Stat Entrance and Chennai Mathematical Institute's B.Sc. Math Entrance. They ranked within the first 200 in the entire country in these entrances. Most of these students attended the problem solving workshops regularly, which happen 5 days every week. CMI B.Sc. Math Entrance […]