While eating out, Mike and Joe each tipped their server 2 dollars. Mike tipped \(10 %\) of his bill and Joe tipped \(20 %\) of his bill. What was the difference, in dollars between their bills?
For each pair of real numbers \(a \neq b\), define the operation ★ as \((a \star b)=\frac{a+b}{a-b}\) What is the value of \(((1 \star 2) \star 3)\) ?
The equations \(2 x+7=3\) and \(b x-10=-2\) have the same solution for \(x\). What is the value of \(b\) ?
A rectangle with a diagonal of length \(x\) is twice as long as it is wide. What is the area of the rectangle?
A store normally sells windows at \($ 100\) each. This week the store is offering one free window for each purchase of four. Dave needs seven windows and Doug needs eight windows. How many dollars will they save if they purchase the windows together rather than separately?
The average (mean) of 20 numbers is 30 , and the average of 30 other numbers is 20 . What is the average of all 50 numbers?
Josh and Mike live 13 miles apart. Yesterday, Josh started to ride his bicycle toward Mike's house. A little later Mike started to ride his bicycle toward Josh's house. When they met, Josh had ridden for twice the length of time as Mike and at four-fifths of Mike's rate. How many miles had Mike ridden when they met?
Square \(E F G H\) is inside the square \(A B C D\) so that each side of \(E F G H\) can be extended to pass through a vertex of \(A B C D\). Square \(A B C D\) has side length \(\sqrt{50}\) and \(B E=1\). What is the area of the inner square \(E F G H\) ?
Thee tiles are marked \(X\) and two other tiles are marked \(O\). The five tiles are randomly arranged in a row. What is the probability that the arrangement reads XOXOX ?
There are two values of \(a\) for which the equation \(4 x^{2}+a x+8 x+9=0\) has only one solution for \(x\). What is the sum of these values of \(a\) ?
A wooden cube \(n\) units on a side is painted red on all six faces and then cut into \(n^{3}\) unit cubes. Exactly one-fourth of the total number of faces of the unit cubes are red. What is \(n\) ?
The gure shown is called a trefoil and is constructed by drawing circular sectors about sides of the congruent equilateral triangles. What is the area of a trefoil whose horizontal base has length 2 ? AMC 10 2005
How many positive integers \(n\) satisfy the following condition: \[ (130 n)^{50}>n^{100}>2^{200} ? \]
How many three-digit numbers satisfy the property that the middle digit is the average of the first and the last digits?
How many positive integer cubes divide \(3!\cdot 5!\cdot 7!\) ?
The sum of the digits of a two-digit number is subtracted from the number. The units digit of the result is 6 . How many two-digit numbers have this property?
In the five-sided star shown, the letters \(A, B, C, D\), and \(E\) are replaced by the numbers \(3,5,6,7\), and 9 , although not necessarily in this order. The sums of the numbers at the ends of the line segments \(\overline{A B}, \overline{B C}, \overline{C D}, \overline{D E}\), and \(\overline{E A}\) form an arithmetic sequence, although not necessarily in this order. What is the middle term of the arithmetic sequence?
Team \(A\) and team \(B\) play a series. The first team to win three games wins the series. Each team is equally likely to win each game, there are no ties, and the outcomes of the individual games are independent. If team \(B\) wins the second game and team \(A\) wins the series, what is the probability that team \(B\) wins the first game?
Three one-inch squares are palced with their bases on a line. The center square is lifted out and rotated \(45^{\circ}\), as shown. Then it is centered and lowered into its original location until it touches both of the adjoining squares. How many inches is the point \(B\) from the line on which the bases of the original squares were placed?
An equiangular octagon has four sides of length 1 and four sides of length \(\frac{\sqrt{2}}{2}\), arranged so that no two consecutive sides have the same length. What is the area of the octagon?
For how many positive integers \(n\) does \(1+2+\cdots+n\) evenly divide from \(6 n\) ?
Let \(S\) be the set of the 2005 smallest multiples of 4 , and let \(T\) be the set of the 2005 smallest positive multiples of 6 . How many elements are common to \(S\) and \(T\) ?
Let \(\overline{A B}\) be a diameter of a circle and \(C\) be a point on \(\overline{A B}\) with \(2 \cdot A C=B C\). Let \(D\) and \(E\) be points on the circle such that \(\overline{D C} \perp \overline{A B}\) and \(\overline{D E}\) is a second diameter. What is the ratio of the area of \(\triangle D C E\) to the area of \(\triangle A B D\) ?
For each positive integer \(m>1\), let \(P(m)\) denote the greatest prime factor of \(m\). For how many positive integers \(n\) is it true that both \(P(n)=\sqrt{n}\) and \(P(n+48)=\sqrt{n+48}\) ?
In \(A B C\) we have \(A B=25, B C=39\), and \(A C=42\). Points \(D\) and \(E\) are on \(A B\) and \(A C\) respectively, with \(A D=19\) and \(A E=14\). What is the ratio of the area of triangle \(A D E\) to the area of quadrilateral \(B C E D\) ?
In 2025, 8 students from Cheenta Academy cracked the prestigious Regional Math Olympiad. In this post, we will share some of their success stories and learning strategies. The Regional Mathematics Olympiad (RMO) and the Indian National Mathematics Olympiad (INMO) are two most important mathematics contests in India.These two contests are for the students who are […]
Cheenta Academy proudly celebrates the success of 27 current and former students who qualified for the Indian Olympiad Qualifier in Mathematics (IOQM) 2025, advancing to the next stage — RMO. This accomplishment highlights their perseverance and Cheenta’s ongoing mission to nurture mathematical excellence and research-oriented learning.
Cheenta students shine at the Purple Comet Math Meet 2025 organized by Titu Andreescu and Jonathan Kanewith top national and global ranks.
Celebrate the success of Cheenta students in the Stanford Math Tournament. The Unified Vectors team achieved Top 20 in the Team Round.