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A park has a circular path that surrounds a rectangular garden. The path is 1 km long, and the garden's length is twice its width. If the path's length is equal to the perimeter of the garden plus the length of the longest diagonal of the garden, what is the area of the garden in square meters? |
Determine the largest prime factor of 1821 that is also a divisor of 2732. |
Simplify the vector cross product expression \((\mathbf{A} + \mathbf{B}) \times (\mathbf{A} + \mathbf{C})\). |
When the expression $5x^2+4x-7$ is divided by $x^2+3x+8$, what is the remainder of the division? |
The largest interval in which \( x \) lies satisfying \( x^{13} - x^9 + x^6 - x + 1 > 0 \) is:
(a) \( [0, \infty) \)
(b) \( (-\infty, 0] \)
(c) \( (-\infty, \infty) \)
(d) None |
Suppose the truncated polynomial algebra $F[\alpha]/(\alpha^n)$ over a field $F$ is a Hopf algebra with $\alpha$ primitive. If $\alpha$ is even-dimensional or $F$ has characteristic 2, what condition must $n$ satisfy for the equation $\sum_{0<i<n} \binom{n}{i} \alpha^i \otimes \alpha^{n-i} = 0$ to hold? |
Determine the quotient group $\mathbb{C}/P$, where $\mathbb{C}$ is the multiplicative group of non-zero complex numbers and $P$ is the subgroup of positive real numbers. |
The square of 23 is equal to the square of 22 plus another square. Find the one that may be added to the square of 22 to make the square of 23. |
A basketball player made $x+3$ free throws during a game, scoring $x-2$ points for each shot. A football player scored $2x-1$ touchdowns, worth $x+1$ points each. At the end of their respective games, the basketball player had scored a total of 15 points more than the football player. What is the value of $x$? |
The vectors $\begin{pmatrix} 2 \\ 3 \end{pmatrix}$ and $\begin{pmatrix} 4 \\ 6 \end{pmatrix}$ are parallel, and the distance between their initial points is 5 units. Determine the distance between the lines defined by the vectors $\begin{pmatrix} 2 \\ 3 \end{pmatrix} + s \begin{pmatrix} 2 \\ 4 \end{pmatrix}$ and $\begi... |
The vertices of a cube are located on the surface of a sphere of radius 6 cm. What is the volume of the cube?
Express your answer in cubic centimeters. |
Given a differentiable function \( f: \mathbb{R} \to \mathbb{R} \) such that \( f(1) = 2 \) and \( f(x+y) = 2^x f(y) + 4^y f(x) \) for all \( x, y \in \mathbb{R} \), find the minimum value of \( f(x) \). |
A triangle has sides \(a\), \(b\), and \(c\). Construct another triangle with sides \(\frac{-a + b + c}{2}\), \(\frac{a - b + c}{2}\), and \(\frac{a + b - c}{2}\). For which triangles can this process be repeated arbitrarily many times? |
Burgers’ equation\n\n\[\n\frac{\partial \phi}{\partial t} + \phi \frac{\partial \phi}{\partial x} = c \frac{\partial^2 \phi}{\partial x^2}\n\]\n\nshows diffusion and nonlinear effects in fluid mechanics (see Logan (1994)). Find the equation for permanent waves by putting \(\phi(x, t) = U(x - ct)\), where \(c\) is the c... |
Find the determinant of the matrix
$$
\left[\begin{array}{cccc}
1 & 1 & 1 & 1 \\
x & a & 0 & 0 \\
x & 0 & b & 0 \\
x & 0 & 0 & c
\end{array}\right].
$$ |
Find $\sin \frac{14 \pi}{5}$. |
Find the square root of the sum of 20% of 40 and 25% of 60. |
Circle $C$ with radius $2$ has diameter $\overline{AB}$. Circle D is internally tangent to circle $C$ at $A$. Circle $E$ is internally tangent to circle $C$, externally tangent to circle $D$, and tangent to $\overline{AB}$. The radius of circle $D$ is three times the radius of circle $E$, and can be written in the form... |
Find the value of $x$ given that $2\log_{\sqrt{2}} x = \log_{\sqrt{2}} (2x+1)$. |
z \frac{\partial z}{\partial x} - xy \frac{\partial z}{\partial y} = 2xz; \quad x + y = 2, \quad yz = 1. |
Let $X$ be a topological space and $X_{\infty}$ its one-point compactification. Under what conditions is $X_{\infty}$ second countable? |
A man wishes to raise a 300-lb weight to the top of a wall 20 m high by dragging it up an inclined plane. If his maximum pulling strength is 140 lb, what is the minimum length of the inclined plane he can use? |
Determine the Lebesgue measure of the Luzin set \( A \), which consists of numbers in the interval \([0,1]\) whose continued fraction expansion contains an infinite subsequence of integers such that each integer is divisible by the previous one. |
A yard is divided into 4 squares. The first square is 4 yards by 4 yards. The second square is two times the length and width of the first. The third square is twice the area of the second square. Finally, the area of the fourth square is the square root of the combined area of the first two squares. What is the total ... |
What is the truth value of the conjunction of two propositions A and B, denoted by A & B, when A is true and B is false? |
Expand the function \( f(z) = \frac{z}{1+z^3} \) as a Laurent series containing only negative degree powers and determine the interval of convergence. |
Evaluate the limit: $$\lim_{x\to0}\bigg[1^{1/\sin^2x}+2^{1/\sin^2x}+\ldots+n^{1/\sin^2x}\bigg]^{\sin^2x}$$ |
What are the three equivalent definitions of a Galois extension of a field \( E \) over a field \( F \)? |
A parabola is defined by the equation $y^2 = 4x$. How many points on the parabola have coordinates that are both integers? |
5. Given real numbers $x, y$ satisfy $x+y=1$. Then, the maximum value of $\left(x^{3}+1\right)\left(y^{3}+1\right)$ is $\qquad$ . |
John gives 20 stamps to each of his 5 friends and 5 more to each of his 5 colleagues. How many stamps did he give to his friends and colleagues? |
The area of a rectangle is 216 square meters. If its length is twice its width, what is the length in meters? |
Let \( n \ge 3 \) and \( a_1, a_2, \dots, a_n \) be real numbers satisfying
\[ \min_{1 \le i < j \le n} |a_i - a_j| = 1. \]
Find the minimum value of \( \sum_{k = 1}^n |a_k|^3 \). |
Find all Jordan structures (i.e., Jordan algebras) on algebras of dimension ≤ 3 over real numbers. Verify that they are all obtained from associative algebras. A Jordan algebra is a commutative algebra where the product satisfies the Jordan identity: \( x^2(yx) = (x^2y)x \). |
Calculate the probability that at least 5 out of the next 15 customers buy items worth less than 30 euros, given that 34% of customers buy items worth at least 30 euros. |
For a positive integer \( n \) and a real number \( x \) (where \( 0 \leq x < n \)), define
$$
f(n, x) = (1 - \{x\}) \binom{n}{[x]} + \{x\} \binom{n}{[x] + 1},
$$
where \( [x] \) is the floor function which gives the greatest integer less than or equal to \( x \), and \( \{x\} = x - [x] \).
If integers \( m \) and \( ... |
Find the general solution of the differential equation $ty''(t) + 2y'(t) = 0$ using the substitution $y'(t) = u(t)$. |
Let \( x = \sqrt{a^2 + a + 1} - \sqrt{a^2 - a + 1}, \, a \in \mathbb{R}. \)\n\nFind all possible values of \(x\). |
If 25 men with an efficiency rate of 80% can build a wall 112 metres long in 6 days, working 8 hours per day, what length of a similar wall (with the same height and thickness) can be built by 40 men with an efficiency rate of 90% in 4 days, working 10 hours per day? |
The probability that the equation $x^{2}+x+n=0$ ($n\in[0,1]$) has real roots is ______. |
If $\log_{3}{15} = \log_{5}{x}$, what is $x$? |
For which positive integers n is the average of the squares of the first n positive integers an integer? |
A discrete rod is characterized by the property that it evolves by a Euclidean motion (translation and rotation) under a linear combination of the Heisenberg flow and the tangent flow. The Heisenberg flow is given by:
\[
\partial_{x} \gamma_{k} = \frac{T_{k} \times T_{k-1}}{1 + \langle T_{k}, T_{k-1} \rangle},
\]
and t... |
Define \( f: E^2 \rightarrow E^1 \) by \( f(x, y) = x \sin y \). Find the first and second partial derivatives of \( f \). |
Suppose that in the iteration method (of alternate directions)\n\n\[B \frac{y^{k+1} - y^k}{\tau} + Ay^k = \varphi, \quad k = 0, 1, \ldots\]\n\nwe have \n\n\[A = A_1 + A_2, \quad A_1 A_2 = A_2 A_1\]\n\nand, in addition,\n\n\[\delta_\alpha E \leq A_\alpha \leq \Delta_\alpha E, \quad A_\alpha = A^*_\alpha, \quad \delta_\a... |
A square is divided into four congruent rectangles. If the perimeter of each of these rectangles is 40 inches, what is the perimeter of the square?
Return your final response as 'Final Answer: \boxed{<answer>}', where <answer> is the number or mathematical expression of the solution. |
Determine whether the series $\sum_{n=0}^{\infty}(-1)^{n} \cdot \frac{n^{2}+7}{n^{3}+10}$ converges absolutely, converges conditionally, or diverges. Justify your answer by checking for absolute convergence first, and if necessary, applying the Alternating Series Test. |
Simplify the expression \(42^{25\sqrt{25^{25}}}\). |
The medians of a triangle are equal if the triangle is what kind?
Lesley: |
Circle $S$ is circumscribed around triangle $ABC$. Circle $S'$ touches sides $AB$ and $BC$ and the circle $S$ from the inside at the point $T$. Let $I$ be the center of the inscribed circle into triangle $ABC$. If $\angle ATI = \angle CTI$, find the value of $\frac{AT}{CT}$ given that $AB = 13$, $BC = 14$, and $CA = 15... |
Find the smallest positive integer $n$ such that the last 30 digits of the decimal representation of $n!(n+1)!(2n+1)! - 1$ are all 9s. |
In a galaxy with exactly 1,000,001 stars, let $M$ be the set of distances between any two stars. Calculate the minimum number of distinct members in $M$. |
In a geometric sequence \(\{a_{n}\}\) with all positive terms, there exist two terms \(a_{m}\) and \(a_{n}\) such that \(\sqrt{a_{m} a_{n}}=8 a_{1}\), and it is known that \(a_{9}=a_{8}+2 a_{7}\). Find the minimum value of \(\frac{1}{m}+\frac{4}{n}\). |
2x + 3y = 6\ny = √(x + 1) |
Explain the formula for the number of orbits of \( G \times H \) on \( B^A \), where \( G \) acts on \( A \), \( H \) acts on \( B \), and \( B^A \) is the set of all mappings from \( A \) to \( B \). Include the meaning of the terms \( Z_G \) and \( m_i(\tau) \). |
In elliptic geometry \((\mathbb{P}^2, \mathcal{S})\), derive a formula for the area of an \(n\)-gon given its \(n\) angles \(\alpha_1, \alpha_2, \dots, \alpha_n\). |
Given the function f(x) = a^x (a > 0 and a ≠ 1), if the tangent line of the curve y = f(x) at the point (0, f(0)) is perpendicular to the line y = x + l, find the value of a. |
Given complex numbers z1\=a+2i (a∈R) and z2\=2-i, if $$\frac {z_{1}}{z_{2}}$$ is a purely imaginary number, find the value of |z1|. |
Given that a reflection maps the point $(2, 3)$ to $(-5, -4)$, determine the image of the point $(-1, 6)$ under the same reflection. |
\( \int \frac{x^3}{x^2 + x + \frac{1}{2}} dx \). |
A random number is selected from the interval [0, 8]. What is the probability that the selected number is closer to 2 than to 6? Express your answer as a decimal to the nearest tenth. |
What is the product of $(x-4y)^3(3x-4y)$, in terms of $x$ and $y$? |
Evaluate the expectation \( E\left[\left(\int_0^t B_s \, ds\right)^2\right] \) for a Brownian motion \( B_s \). |
The left and right foci of the hyperbola $M: x^{2}- \frac {y^{2}}{b^{2}}=1$ are denoted as $F_{1}$ and $F_{2}$, respectively, with $|F_{1}F_{2}|=2c$. A circle with the origin $O$ as its center and $c$ as its radius intersects the hyperbola $M$ at point $P$ in the first quadrant. If $|PF_{1}|=c+2$, then the x-coordinate... |
In triangle $ABC$, point $M$ is on segment $BC$ such that $CM:MB = 3:5$. Point $P$ divides segment $AM$ in the ratio $AP:PM = 2:3$. Line $BP$ intersects segment $AC$ at point $Q$. Using vectors, find the ratio $AQ:QC$. Express your answer as a simplified ratio. |
Compute $\cos 75^\circ$ using a different angle sum, specifically $\cos (60^\circ + 15^\circ)$. |
At a conference, each speaker gave exactly two presentations and each presentation was attended by exactly three speakers. If 15 presentations were given in total, how many speakers attended the conference? |
Determine the maximum value of $|z^3 - z + 2|$ for $z \in \mathbb{C}$ such that $|z| = 1$. |
Determine the shape of $\triangle ABC$ if the three interior angles $A$, $B$, $C$ satisfy $\cos 2A - \cos 2B = 2\sin^2C$.
(Hint: If necessary, you can also directly use the reading material and conclusions from question 19.) |
Return your final response within \boxed{}. Four people are sitting at four sides of a table, and they are dividing a 32-card Hungarian deck equally among themselves. If one selected player does not receive any aces, what is the probability that the player sitting opposite them also has no aces among their 8 cards? |
Find a conformal map \( w(z) \) of the right half-disk \( \{ \text{Re}(z), |z|<1 \} \) onto the upper half-plane that maps \( -i \) to \( 0 \), \( +i \) to \( \infty \), and \( 0 \) to \( -1 \). What is \( w(1) \)? |
Pick a prime number less than $30$. Square the prime number. The result is a number between two consecutive integers,Write this number as a mixed number in which the fractional part is the integer obtained by dividing the prime by two. How far off is this estimate? Express your answer as a decimal to the nearest tenth. |
Solve the nonlinear partial differential equation using the method of characteristic curves: $$\left\{\begin{matrix} -xu_x+uu_y=y & \\ u(x,2x)=0& \end{matrix}\right.$$ |
How many natural numbers $n$ less than $10^{6}$ are there such that the sum of the digits of $n$ is even, and the sum of the digits of $(n+1)$ is also even
|
Determine the inverse function of $y = \sin x$, where $x \in \left[- \frac{\pi}{2}, \frac{\pi}{2}\right]$. |
A particle is located on the coordinate plane at $(5,0)$ . Define a move for the particle as a counterclockwise rotation of $\pi/4$ radians about the origin followed by a translation of $10$ units in the positive $x$ -direction. Given that the particle's position after $150$ moves is $(p,q)$ , find the greatest integer... |
Let \(ABCD\) be a trapezoid with \(AD \parallel BC\) and \(\angle BCD < \angle ABC < 90^\circ\). Let \(E\) be the intersection point of the diagonals \(AC\) and \(BD\). The circumcircle \(\omega\) of \(\triangle BEC\) intersects the segment \(CD\) at \(X\). The lines \(AX\) and \(BC\) intersect at \(Y\), while the line... |
In the Cartesian coordinate system, circle \( C_1 \) and circle \( C_2 \) intersect at points \( P \) and \( Q \), where the coordinates of point \( P \) are \( (3, 2) \). The product of the radii of the two circles is \( \frac{13}{2} \). If the line \( y = kx \) (where \( k > 0 \)) is tangent to both circles \( C_1 \)... |
Solve Laplace's equation inside a circular cylinder subject to the boundary conditions:\n\n(a) \n\[u(r, \theta, 0) = \alpha(r, \theta), \quad u(r, \theta, H) = 0, \quad u(a, \theta, z) = 0\]\n\n(b) \n\[u(r, \theta, 0) = \alpha(r) \sin 7\theta, \quad u(r, \theta, H) = 0, \quad u(a, \theta, z) = 0\]\n\n(c) \n\[u(r, \t... |
Evaluate the sum \( S = \sum_{n=0}^{\infty}\frac{(n!)^{2}3^{n}}{(2n+1)!} \) and express your answer in the form \( \frac{a\pi}{b\sqrt{c}} \), where \( a, b, \) and \( c \) are positive integers and \( c \) is square-free. Find \( a + b + c \). |
Every day, you put either 1 or 2 dollars into a piggy bank, each with a probability of \( \frac{1}{2} \). What is the expected number of 2-dollar bills in the piggy bank when the total amount reaches at least 100 dollars for the first time? |
The polynomial \(x^3 - 2004x^2 + mx + n\) has integer coefficients and three distinct positive zeros. Exactly one of these is an integer, and it is the sum of the other two. How many values of \(n\) are possible? |
Consider the polynomial \( f = x^4 - 2 \in \mathbb{Q}[x] \). Why does Theorem 44.5 imply that the splitting field of \( f \) over \(\mathbb{Q}\) is a simple extension of \(\mathbb{Q}\)? Give such a description (i.e., express the splitting field as \(\mathbb{Q}(\mu)\) for some algebraic element \(\mu\)), justifying your... |
Find all right triangles such that their side lengths form an arithmetic sequence. |
The elliptic curve \(Y_{1}^{2} = (X_{1} - 2)(X_{1}^{2} - 28 X_{1} + 4)\) arises from attempting to solve the equation \(x^6 - y^4 = z^2\) in nonzero coprime integers \(x, y, z\). Using the mwrank program or 2-descent methods, find all rational points on this elliptic curve and explain why they do not yield a solution t... |
Given a triangle \(ABC\) and two points \(P\) and \(Q\), let \(Pa, Pb, Pc\) be the reflections of \(P\) in \(BC, CA, AB\) respectively, and \(Qa, Qb, Qc\) be the reflections of \(Q\) in \(BC, CA, AB\) respectively. Let \(Oa, Ob, Oc\) be the circumcenters of triangles \(PaQbQc\), \(PbQcQa\), and \(PcQaQb\) respectively.... |
Which of the following equations have the same graph?
$I.\quad y=x-3 \qquad II.\quad y=\frac{x^2-9}{x+3}\qquad III.\quad (x+3)y=x^2-9$
$\text{(A) I and II only} \quad \text{(B) I and III only} \quad \text{(C) II and III only} \quad \text{(D) I,II,and III} \quad \text{(E) None. All of the equations have different graphs... |
Three trucks need to transport $k$ full containers, $k$ half-full containers, and $k$ empty containers in such a way that each truck is loaded identically, and each has the same number of containers. How many ways can this be done for $k=7$, if the trucks and the containers of the same fill level are not considered dis... |
A rectangular playground has a perimeter of 1200 m and a breadth of 500 m. Additionally, it has a diagonal path of length 850 m. Find the length of the rectangular playground. |
Given the solution set for the inequality \( a x^{2} + b x + c > 0 \) is \( \{ x \mid -4 < x < 7 \} \), find the solution set for the inequality \( c x^{2} - b x + a > 0 \). |
Return your final response within \boxed{}. Let $\triangle ABC$ be a triangle with circumcenter $O$ satisfying $AB=13$ , $BC = 15$ , and $AC = 14$ . Suppose there is a point $P$ such that $PB \perp BC$ and $PA \perp AB$ . Let $X$ be a point on $AC$ such that $BX \perp OP$ . What is the ratio $AX/XC$... |
What is the maximum volume of a cylinder that can be inscribed in a sphere with radius 4 cm? |
Find the least value of \( n \geq 2 \) such that among any \( n^{2018} \) consecutive positive integers, there exists a positive integer \( m \) such that \( 2017^n \) divides \( s(m^2) \), where \( s(m^2) \) denotes the sum of the digits of \( m^2 \). |
Given $\cos x = -\frac{1}{3}$, and $x$ is an angle in the third quadrant, find $\tan 2x$. |
Let the function $f(x)$ be defined by
$$f(x)=\begin{cases}
x+5 & \text{if } x\geq -2,\\
x^2 & \text{if } x < -2.\\
\end{cases}$$
Determine the value of $a$ for which the equation $f(x)=a$ has exactly three solutions. |
The following table gives the velocity data for the space shuttle Endeavour between liftoff and the jettisoning of the solid rocket boosters. Use a graphing calculator or computer to model this data by a third-degree polynomial.
| Event | Time (s) | Velocity (ft/s) |
| :--- | :---: | :---: |
| Launch | 0 | 0 |
| Begin... |
(12 points) With a budget of 2000 yuan to purchase tables at 50 yuan each and chairs at 20 yuan each, the goal is to maximize the total number of tables and chairs. However, the number of chairs should not be less than the number of tables and should not exceed 1.5 times the number of tables. How many tables and chairs... |
A household has the following monthly income and expenses:
- **Income:** State pension (R 1140), disability grant (R 1140), salary (R 5250).
- **Expenses:** Rent (R 2300), transport (R 520), cell phone (R 200), pre-paid electricity (R 800), water bill (R 350), TV contract (R 250), loan repayment (R 310), furniture ... |
Find the natural numbers \((n, m)\) such that the following system of equations is satisfied:
\[\begin{cases} 13n - 9m = 110 \\ \text{lcm}(n, m) - \text{gcd}(n, m) = 3n + 12 \end{cases}\] |
Using only the digits 1, 2, 3, 4, and 5, two players A and B compose a 2005-digit number N by selecting one digit at a time, with A starting first. The last player to play wins if and only if N is divisible by 9. Who will win if both players play optimally? |
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CONCORD training prompts
1600 math prompts used to train the CONCORD models from
CONCORD: Label-Free Calibration of Verbalized LLM Confidence via Rollout Consistency. Maximilian Schall, Gerard de Melo, Padhraic Smyth. COLM 2026.
Sampled uniformly across difficulty levels from
OpenDataArena/ODA-Math-460k
with seed 42. Only the prompt column (question) is retained — CONCORD is
label-free, so reference answers are intentionally not included.
- Source:
OpenDataArena/ODA-Math-460k - Subset size: 1600
- Columns:
question - Sampling: uniform over
difficulty, seed 42
If you need reference answers, difficulty labels, or the full dataset, use the original source directly.
Citation
@inproceedings{schall2026concord,
title = {{CONCORD}: Label-Free Calibration of Verbalized {LLM} Confidence via Rollout Consistency},
author = {Schall, Maximilian and de Melo, Gerard and Smyth, Padhraic},
booktitle = {Conference on Language Modeling (COLM)},
year = {2026}
}
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