id int64 1 92 | category stringclasses 6 values | question stringlengths 43 841 | answer null |
|---|---|---|---|
1 | Algebra | Compute the number of solutions x \in [ 0 , 2 \pi ] to cos ( x^2) + sin ( x^2) = 0 . | null |
2 | Algebra | Suppose that for some angle 0 \leq \theta \leq \pi / 4 , the roots of x^2 + a x + \frac{3}{10} are sin \theta and cos \theta . If the roots of p ( x ) = x^2 + c x + d are sin ( 2 \theta ) and cos ( 2 \theta ) , what is the value of p ( 1 ) ? | null |
3 | Algebra | Let q ( x ) = x^3 - 9 x^2 + 18x + 27. Compute q ( −10) + q ( −8) + q ( −6) + … + q ( 16) . | null |
4 | Algebra | Compute 5 1 \cdot 2 \cdot 3 \cdot 4+7 2 \cdot 3 \cdot 4 \cdot 5+ … +99 48 \cdot 49 \cdot 50 \cdot 51. | null |
5 | Algebra | Ashley writes the concatenation of \lfloor 2.5^1\rfloor , \lfloor 2.5^2\rfloor , … , \lfloor 2.5^1 0 0 0\rfloor on the board. Her number is 199667 digits long. No w , Bob writes the concatenation of 4^1, 4^2, . . . , 4^{1000} on the board. Compute the number of digits in Bob ’s number . | null |
6 | Algebra | Let f ( x , y ) = x y and g ( x , y ) = x^2 - y^2. If a counter clockwise rotation of \theta radians about the origin sends g ( x , y ) = a to f ( x , y ) = b , compute the value of a b tan \theta. | null |
7 | Algebra | Find the number of lines of symmetr y that pass through the origin for | x y ( x + y ) ( x − y ) | = 1 . | null |
8 | Algebra | Let a1 , a^2 , a^3 , and a4 be non-negative real numbers such that a^2 1 + a^2 2 + a^2 3 + a^24 = 1 . Compute the minimum possible value of 6 a^3 1 + 8 a^3 2 + 1 2 a^3 3 + 24 a^3 4 . | null |
9 | Algebra | Compute \sum_{100} a = 0\sum_{100} b = 01 1 + cos (2 \pi (a - b)/101). | null |
10 | Algebra | Call a polynomial x8+ b_7x^7+ \cdots + b_1x^1+ 1 binary if each bi is either 0 or 1 . Compute the number of binary polynomials that have at least one real root. | null |
11 | Calculus | Compute lim x → \infty\int_x 0et^2d t ex2 . | null |
12 | Calculus | Consider the region specified by the union of the inequalities 1 \geq y \geq x2 and 1 \geq x \geq y^2. What is the volume of the solid create d by rotating this region about the x -axis? | null |
13 | Calculus | Compute the limit lim x → 0g'(x) x6, where g ( x ) = \int_{x^4}^0x t e^{t/x} x^2 + t^2d t . | null |
14 | Calculus | Consider the pair of ladders shown in the below image . The bottom ladder ( dashed line) connects the points ( 0 , 4 ) and ( 4 , 0 ) . The top ladder ( dotted line) is attached to the bottom ladder at ( 2 , 2 ) and touches the wall at ( 0 , 6 ) . This pair of ladders begins to slide down the wall, such that the top ladder remains attached to the midpoint of the bottom ladder . When the bottom ladder touches the wall at the point ( 0 , 2 ) , the end at which it touches the wall is moving downward at the rate of 2 units per second. At that point in time , what is the rate at which the wall end of the top ladder is moving downward, in units per second? | null |
15 | Calculus | Charlie chooses a real number c > 0 . Bob chooses a real number b from the interval ( 0 , 2 c ) uniformly at random. Alice chooses a real number \tilde a from the interval ( 0 , 2\sqrtc ) uniformly at random. Let a = \tilde a2. What is the probability that the roots of a x^2 + b x + c have nonzero imaginary parts and have real parts with absolute value greater than 1 ? | null |
16 | Calculus | Let f0 ( x ) = max ( | x | , cos ( x ) ) , and fk + 1 ( x ) = max ( | x | − fk ( x ) , fk ( x ) − cos ( x ) ) . Compute lim n → \infty\int_{-\pi/2}^{\pi/2}( fn − 1 ( x ) + fn ( x ) ) d x . | null |
17 | Calculus | If f ( x ) is a non-negative differentiable function defined over positive real numbers that satisfies f ( 1 ) = \frac{25}{16} and f'(x) = 2 x− 1f ( x ) + x2 \sqrt f ( x ) , compute f ( 2 ) . St anford Ma th Tournament Calculus April 13, 2024 | null |
18 | Calculus | Compute \int_1^0cos (\pi 4+\frac{10^4 - 1}{2}arccos ( x ) ) d x . | null |
19 | Calculus | The integral \intf ( 2024 ) 0f− 1( x ) d x , where f ( x ) = \int_x 0e− t^2d t , can be written in the form A ( 1 − e− B) for positive rational constants A and B . Compute A + \lfloor \log_{10} B \rfloor . | null |
20 | Calculus | Compute \sum_{\infty} m = 0\sum_{\infty} n = 0(\frac{1}{4})^{m+n} ( 2 n + 1 ) ( m + n + 1 ). | null |
21 | General | Find the volume of the pyramid with vertices at the coordinates ( 1 , 0 , 0 ) , ( 0 , 1 , 0 ) , ( 0 , 0 , 1 ) , ( 0 , 0 , 0 ) . | null |
22 | General | Consider the following system of equations: w + x + y = 8 y + z = 1 0 w + x + z = 1 2 . Find w + x + y + z . | null |
23 | General | What is the number of 5 -digit numbers that have strictly decreasing digits from left to right? | null |
24 | General | Call a number balanced if it is divisible by p + 1 0 where p is its smallest prime divisor . How many numbers from 1 to 100 , inclusive , are balanced? | null |
25 | General | Let \mathcal{A} be the region in the x y -plane bounded by y = 0 , y = x , and y = 2 − x . \mathcal{A} includes the area enclosed by these boundaries, as well as the boundaries themselves. What is the maximum possible radius of a circle that lies in \mathcal{A} ? | null |
26 | General | Find the area enclosed by the relation: | x + y | + | x − y | = 1 6 . | null |
27 | General | What is the probability of obtaining a sum of 9 by rolling 4 six-side d dice? | null |
28 | General | For some real constant c , the roots of the quadratic x^2 + c x − 2024 are r and s . If the quadratic x^2 + r x + s has one distinct root t (not necessarily real), find t . | null |
29 | General | Call two positive integers similar if their prime factorization have the same number of distinct prime divisors, and when ordered in some way , the exponents match. For example , 250 and 24 are similar be cause 250 = 5^3 \cdot 2 , and 24 = 2^3 \cdot 3 . How many positive integers less than or equal to 2 0 0 are similar to 1 8 (including itself)? | null |
30 | General | What is the sum of the possible values of c such that the polynomial x^2 - 4 0 x + c = 0 has positive integer roots (possibly equal to each other )? St anford Ma th Tournament General April 13, 2024 | null |
31 | General | Let \triangle A B C be an equilateral triangle with side length 6 . Three circles of radius 6 are centered at A , B , and C . Compute the radius of the circle that is centered at the center of \triangle A B C , is internally tangent to these three circles, and lies in the interior of the three circles. | null |
32 | General | Compute the largest positive integer x less than 1000 that satisfies x^2 \equiv 24 ( mod 1000 ) . | null |
33 | General | Jana is decorating her room by hanging zero or more strings of lights. She has 5 collinear attachment points ( A, B, C, D , and E), and she can connect any two attachment points with a semicircular string of lights ( direction hanging downward), as long as no two strings cross. In how many different patterns can she hang the lights? | null |
34 | General | Let 0 . db = 0 . d d d . . .b denote a repeating decimal written in base b . If 0 . 4a + 0 . 7b = 1 for positive integers a and b such that a \neq b , what is the minimum possible value of a + b ? | null |
35 | General | Perry bakes a pineapple upside-down cake with 3 slices of pineapple on top , each in the shape of regular dodecagons of side length 1 . The pineapple slices overlap each other as shown in the figure . Compute the length of the cut A B that Perry makes to slice the cake in half. | null |
36 | General | Let P ( n ) represent the number of real roots x for the equation xn+ xn − 1+ . . . + x1+ 1 = 0 . Compute P ( 1 ) + P ( 2 ) + P ( 3 ) + . . . + P ( 2024 ) . St anford Ma th Tournament General April 13, 2024 | null |
37 | General | Nacho is building a sandcastle . Each time he adds a scoop of sand, he has a 5 6 chance that the sandcastle will increase by 1 inch in height. Nacho is a clumsy engineer , so each time the height doesn’t increase , the sandcastle topples and loses 1 3 of its current height. Suppose Nacho starts his sandcastle at height H . What H should he choose so that after any number of scoops, the expected height of his sandcastle is still H ? | null |
38 | General | Consider the horizontal line that intersects the ellipse x^2/9+ y^2 = 1 at points A and B above the x - axis such that \angle A O B = 120^\circ, where point O is the origin. Compute the area of the region of the ellipse that lies above this line . | null |
39 | General | Let a1 , a^2 , … be a strictly increasing sequence of positive integers such that a^3 k − 2 is divisible by 8 and a^3 k is divisible by 9 for all positive integers k . Find the largest possible positive integer i such that ai > 2024 and ai − 1 \leq 2024 . | null |
40 | General | I have 3 red balls, 3 blue balls, and 3 yellow balls. These 9 balls are randomly arranged on a 3 \times 3 grid. Let a 3-in-a-row denote when 3 balls of the same color are aligned in a line in the grid ( a row , column, or diagonal consisting of 3 balls of the same color ). What is the expected number of 3-in-a- rows that will show up in the grid? | null |
41 | General | How many positive integers n are there such that the powers of 2024 ( mod n ) repeat in a cycle of length 2 ? In other words, how many positive integers n are there such that 2024^0 ( mod n ) \equiv 2024^2 ( mod n ) \equiv 2024^4 ( mod n ) . . . and 2024^1 ( mod n ) \equiv 2024^3 ( mod n ) \equiv 2024^5 ( mod n ) . . . but 2024^0 ( mod n ) \equiv 2024^1 ( mod n ) ? | null |
42 | General | Consider a quadratic function P ( x ) = a x^2 + b x + c with distinct positive roots r1 and r2 , and a second polynomial Q ( x ) = c x^2 + b x + a with roots r3 and r4 . John writes four numbers on the whiteboard: r1 , r2 , 4 r3 and 4 r4 . What is the smallest possible integer value of the sum of the numbers John wrote down? | null |
43 | General | Let A B C D E be a regular pentagon with side length 1 . Circles \omegaB , \omegaC , and \omegaD are centered at B , C , and D respectively , each with radius 1 . \omegaB intersects \omegaC inside A B C D E at point F , and \omegaC intersects \omegaD inside A B C D E at point G . Compute the ratio of the measure of \angle A F B to the measure of \angle A G B . | null |
44 | General | Let F be a set of subsets of { 1 , 2 , 3 } . F is called distinguishing if each of 1 , 2 , and 3 are distinguishable from each other—that is, 1 , 2 , and 3 are each in a distinct set of subsets from each other . For example F = { { 1 , 3 } , { 2 , 3 } } is distinguishing be cause 1 is in { 1 , 3 } , 2 is in { 2 , 3 } , and 3 is in { 1 , 3 } and { 2 , 3 } . F = { { 1 , 2 } , { 2 } } is also distinguishing : 1 is in { 1 , 2 } , 2 is in { 1 , 2 } and { 2 } , and 3 is in none of the subsets. On the other hand, F = { { 1 } , { 2 , 3 } } is not distinguishing . Both 2 and 3 are only in { 2 , 3 } , so they cannot be distinguished from each other . How many distinguishing sets of subsets of { 1 , 2 , 3 } are there? St anford Ma th Tournament General April 13, 2024 | null |
45 | General | For the numbers 4^4, 4^{44}, 4^{444}, … , 4^{44...4} where the last exponent is 2024 digits long, Quincy writes down the remainders when they are divided by 2024 . Compute the sum of the distinct numbers that Quincy writes. | null |
46 | Discrete | Some of the students pair up . Each student can be in at most one pair . | null |
47 | Discrete | The pairs swap phones according to some swap order (i.e . an ordering of the pairs). For a given assignment \pi of the 2 n students to the phones they originally picked up , let r ( \pi ) be the minimum number of rounds required for the students to each receive back their own phone , assuming the students make swaps optimally . Let s ( \pi ) be the number of ways to swap phones ( determine d by pairings and swap orders over all rounds) achieving r ( \pi ) rounds. Let M ( n ) be the maximum value of r ( \pi ) over all assignments \pi , and let f ( n ) be the sum of s ( \pi ) over all \pi with r ( \pi ) < M ( n ) . Then, there exists a unique ordered pair ( a , b ) with a > 0 and b > 0 such that limn → \inftyf ( n ) · an ( 2 n ) \neq b . Compute ( a , b ) . Note: It may be helpful to know that ex= \sum_{\infty} k=0} \frac{x^k}{k!}. | null |
48 | Discrete | Let F be a set of subsets of { 1 , 2 , 3 } . F is called distinguishing if each of 1 , 2 , and 3 are distinguishable from each other—that is, 1 , 2 , and 3 are each in a distinct set of subsets from each other . For example F = { { 1 , 3 } , { 2 , 3 } } is distinguishing be cause 1 is in { 1 , 3 } , 2 is in { 2 , 3 } , and 3 is in { 1 , 3 } and { 2 , 3 } . F = { { 1 , 2 } , { 2 } } is also distinguishing : 1 is in { 1 , 2 } , 2 is in { 1 , 2 } and { 2 } , and 3 is in none of the subsets. On the other hand, F = { { 1 } , { 2 , 3 } } is not distinguishing . Both 2 and 3 are only in { 2 , 3 } , so they cannot be distinguished from each other . How many distinguishing sets of subsets of { 1 , 2 , 3 } are there? | null |
49 | Discrete | Each vertex and edge of an equilateral triangle is randomly labeled with a distinct integer from 1 to 1 0 , inclusive . Compute the probability that the number on each edge is the sum of those on its vertices. | null |
50 | Discrete | We define the spillage of a number as s ( x ) = \lfloor \frac{x}{100\rfloor , that is, the largest integer that is at most x 100. The spillage of a list of numbers [ a1 , a^2 , … , an ] is the sum of left to right spillages : s ( a1 ) + s ( a1 a^2 ) + s ( a1 a^2 a^3 ) + … + s ( a1 a^2 \cdots an ) . Let M be the minimum possible spillage of [ 1 , 2 , … , 1 0 ] over all the permutations of this list. How many of these permutations achieve M ? | null |
51 | Discrete | Alice is playing with magnets on her fridge . She has 7 magnets, with the numbers 1 , 2 , 3 , 4 , 5 , 6 , 7 , in that order in a row , and she also has two magnets with a " + " sign, two magnets with a " − " sign, and two magnets with a " × " sign. She randomly puts these six operation magnets between her 7 number magnets, with one operation between every two consecutive numbers, and evaluates the resulting expression (following the order of operations). What is the expected value of her result? | null |
52 | Discrete | Let 1 \leq A \leq 119 and 1 \leq B \leq 139 be two integers such that A/60 and B/70 are fractions in simplest form, yet, when adding A/60 and B/70 by rewriting both fractions with their lowest common denominator and adding the resulting numerators, the new fraction can be simplified. Find how many ordered pairs ( A , B ) are possible . | null |
53 | Discrete | Call a polynomial P cool if it has degree less than 257 , each of its coefficients are nonnegative integers less than 257 , and \sum2 5 6 k = 0P ( kj) is divisible by 257 for all positive integers j . How many cool polynomials are there? ( Assume that the polynomial P ( x ) = 0 has degree less than 257 . ) | null |
54 | Discrete | Let f : \mathbb{N} → \mathbb{R} be a function which satisfies \frac{1}{64}n^2 = \sum_{d|n}f ( d ) f(n/d) . What is the least integer n for which f ( n ) is an integer? St anford Ma th Tournament Discrete April 13, 2024 | null |
55 | Discrete | There are 2 n students taking an exam, and at the beginning they all put their phones into a pile . When leaving, each person takes an arbitrary phone from the pile . Unfortunately , it might be the case that some students did not get back their own phone! To get back the correct phones, the students come up with the following strategy . They repeat the following round as many times as needed: | null |
56 | Guts | Viktor wants to build a big sandcastle with a triangle base . What is the maximum area of a right triangle with hypotenuse 1 0 ? | null |
57 | Guts | For his 21st birthday , Arpit would like to play a game of 2 1 . He would like to achieve 2 1 total points by drawing three cards and adding up their point values, with the third card’s point value being worth twice as much (multiplied by two in the sum). If there are infinite cards with point values 1 through 1 4 , how many ways are there for him to get to 2 1 ? Note that the order of the cards drawn matters. | null |
58 | Guts | Compute the number of positive integers n less than 100 such that n^2 divides n ! . | null |
59 | Guts | The image to the right is comprised of black and white interlocking shapes that are similar to each other . Each shape ’s perimeter is composed of one " outer" semicircular arc and two smaller "inner" semicircular arcs. The largest shape , which is white , has an outer radius of length 1 and an inner radius of length 1/2 . If the pattern depicted continues infinitely , what is the positive difference between the total area of the white shapes and the total area of the black shapes? | null |
60 | Guts | Dean is at the beach making sandcastles too , but there ’s a problem — he ’s ambidextrous! His sandcastles always end up looking the same from the left and right. What is the largest 4 -digit palindrome that can be written as a sum of three 3 -digit palindromes? | null |
61 | Guts | Misha is the sandcastle building go d. 2024 Greek go ds and go ddesses (numbered as 1 to 2024 from most to least important) are coming together for a banquet. You are deity number 2024 . There are 2024 seats labeled with the deities’ numbers, and the go ds enter in order from least to most important. When you enter , you choose a random seat to sit in ( which may be your designated seat). When go d i enters, if their seat is empty they sit in it. Otherwise (if you are in it), they pay you i + 1 prayers to move out of it so that they can sit. You then take an unoccupied seat at random. Compute the expected total number of prayers you earn through this procedure . | null |
62 | Guts | Kat told Viktor that equilateral triangles make better sandcastles. In equilateral triangle A B C , points D , E , and F are chosen on line segments B C , C A , and A B such that \angle F C A = 2 \angle E B C = 4 \angle D A B . Line A D meets C F at X and B E at Y . Given that the four points C , E , X , Y are concyclic, compute \angle F C A . St anford Ma th Tournament Guts April 13, 2024 | null |
63 | Guts | How many positive integers n are there such that the following equation has at least one real solution in x ? x^4 + 4 x^3 + 24 x^2 + 4 0 x + n = 0 | null |
64 | Guts | Eric comes and destroys all the sandcastles. He gives builders this problem instead: Given that 3^{36} + 3^{25} + 3^{13} + 1 has three prime factors, compute its largest prime factor . | null |
65 | Guts | Consider the triangle A B C where A C = 1 and A B = 1 . G is the centroid of A B C . Points D , F , L , H , and I are the midpoints of A C , B C , A G , G B , and C G respectively . Let M be the point where C L intersects B D and let K be the point where C H intersects A F . Compute the ratio of the area of pentagon I M L H K to the area of triangle \triangle A B C . | null |
66 | Guts | Let T be a triangle with the largest possible area whose vertices all have coordinates of the form ( p , q ) such that p , q are prime numbers less than 100 . How many lattice points are either contained in Tor lie on the boundary of T ? | null |
67 | Guts | What is the smallest positive integer with the pr operty that the sum of its pr oper divisors is at least twice as great as itself? (The pr oper divisors of a number are the positive divisors of the number excluding the number itself.) | null |
68 | Guts | Compute the remainder when (10!)^{20} is divided by 2024 . | null |
69 | Guts | A right square pyramid with height 1 2 and a base of side length 1 0 is inscribe d in sphere S . Compute the largest possible radius of a sphere that lies inside S and is tangent to one of the lateral faces of the pyramid. | null |
70 | Guts | For any integer n \geq 2 , with prime factorization n = p_a1 1 pa^2 2 … p_{a_k} k , we let f ( n ) = \sumk i = 1pi ai . For example , since 9 0 = 21· 32· 51, we have f ( 9 0 ) = 2 · 1 + 3 · 2 + 5 · 1 = 1 3 . Let m be the minimum value that f ( n ) can take on for all integers n > 2024 . Find the smallest integer k \geq 2 such that f ( k ) = m . | null |
71 | Guts | Compute \lfloor2 0 2 3 4202\rfloor + \lfloor2 0 2 3 · 2 4202\rfloor + … + \lfloor2 0 2 3 · 4201 4202\rfloor . | null |
72 | Guts | Triangle A B C has side lengths A B = 24 and A C = 2 2 , and the radius of its circumcircle is 1 3 . Compute the sum of the possible lengths of B C . | null |
73 | Guts | Consider the following rule for moves on the two-dimensional integer lattice: for each coordinate ( b , c ) that you are on, move to ( b + 1 , c ) if 0 = x^2 + b x + c has no real solution, and move to ( b , c + 1 ) otherwise . If you begin at ( 0 , 0 ) , what coordinates do you land on after 2024 moves? | null |
74 | Guts | What is the largest composite number n such that the sum of the digits of n is larger than the greatest divisor of n , excluding n itself? | null |
75 | Guts | Consider cutting the ellipse y2+x^2/9= 1 by the line y =\sqrt{7} x + 4 . What is the largest area bounded by the ellipse and the line? | null |
76 | Guts | Consider the 4 \times 8 grid of points below that represents the Stanford campus. Stanford has developed a way to teleport Main Quad (represented by the rectangle on the lattice) anywhere on campus such that 4 points are contained within the rectangle and none of these points are dorms (the dimensions of Main Quad remain the same as in the figure). The bottom-left and top-right corners of the grid are dorms. St anford Ma th Tournament Guts April 13, 2024 Abby wants to bike from Schiff to Lantana, but does not want to pass through any points in Main Quad. If Abby only moves from one lattice point to another in the up and right directions ( and Main Quad does not move as she bikes), compute the sum of the number of paths she can take for all possible positions of Main Quad. | null |
77 | Guts | Note: this round consists of a cycle , where each answer is the input into the next problem. Let \mathcal{A} be the answer to problem 24. Let A B C be a triangle with area \mathcal{A} and \angle A B C = 90^\circ. Let M and N be the midpoints of B C and B A , respectively . Let P be a point on the circumcircle of A B C such that arc A P C is distinct from arc A B C , and let P ' be the point of intersection of line P M and the circumcircle of \triangle A B C such that P \neq P ' . Let T be the intersection of P A and P ' B . If T also lies on line M N and tan \angle B A C = 3 , compute the area of \triangle A B T . | null |
78 | Guts | Let R be the answer to problem 22 and set N = 100 + R . Let f : { 2 , 3 , . . . , N } → { 2 , 3 , . . . , N } be any function such that there are exactly N + 1 5 ordered pair solutions to the equation f ( x ) − f ( y ) = 0 . Suppose F is the collection of these functions f which maximize \log_2 f(2) \log_3 f(3) \cdots \log_N f(N) . Over all f \in F , compute the maximum possible value of \sum_{i=2}^N (f(i) - i) . | null |
79 | Guts | Let M be the answer to problem 23. Compute the number of integers 0 \leq k \leq 2196 − M such that 2196 ! M ! k ! ( 2196 − M − k ) !\equiv 0 ( mod 1 3 ) . | null |
80 | Guts | Frank composes a random 15-note melody where each note is either A, B, or C. What is the probability that no sequence of 5 consecutive notes occurs more than once in the melody he composes? (For example , in the melody ABCABCABCABCABC the sequence ABCAB occurs more than once .) For an estimate of E , you will get max ( 0 , 2 5 − \lceil 5 0 0 | E − X | \rceil ) points, where X is the true answer . | null |
81 | Guts | A prime number is said to be toothless if none of its digits are 2 , 4 , or 8 . Estimate the number of toothless primes with at most 8 digits. For an estimate of E , you will get max ( 0 , 2 5 − \lceil| E − X | 4 0 0 0\rceil ) points, where X is the true answer . St anford Ma th Tournament Guts April 13, 2024 | null |
82 | Guts | Two binary sequences a , b are uniformly randomly chosen from all binary sequences of length 2 0 0 . At each step , a random digit of a is flipped, and the digits of b are uniformly randomly permuted. Let X be the expected number of steps before a and b are the same . Estimate X − 2^{200}. For an estimate of E , you will get max ( 0 , 2 5 − \lceil| E − S | 1 0\rceil ) points, where S is the true value of X − 2^{200}. | null |
83 | Geometry | Let A B C D be a square , and let P be a point chosen on segment A C . There is a point X on segment B C such that P X = P B = 3 7 and B X = 24 . Compute the side length of A B C D . | null |
84 | Geometry | Let \triangle A B C be an equilateral triangle with side length 6 . Three circles of radius 6 are centered at A , B , and C . Compute the radius of the circle that is centered at the center of \triangle A B C , is internally tangent to these three circles, and lies in the interior of the three circles. | null |
85 | Geometry | Let \omega be the circle inscribe d in regular he xagon A B C D E F with side length 1 , and let the midpoint of side B C be M . Segment A M intersects \omega at point P \neq M . Compute the length of A P . | null |
86 | Geometry | Let \triangle O A B and \triangle O A ' B ' be equilateral triangles such that \angle A O A ' = 90^\circ, \angle B O B ' = 90^\circ, and \angle A O B ' is obtuse . Given that the side length of \triangle O A ' B ' is 1 and the circumradius of \triangle O A B ' is\sqrt{6} 1 , compute the side length of \triangle O A B . | null |
87 | Geometry | In right triangle \triangle A B C with right angle at B , let I be the incenter and G the centroid. Let the foot of the perpendicular from I to A B be D and the foot of the perpendicular from G to C B be E . Line l is drawn such that l is parallel to D E and passes through B . Line I D meets l at X , and line G E meets l at Y . Given that A B = 8 and C B = 1 5 , compute the length X Y . | null |
88 | Geometry | Let A B C D E be a regular pentagon with side length 1 . Circles \omegaB , \omegaC , and \omegaD are centered at B , C , and D respectively , each with radius 1 . \omegaB intersects \omegaC inside A B C D E at point F , and \omegaC intersects \omegaD inside A B C D E at point G . Compute the ratio of the measure of \angle A F B to the measure of \angle A G B . | null |
89 | Geometry | Consider the horizontal line that intersects the ellipse x^2/9+ y^2 = 1 at points A and B above the x - axis such that \angle A O B = 120^\circ, where point O is the origin. Compute the area of the region of the ellipse that lies above this line . | null |
90 | Geometry | Points A and B lie on a circle centered at O such that A B = 1 4 . The perpendicular bisector of A B intersects \odot O at point C such that O lies in the interior of \triangle A B C and A C = 35\sqrt{2} . Lines B O and A C intersect at point D . Compute the ratio of the area of \triangle D O C to the area of \triangle D B C . | null |
91 | Geometry | Consider a prism with regular he xagon bases. We form an antiprism by removing the lateral faces, rotating one of the bases 30^\circ about the axis passing through the centers of the bases, and forming 1 2 triangular faces between the bases where each triangular face consists of one vertex of one of the bases and the two closest vertices of the other base . Compute the ratio of the volume of the antiprism to the volume of the original prism. St anford Ma th Tournament Geometry April 13, 2024 | null |
92 | Geometry | Given a triangle A B C , let the tangent lines to the circumcircle of \triangle A B C at points A and B intersect at point T . Line C T intersects the circumcircle for a second time at point D . Let the projections of D onto A B , B C , A C be M , N , P respectively . From M draw a line perpendicular to N P intersecting B C at E . If E C = 5 , E B = 1 1 , and \angle C E P = 120^\circ, compute the length of C P . | null |
Stanford Math Tournament (SMT) 2024 Dataset
Source: Official 2024 Stanford Math Tournament test and solution PDFs
Format: CSV
Filename: smt_2024_dataset_final.csv
🧭 Overview
This dataset contains verified question–answer pairs from the 2024 Stanford Math Tournament (SMT).
All entries are text-only, derived from the official problem and solution booklets.
Problems requiring diagrams or written proofs (e.g., Power Round, geometry figures) were excluded.
Each record includes:
| Field | Description |
|---|---|
id |
Unique problem ID |
category |
Contest section (General, Algebra, Calculus, Discrete, or Guts) |
question |
Cleaned text of the problem statement |
answer |
Final numeric or symbolic solution from the official key |
📊 Dataset Statistics
- Algebra: 10 problems
- Calculus: 10 problems
- Discrete: 10 problems
- General: 25 problems
- Geometry: 10 problems
- Guts: 27 problems
Total: 92 verified question–answer pairs
⚙️ Preparation
- Extracted text using PDF parsing from official SMT 2024 problem and solution packets.
- Matched by numeric index and cleaned of formatting noise.
- Excluded any entries without explicit final answers or requiring diagram-specific data.
- Normalized answers to concise numeric or algebraic expressions.
✅ Intended Use
- Mathematical reasoning benchmark for LLMs and tutoring systems
- Prompt-engineering evaluation (step-by-step or direct-answer tasks)
- QA retrieval and auto-grader calibration across symbolic math domains
🧾 Citation
If you use this dataset, please credit:
Stanford Math Tournament 2024 — Cleaned Dataset (compiled from official test PDFs).
Prepared by Natasha Mayorga, 2025.
Official SMT site: https://www.stanfordmathtournament.com/past-tests/2024
📁 File Info
smt_2024_dataset_cleaned.csv- Encoding: UTF-8
- Columns:
category,index,question,answer
Last updated: 2025-10-08
License: Derived from public educational material; redistribution for research and educational use only.
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