question_id string | subfield string | context string | question string | images images list | final_answer list | is_multiple_answer bool | unit string | answer_type string | error string | source string |
|---|---|---|---|---|---|---|---|---|---|---|
1735 | Geometry | null | Three circular arcs $\gamma_{1}, \gamma_{2}$, and $\gamma_{3}$ connect the points $A$ and $C$. These arcs lie in the same half-plane defined by line $A C$ in such a way that $\operatorname{arc} \gamma_{2}$ lies between the $\operatorname{arcs} \gamma_{1}$ and $\gamma_{3}$. Point $B$ lies on the segment $A C$. Let $h_{1... | null | true | null | null | null | TP_MM_maths_en_COMP | |
1975 | Combinatorics | null | Construct a tetromino by attaching two $2 \times 1$ dominoes along their longer sides such that the midpoint of the longer side of one domino is a corner of the other domino. This construction yields two kinds of tetrominoes with opposite orientations. Let us call them Sand Z-tetrominoes, respectively.
S-tetrominoes... | null | true | null | null | null | TP_MM_maths_en_COMP | |
2039 | Combinatorics | null | An anti-Pascal pyramid is a finite set of numbers, placed in a triangle-shaped array so that the first row of the array contains one number, the second row contains two numbers, the third row contains three numbers and so on; and, except for the numbers in the bottom row, each number equals the absolute value of the di... | null | true | null | null | null | TP_MM_maths_en_COMP | |
2184 | Geometry | null | Let $A B C D$ be a cyclic quadrilateral, and let diagonals $A C$ and $B D$ intersect at $X$. Let $C_{1}, D_{1}$ and $M$ be the midpoints of segments $C X$, $D X$ and $C D$, respectively. Lines $A D_{1}$ and $B C_{1}$ intersect at $Y$, and line $M Y$ intersects diagonals $A C$ and $B D$ at different points $E$ and $F$, ... | null | true | null | null | null | TP_MM_maths_en_COMP | |
2227 | Geometry | null | Let $H$ be the orthocenter and $G$ be the centroid of acute-angled triangle $\triangle A B C$ with $A B \neq A C$. The line $A G$ intersects the circumcircle of $\triangle A B C$ at $A$ and $P$. Let $P^{\prime}$ be the reflection of $P$ in the line $B C$. Prove that $\angle C A B=60^{\circ}$ if and only if $H G=G P^{\p... | null | true | null | null | null | TP_MM_maths_en_COMP | |
2270 | Geometry | null | In the diagram, two circles are tangent to each other at point $B$. A straight line is drawn through $B$ cutting the two circles at $A$ and $C$, as shown. Tangent lines are drawn to the circles at $A$ and $C$. Prove that these two tangent lines are parallel.
| null | true | null | null | null | TP_MM_maths_en_COMP | |
2274 | Combinatorics | null | A school has a row of $n$ open lockers, numbered 1 through $n$. After arriving at school one day, Josephine starts at the beginning of the row and closes every second locker until reaching the end of the row, as shown in the example below. Then on her way back, she closes every second locker that is still open. She con... | null | true | null | null | null | TP_MM_maths_en_COMP | |
2348 | Geometry | null | In the diagram, $C$ lies on $B D$. Also, $\triangle A B C$ and $\triangle E C D$ are equilateral triangles. If $M$ is the midpoint of $B E$ and $N$ is the midpoint of $A D$, prove that $\triangle M N C$ is equilateral.
| null | true | null | null | null | TP_MM_maths_en_COMP | |
2363 | Geometry | null | In parallelogram $A B C D, A B=a$ and $B C=b$, where $a>b$. The points of intersection of the angle bisectors are the vertices of quadrilateral $P Q R S$.
Prove that $P Q R S$ is a rectangle. | null | true | null | null | null | TP_MM_maths_en_COMP | |
2364 | Geometry | null | In parallelogram $A B C D, A B=a$ and $B C=b$, where $a>b$. The points of intersection of the angle bisectors are the vertices of quadrilateral $P Q R S$.
Prove that $P R=a-b$. | null | true | null | null | null | TP_MM_maths_en_COMP | |
2367 | Geometry | null | An equilateral triangle $A B C$ has side length 2 . A square, $P Q R S$, is such that $P$ lies on $A B, Q$ lies on $B C$, and $R$ and $S$ lie on $A C$ as shown. The points $P, Q, R$, and $S$ move so that $P, Q$ and $R$ always remain on the sides of the triangle and $S$ moves from $A C$ to $A B$ through the interior of ... | null | true | null | null | null | TP_MM_maths_en_COMP | |
2398 | Geometry | null | In the diagram, line segment $F C G$ passes through vertex $C$ of square $A B C D$, with $F$ lying on $A B$ extended and $G$ lying on $A D$ extended. Prove that $\frac{1}{A B}=\frac{1}{A F}+\frac{1}{A G}$.
| null | true | null | null | null | TP_MM_maths_en_COMP | |
2448 | Geometry | null | A circle with its centre on the $y$-axis intersects the graph of $y=|x|$ at the origin, $O$, and exactly two other distinct points, $A$ and $B$, as shown. Prove that the ratio of the area of triangle $A B O$ to the area of the circle is always $1: \pi$.
| null | true | null | null | null | TP_MM_maths_en_COMP | |
2449 | Geometry | null | In the diagram, triangle $A B C$ has a right angle at $B$ and $M$ is the midpoint of $B C$. A circle is drawn using $B C$ as its diameter. $P$ is the point of intersection of the circle with $A C$. The tangent to the circle at $P$ cuts $A B$ at $Q$. Prove that $Q M$ is parallel to $A C$.
| null | true | null | null | null | TP_MM_maths_en_COMP | |
2462 | Geometry | null | A large square $A B C D$ is drawn, with a second smaller square $P Q R S$ completely inside it so that the squares do not touch. Line segments $A P, B Q, C R$, and $D S$ are drawn, dividing the region between the squares into four nonoverlapping convex quadrilaterals, as shown. If the sides of $P Q R S$ are not paralle... | null | true | null | null | null | TP_MM_maths_en_COMP | |
2474 | Geometry | null | In triangle $A B C, \angle A B C=90^{\circ}$. Rectangle $D E F G$ is inscribed in $\triangle A B C$, as shown. Squares $J K G H$ and $M L F N$ are inscribed in $\triangle A G D$ and $\triangle C F E$, respectively. If the side length of $J H G K$ is $v$, the side length of $M L F N$ is $w$, and $D G=u$, prove that $u=v... | null | true | null | null | null | TP_MM_maths_en_COMP | |
2488 | Geometry | null | In the diagram, quadrilateral $A B C D$ has points $M$ and $N$ on $A B$ and $D C$, respectively, with $\frac{A M}{A B}=\frac{N C}{D C}$. Line segments $A N$ and $D M$ intersect at $P$, while $B N$ and $C M$ intersect at $Q$. Prove that the area of quadrilateral $P M Q N$ equals the sum of the areas of $\triangle A P D$... | null | true | null | null | null | TP_MM_maths_en_COMP | |
2499 | Geometry | null | In the diagram, $A B$ and $B C$ are chords of the circle with $A B<B C$. If $D$ is the point on the circle such that $A D$ is perpendicular to $B C$ and $E$ is the point on the circle such that $D E$ is parallel to $B C$, carefully prove, explaining all steps, that $\angle E A C+\angle A B C=90^{\circ}$.
| null | true | null | null | null | TP_MM_maths_en_COMP | |
2516 | Number Theory | null | Suppose that $m$ and $n$ are positive integers with $m \geq 2$. The $(m, n)$-sawtooth sequence is a sequence of consecutive integers that starts with 1 and has $n$ teeth, where each tooth starts with 2, goes up to $m$ and back down to 1 . For example, the $(3,4)$-sawtooth sequence is
The $(3,4)$-sawtooth sequence in... | null | true | null | null | null | TP_MM_maths_en_COMP | |
2526 | Geometry | null | In the diagram, $A B C D$ is a square. Points $E$ and $F$ are chosen on $A C$ so that $\angle E D F=45^{\circ}$. If $A E=x, E F=y$, and $F C=z$, prove that $y^{2}=x^{2}+z^{2}$.
| null | true | null | null | null | TP_MM_maths_en_COMP | |
2538 | Geometry | null | In the diagram, $A B$ is tangent to the circle with centre $O$ and radius $r$. The length of $A B$ is $p$. Point $C$ is on the circle and $D$ is inside the circle so that $B C D$ is a straight line, as shown. If $B C=C D=D O=q$, prove that $q^{2}+r^{2}=p^{2}$.
| null | true | null | null | null | TP_MM_maths_en_COMP | |
2542 | Algebra | null | Suppose there are $n$ plates equally spaced around a circular table. Ross wishes to place an identical gift on each of $k$ plates, so that no two neighbouring plates have gifts. Let $f(n, k)$ represent the number of ways in which he can place the gifts. For example $f(6,3)=2$, as shown below.
Throughout this problem,... | null | true | null | null | null | TP_MM_maths_en_COMP | |
2547 | Geometry | null | In trapezoid $A B C D, B C$ is parallel to $A D$ and $B C$ is perpendicular to $A B$. Also, the lengths of $A D, A B$ and $B C$, in that order, form a geometric sequence. Prove that $A C$ is perpendicular to $B D$.
(A geometric sequence is a sequence in which each term after the first is obtained from the previous ter... | null | true | null | null | null | TP_MM_maths_en_COMP | |
2561 | Geometry | null | In the graph, the parabola $y=x^{2}$ has been translated to the position shown.
Prove that $d e=f$.
| null | true | null | null | null | TP_MM_maths_en_COMP | |
2562 | Geometry | null | In quadrilateral $K W A D$, the midpoints of $K W$ and $A D$ are $M$ and $N$ respectively. If $M N=\frac{1}{2}(A W+D K)$, prove that $WA$ is parallel to $K D$.
| null | true | null | null | null | TP_MM_maths_en_COMP | |
2563 | Number Theory | null | Consider the first $2 n$ natural numbers. Pair off the numbers, as shown, and multiply the two members of each pair. Prove that there is no value of $n$ for which two of the $n$ products are equal.
| null | true | null | null | null | TP_MM_maths_en_COMP | |
2803 | Combinatorics | null | This Question involves one Robber and one or more Cops. After robbing a bank, the Robber retreats to a network of hideouts, represented by dots in the diagram below. Every day, the Robber stays holed up in a single hideout, and every night, the Robber moves to an adjacent hideout. Two hideouts are adjacent if and only ... | null | true | null | null | null | TP_MM_maths_en_COMP | |
2804 | Combinatorics | null | This Question involves one Robber and one or more Cops. After robbing a bank, the Robber retreats to a network of hideouts, represented by dots in the diagram below. Every day, the Robber stays holed up in a single hideout, and every night, the Robber moves to an adjacent hideout. Two hideouts are adjacent if and only ... | null | true | null | null | null | TP_MM_maths_en_COMP | |
2805 | Combinatorics | null | This Question involves one Robber and one or more Cops. After robbing a bank, the Robber retreats to a network of hideouts, represented by dots in the diagram below. Every day, the Robber stays holed up in a single hideout, and every night, the Robber moves to an adjacent hideout. Two hideouts are adjacent if and only ... | null | true | null | null | null | TP_MM_maths_en_COMP | |
2807 | Combinatorics | null | This Question involves one Robber and one or more Cops. After robbing a bank, the Robber retreats to a network of hideouts, represented by dots in the diagram below. Every day, the Robber stays holed up in a single hideout, and every night, the Robber moves to an adjacent hideout. Two hideouts are adjacent if and only ... | null | true | null | null | null | TP_MM_maths_en_COMP | |
2809 | Combinatorics | null | This Question involves one Robber and one or more Cops. After robbing a bank, the Robber retreats to a network of hideouts, represented by dots in the diagram below. Every day, the Robber stays holed up in a single hideout, and every night, the Robber moves to an adjacent hideout. Two hideouts are adjacent if and only ... | null | true | null | null | null | TP_MM_maths_en_COMP | |
2811 | Combinatorics | null | This Question involves one Robber and one or more Cops. After robbing a bank, the Robber retreats to a network of hideouts, represented by dots in the diagram below. Every day, the Robber stays holed up in a single hideout, and every night, the Robber moves to an adjacent hideout. Two hideouts are adjacent if and only ... | null | true | null | null | null | TP_MM_maths_en_COMP | |
2812 | Combinatorics | null | This Question involves one Robber and one or more Cops. After robbing a bank, the Robber retreats to a network of hideouts, represented by dots in the diagram below. Every day, the Robber stays holed up in a single hideout, and every night, the Robber moves to an adjacent hideout. Two hideouts are adjacent if and only ... | null | true | null | null | null | TP_MM_maths_en_COMP | |
2813 | Combinatorics | null | This Question involves one Robber and one or more Cops. After robbing a bank, the Robber retreats to a network of hideouts, represented by dots in the diagram below. Every day, the Robber stays holed up in a single hideout, and every night, the Robber moves to an adjacent hideout. Two hideouts are adjacent if and only ... | null | true | null | null | null | TP_MM_maths_en_COMP | |
2814 | Combinatorics | null | This Question involves one Robber and one or more Cops. After robbing a bank, the Robber retreats to a network of hideouts, represented by dots in the diagram below. Every day, the Robber stays holed up in a single hideout, and every night, the Robber moves to an adjacent hideout. Two hideouts are adjacent if and only ... | null | true | null | null | null | TP_MM_maths_en_COMP | |
2815 | Combinatorics | null | This Question involves one Robber and one or more Cops. After robbing a bank, the Robber retreats to a network of hideouts, represented by dots in the diagram below. Every day, the Robber stays holed up in a single hideout, and every night, the Robber moves to an adjacent hideout. Two hideouts are adjacent if and only ... | null | true | null | null | null | TP_MM_maths_en_COMP | |
2816 | Combinatorics | null | This Question involves one Robber and one or more Cops. After robbing a bank, the Robber retreats to a network of hideouts, represented by dots in the diagram below. Every day, the Robber stays holed up in a single hideout, and every night, the Robber moves to an adjacent hideout. Two hideouts are adjacent if and only ... | null | true | null | null | null | TP_MM_maths_en_COMP | |
2873 | Combinatorics | null | An $\boldsymbol{n}$-label is a permutation of the numbers 1 through $n$. For example, $J=35214$ is a 5 -label and $K=132$ is a 3 -label. For a fixed positive integer $p$, where $p \leq n$, consider consecutive blocks of $p$ numbers in an $n$-label. For example, when $p=3$ and $L=263415$, the blocks are 263,634,341, and... | null | true | null | null | null | TP_MM_maths_en_COMP | |
2874 | Combinatorics | null | An $\boldsymbol{n}$-label is a permutation of the numbers 1 through $n$. For example, $J=35214$ is a 5 -label and $K=132$ is a 3 -label. For a fixed positive integer $p$, where $p \leq n$, consider consecutive blocks of $p$ numbers in an $n$-label. For example, when $p=3$ and $L=263415$, the blocks are 263,634,341, and... | null | true | null | null | null | TP_MM_maths_en_COMP | |
2877 | Combinatorics | null | An $\boldsymbol{n}$-label is a permutation of the numbers 1 through $n$. For example, $J=35214$ is a 5 -label and $K=132$ is a 3 -label. For a fixed positive integer $p$, where $p \leq n$, consider consecutive blocks of $p$ numbers in an $n$-label. For example, when $p=3$ and $L=263415$, the blocks are 263,634,341, and... | null | true | null | null | null | TP_MM_maths_en_COMP | |
2878 | Combinatorics | null | An $\boldsymbol{n}$-label is a permutation of the numbers 1 through $n$. For example, $J=35214$ is a 5 -label and $K=132$ is a 3 -label. For a fixed positive integer $p$, where $p \leq n$, consider consecutive blocks of $p$ numbers in an $n$-label. For example, when $p=3$ and $L=263415$, the blocks are 263,634,341, and... | null | true | null | null | null | TP_MM_maths_en_COMP | |
2881 | Combinatorics | null | An $\boldsymbol{n}$-label is a permutation of the numbers 1 through $n$. For example, $J=35214$ is a 5 -label and $K=132$ is a 3 -label. For a fixed positive integer $p$, where $p \leq n$, consider consecutive blocks of $p$ numbers in an $n$-label. For example, when $p=3$ and $L=263415$, the blocks are 263,634,341, and... | null | true | null | null | null | TP_MM_maths_en_COMP | |
2883 | Combinatorics | null | An $\boldsymbol{n}$-label is a permutation of the numbers 1 through $n$. For example, $J=35214$ is a 5 -label and $K=132$ is a 3 -label. For a fixed positive integer $p$, where $p \leq n$, consider consecutive blocks of $p$ numbers in an $n$-label. For example, when $p=3$ and $L=263415$, the blocks are 263,634,341, and... | null | true | null | null | null | TP_MM_maths_en_COMP | |
2928 | Geometry | null | A king strapped for cash is forced to sell off his kingdom $U=\left\{(x, y): x^{2}+y^{2} \leq 1\right\}$. He sells the two circular plots $C$ and $C^{\prime}$ centered at $\left( \pm \frac{1}{2}, 0\right)$ with radius $\frac{1}{2}$. The retained parts of the kingdom form two regions, each bordered by three arcs of circ... | null | true | null | null | null | TP_MM_maths_en_COMP | |
2934 | Geometry | null | A king strapped for cash is forced to sell off his kingdom $U=\left\{(x, y): x^{2}+y^{2} \leq 1\right\}$. He sells the two circular plots $C$ and $C^{\prime}$ centered at $\left( \pm \frac{1}{2}, 0\right)$ with radius $\frac{1}{2}$. The retained parts of the kingdom form two regions, each bordered by three arcs of circ... | null | true | null | null | null | TP_MM_maths_en_COMP | |
2935 | Geometry | null | A king strapped for cash is forced to sell off his kingdom $U=\left\{(x, y): x^{2}+y^{2} \leq 1\right\}$. He sells the two circular plots $C$ and $C^{\prime}$ centered at $\left( \pm \frac{1}{2}, 0\right)$ with radius $\frac{1}{2}$. The retained parts of the kingdom form two regions, each bordered by three arcs of circ... | null | true | null | null | null | TP_MM_maths_en_COMP | |
2936 | Geometry | null | A king strapped for cash is forced to sell off his kingdom $U=\left\{(x, y): x^{2}+y^{2} \leq 1\right\}$. He sells the two circular plots $C$ and $C^{\prime}$ centered at $\left( \pm \frac{1}{2}, 0\right)$ with radius $\frac{1}{2}$. The retained parts of the kingdom form two regions, each bordered by three arcs of circ... | null | true | null | null | null | TP_MM_maths_en_COMP | |
2937 | Geometry | null | A king strapped for cash is forced to sell off his kingdom $U=\left\{(x, y): x^{2}+y^{2} \leq 1\right\}$. He sells the two circular plots $C$ and $C^{\prime}$ centered at $\left( \pm \frac{1}{2}, 0\right)$ with radius $\frac{1}{2}$. The retained parts of the kingdom form two regions, each bordered by three arcs of circ... | null | true | null | null | null | TP_MM_maths_en_COMP | |
2938 | Geometry | null | A king strapped for cash is forced to sell off his kingdom $U=\left\{(x, y): x^{2}+y^{2} \leq 1\right\}$. He sells the two circular plots $C$ and $C^{\prime}$ centered at $\left( \pm \frac{1}{2}, 0\right)$ with radius $\frac{1}{2}$. The retained parts of the kingdom form two regions, each bordered by three arcs of circ... | null | true | null | null | null | TP_MM_maths_en_COMP | |
2939 | Geometry | null | A king strapped for cash is forced to sell off his kingdom $U=\left\{(x, y): x^{2}+y^{2} \leq 1\right\}$. He sells the two circular plots $C$ and $C^{\prime}$ centered at $\left( \pm \frac{1}{2}, 0\right)$ with radius $\frac{1}{2}$. The retained parts of the kingdom form two regions, each bordered by three arcs of circ... | null | true | null | null | null | TP_MM_maths_en_COMP | |
2941 | Geometry | null | A king strapped for cash is forced to sell off his kingdom $U=\left\{(x, y): x^{2}+y^{2} \leq 1\right\}$. He sells the two circular plots $C$ and $C^{\prime}$ centered at $\left( \pm \frac{1}{2}, 0\right)$ with radius $\frac{1}{2}$. The retained parts of the kingdom form two regions, each bordered by three arcs of circ... | null | true | null | null | null | TP_MM_maths_en_COMP | |
2942 | Geometry | null | A king strapped for cash is forced to sell off his kingdom $U=\left\{(x, y): x^{2}+y^{2} \leq 1\right\}$. He sells the two circular plots $C$ and $C^{\prime}$ centered at $\left( \pm \frac{1}{2}, 0\right)$ with radius $\frac{1}{2}$. The retained parts of the kingdom form two regions, each bordered by three arcs of circ... | null | true | null | null | null | TP_MM_maths_en_COMP | |
2943 | Combinatorics | null | A king strapped for cash is forced to sell off his kingdom $U=\left\{(x, y): x^{2}+y^{2} \leq 1\right\}$. He sells the two circular plots $C$ and $C^{\prime}$ centered at $\left( \pm \frac{1}{2}, 0\right)$ with radius $\frac{1}{2}$. The retained parts of the kingdom form two regions, each bordered by three arcs of circ... | null | true | null | null | null | TP_MM_maths_en_COMP | |
2944 | Combinatorics | null | A king strapped for cash is forced to sell off his kingdom $U=\left\{(x, y): x^{2}+y^{2} \leq 1\right\}$. He sells the two circular plots $C$ and $C^{\prime}$ centered at $\left( \pm \frac{1}{2}, 0\right)$ with radius $\frac{1}{2}$. The retained parts of the kingdom form two regions, each bordered by three arcs of circ... | null | true | null | null | null | TP_MM_maths_en_COMP | |
2945 | Combinatorics | null | A king strapped for cash is forced to sell off his kingdom $U=\left\{(x, y): x^{2}+y^{2} \leq 1\right\}$. He sells the two circular plots $C$ and $C^{\prime}$ centered at $\left( \pm \frac{1}{2}, 0\right)$ with radius $\frac{1}{2}$. The retained parts of the kingdom form two regions, each bordered by three arcs of circ... | null | true | null | null | null | TP_MM_maths_en_COMP | |
2946 | Combinatorics | null | A king strapped for cash is forced to sell off his kingdom $U=\left\{(x, y): x^{2}+y^{2} \leq 1\right\}$. He sells the two circular plots $C$ and $C^{\prime}$ centered at $\left( \pm \frac{1}{2}, 0\right)$ with radius $\frac{1}{2}$. The retained parts of the kingdom form two regions, each bordered by three arcs of circ... | null | true | null | null | null | TP_MM_maths_en_COMP | |
3052 | Combinatorics | null | The arrangement of numbers known as Pascal's Triangle has fascinated mathematicians for centuries. In fact, about 700 years before Pascal, the Indian mathematician Halayudha wrote about it in his commentaries to a then-1000-year-old treatise on verse structure by the Indian poet and mathematician Pingala, who called it... | null | true | null | null | null | TP_MM_maths_en_COMP | |
3054 | Combinatorics | null | The arrangement of numbers known as Pascal's Triangle has fascinated mathematicians for centuries. In fact, about 700 years before Pascal, the Indian mathematician Halayudha wrote about it in his commentaries to a then-1000-year-old treatise on verse structure by the Indian poet and mathematician Pingala, who called it... | null | true | null | null | null | TP_MM_maths_en_COMP | |
3055 | Combinatorics | null | The arrangement of numbers known as Pascal's Triangle has fascinated mathematicians for centuries. In fact, about 700 years before Pascal, the Indian mathematician Halayudha wrote about it in his commentaries to a then-1000-year-old treatise on verse structure by the Indian poet and mathematician Pingala, who called it... | null | true | null | null | null | TP_MM_maths_en_COMP | |
3056 | Combinatorics | null | The arrangement of numbers known as Pascal's Triangle has fascinated mathematicians for centuries. In fact, about 700 years before Pascal, the Indian mathematician Halayudha wrote about it in his commentaries to a then-1000-year-old treatise on verse structure by the Indian poet and mathematician Pingala, who called it... | null | true | null | null | null | TP_MM_maths_en_COMP | |
3059 | Combinatorics | null | The arrangement of numbers known as Pascal's Triangle has fascinated mathematicians for centuries. In fact, about 700 years before Pascal, the Indian mathematician Halayudha wrote about it in his commentaries to a then-1000-year-old treatise on verse structure by the Indian poet and mathematician Pingala, who called it... | null | true | null | null | null | TP_MM_maths_en_COMP | |
3072 | Combinatorics | null | Leibniz's Harmonic Triangle: Consider the triangle formed by the rule
$$
\begin{cases}\operatorname{Le}(n, 0)=\frac{1}{n+1} & \text { for all } n \\ \operatorname{Le}(n, n)=\frac{1}{n+1} & \text { for all } n \\ \operatorname{Le}(n, k)=\operatorname{Le}(n+1, k)+\operatorname{Le}(n+1, k+1) & \text { for all } n \text {... | null | true | null | null | null | TP_MM_maths_en_COMP | |
938 | Mechanics | 4. A complex dance
In this problem, we will solve a number of differential equations corresponding to very different physical phenomena that are unified by the idea of oscillation. Oscillations are captured elegantly by extending our notion of numbers to include the imaginary unit number $i$, strangely defined to obe... | (a) The usual form of Newton's second law $(\vec{F}=m \vec{a})$ breaks down when we go into a rotating frame, where both the centrifugal and Coriolis forces become important to account for. Newton's second law then takes the form
$$
\vec{F}=m(\vec{a}+2 \vec{v} \times \vec{\Omega}+\vec{\Omega} \times(\vec{\Omega} \time... | null | false | null | null | null | TP_TO_physics_en_COMP | |
943 | Mechanics | 4. A complex dance
In this problem, we will solve a number of differential equations corresponding to very different physical phenomena that are unified by the idea of oscillation. Oscillations are captured elegantly by extending our notion of numbers to include the imaginary unit number $i$, strangely defined to obe... | (f) If the energy of a wave is $E=\hbar \omega$ and the momentum is $p=\hbar k$, show that the dispersion relation found in part (e) resembles the classical expectation for the kinetic energy of a particle, $\mathrm{E}=\mathrm{mv}^{2} / \mathbf{2}$. | null | false | null | null | null | TP_TO_physics_en_COMP | |
946 | Electromagnetism | 5. Polarization and Oscillation
In this problem, we will understand the polarization of metallic bodies and the method of images that simplifies the math in certain geometrical configurations.
Throughout the problem, suppose that metals are excellent conductors and they polarize significantly faster than the classic... | (b) Laplace's equation is a second order differential equation
$$
\nabla^{2} \phi=\frac{\partial^{2} \phi}{\partial x^{2}}+\frac{\partial^{2} \phi}{\partial y^{2}}+\frac{\partial^{2} \phi}{\partial z^{2}}=0
\tag{8}
$$
Solutions to this equation are called harmonic functions. One of the most important properties satis... | null | false | null | null | null | TP_TO_physics_en_COMP | |
963 | Modern Physics | 4. Lorentz Boost
In Newtonian kinematics, inertial frames moving relatively to each other are related by the following transformations called Galilean boosts:
$$
\begin{aligned}
x^{\prime} & =x-v t \\
t^{\prime} & =t
\end{aligned}
$$
In relativistic kinematics, inertial frames are similarly related by the Lorentz bo... | (a) The homogeneity of space and time imply that the laws of physics are the same no matter where in space and time you are. In other words, they do not depend on a choice of origin for coordinates $x$ and $t$. Use this fact to show that $\frac{\partial X}{\partial x}$ is independent of the position $x$ and $\frac{\par... | null | false | null | null | null | TP_TO_physics_en_COMP | |
964 | Modern Physics | 4. Lorentz Boost
In Newtonian kinematics, inertial frames moving relatively to each other are related by the following transformations called Galilean boosts:
$$
\begin{aligned}
x^{\prime} & =x-v t \\
t^{\prime} & =t
\end{aligned}
$$
In relativistic kinematics, inertial frames are similarly related by the Lorentz bo... | (b) The isotropy of space implies that there is no preferred direction in the universe, i.e., that the laws of physics are the same in all directions. Use this to study the general coordinate transformations $X, T$ after setting $x \rightarrow-x$ and $x^{\prime} \rightarrow-x^{\prime}$ and conclude that $A(v), D(v)$ ar... | null | false | null | null | null | TP_TO_physics_en_COMP | |
965 | Modern Physics | 4. Lorentz Boost
In Newtonian kinematics, inertial frames moving relatively to each other are related by the following transformations called Galilean boosts:
$$
\begin{aligned}
x^{\prime} & =x-v t \\
t^{\prime} & =t
\end{aligned}
$$
In relativistic kinematics, inertial frames are similarly related by the Lorentz bo... | (c) The principle of relativity implies that the laws of physics are agreed upon by observers in inertial frames. This implies that the general coordinate transformations $X, T$ are invertible and their inverses have the same functional form as $X, T$ after setting $v \rightarrow-v$. Use this fact to show the following... | null | false | null | null | null | TP_TO_physics_en_COMP | |
966 | Modern Physics | 4. Lorentz Boost
In Newtonian kinematics, inertial frames moving relatively to each other are related by the following transformations called Galilean boosts:
$$
\begin{aligned}
x^{\prime} & =x-v t \\
t^{\prime} & =t
\end{aligned}
$$
In relativistic kinematics, inertial frames are similarly related by the Lorentz bo... | (d) Use the previous results and the fact that the location of the $F^{\prime}$ frame may be given by $x=v t$ in the $F$ frame to conclude that the coordinate transformations have the following form:
$$
\begin{aligned}
x^{\prime} & =A(v) x-v A(v) t \\
t^{\prime} & =-\left(\frac{A(v)^{2}-1}{v A(v)}\right) x+A(v) t
\end... | null | false | null | null | null | TP_TO_physics_en_COMP | |
967 | Modern Physics | 4. Lorentz Boost
In Newtonian kinematics, inertial frames moving relatively to each other are related by the following transformations called Galilean boosts:
$$
\begin{aligned}
x^{\prime} & =x-v t \\
t^{\prime} & =t
\end{aligned}
$$
In relativistic kinematics, inertial frames are similarly related by the Lorentz bo... | (e) Assume that a composition of boosts results in a boost of the same functional form. Use this fact and all the previous results you have derived about these generalized boosts to conclude that
$$
\frac{A(v)^{2}-1}{v^{2} A(v)}=\kappa .
$$
for an arbitrary constant $\kappa$. | null | false | null | null | null | TP_TO_physics_en_COMP | |
968 | Modern Physics | 4. Lorentz Boost
In Newtonian kinematics, inertial frames moving relatively to each other are related by the following transformations called Galilean boosts:
$$
\begin{aligned}
x^{\prime} & =x-v t \\
t^{\prime} & =t
\end{aligned}
$$
In relativistic kinematics, inertial frames are similarly related by the Lorentz bo... | (f) (1 point) Show that $\kappa$ has dimensions of (velocity $)^{-2}$, and show that the generalized boost now has the form
$$
\begin{aligned}
x^{\prime} & =\frac{1}{\sqrt{1-\kappa v^{2}}}(x-v t) \\
t^{\prime} & =\frac{1}{\sqrt{1-\kappa v^{2}}}(t-\kappa v x)
\end{aligned}
$$ | null | false | null | null | null | TP_TO_physics_en_COMP | |
969 | Modern Physics | 4. Lorentz Boost
In Newtonian kinematics, inertial frames moving relatively to each other are related by the following transformations called Galilean boosts:
$$
\begin{aligned}
x^{\prime} & =x-v t \\
t^{\prime} & =t
\end{aligned}
$$
In relativistic kinematics, inertial frames are similarly related by the Lorentz bo... | (g) Assume that $v$ may be infinite. Argue that $\kappa=0$ and show that you recover the Galilean boost. Under this assumption, explain using a Galilean boost why this implies that a particle may travel arbitrarily fast. | null | false | null | null | null | TP_TO_physics_en_COMP | |
970 | Modern Physics | 4. Lorentz Boost
In Newtonian kinematics, inertial frames moving relatively to each other are related by the following transformations called Galilean boosts:
$$
\begin{aligned}
x^{\prime} & =x-v t \\
t^{\prime} & =t
\end{aligned}
$$
In relativistic kinematics, inertial frames are similarly related by the Lorentz bo... | (h) Assume that $v$ must be smaller than a finite value. Show that $1 / \sqrt{\kappa}$ is the maximum allowable speed, and that this speed is frame invariant, i.e., $\frac{d x^{\prime}}{d t^{\prime}}=\frac{d x}{d t}$ for something moving at speed $1 / \sqrt{\kappa}$. Experiment has shown that this speed is $c$, the spe... | null | false | null | null | null | TP_TO_physics_en_COMP | |
975 | Electromagnetism | 2. Johnson-Nyquist noise
In this problem we study thermal noise in electrical circuits. The goal is to derive the JohnsonNyquist spectral (per-frequency, $f$ ) density of noise produced by a resistor, $R$ :
$$
\frac{d\left\langle V^{2}\right\rangle}{d f}=4 k T R
\tag{2}
$$
Here, \langle\rangle denotes an average ove... | (a) The electromagnetic modes travel through the ends, $x=0$ and $x=L$, of the resistor. Show that the wavevectors corresponding to periodic waves on the interval $[0, L]$ are $k_{n}=\frac{2 \pi n}{L}$.
Then, show that the number of states per angular frequency is $\frac{d n}{d \omega_{n}}=\frac{L}{2 \pi c^{\prime}}$. | null | false | null | null | null | TP_TO_physics_en_COMP | |
976 | Electromagnetism | 2. Johnson-Nyquist noise
In this problem we study thermal noise in electrical circuits. The goal is to derive the JohnsonNyquist spectral (per-frequency, $f$ ) density of noise produced by a resistor, $R$ :
$$
\frac{d\left\langle V^{2}\right\rangle}{d f}=4 k T R
\tag{2}
$$
Here, \langle\rangle denotes an average ove... | (b) Each mode $n$ in the resistor can be thought of as a species of particle, called a bosonic collective mode. This particle obeys Bose-Einstein statistics: the average number of particles $\left\langle N_{n}\right\rangle$ in the mode $n$ is
$$
\left\langle N_{n}\right\rangle=\frac{1}{\exp \frac{\hbar \omega_{n}}{k T... | null | false | null | null | null | TP_TO_physics_en_COMP | |
977 | Electromagnetism | 2. Johnson-Nyquist noise
In this problem we study thermal noise in electrical circuits. The goal is to derive the JohnsonNyquist spectral (per-frequency, $f$ ) density of noise produced by a resistor, $R$ :
$$
\frac{d\left\langle V^{2}\right\rangle}{d f}=4 k T R
\tag{2}
$$
Here, \langle\rangle denotes an average ove... | (c) By analogy to the photon, explain why the energy of each particle in the mode $n$ is $\hbar \omega_{n}$. | null | false | null | null | null | TP_TO_physics_en_COMP | |
978 | Electromagnetism | 2. Johnson-Nyquist noise
In this problem we study thermal noise in electrical circuits. The goal is to derive the JohnsonNyquist spectral (per-frequency, $f$ ) density of noise produced by a resistor, $R$ :
$$
\frac{d\left\langle V^{2}\right\rangle}{d f}=4 k T R
\tag{2}
$$
Here, \langle\rangle denotes an average ove... | (d) Using parts (a), (b), and (c), show that the average power delivered to the resistor (or produced by the resistor) per frequency interval is
$$
P[f, f+d f] \approx k T d f .
\tag{6}
$$
Here, $f=\omega / 2 \pi$ is the frequency. $P[f, f+d f]$ is known as the available noise power of the resistor. (Hint: Power is d... | null | false | null | null | null | TP_TO_physics_en_COMP | |
979 | Electromagnetism | 2. Johnson-Nyquist noise
In this problem we study thermal noise in electrical circuits. The goal is to derive the JohnsonNyquist spectral (per-frequency, $f$ ) density of noise produced by a resistor, $R$ :
$$
\frac{d\left\langle V^{2}\right\rangle}{d f}=4 k T R
\tag{2}
$$
Here, \langle\rangle denotes an average ove... | (a) Assume that resistors $R$ and $r$ are in series with a voltage $V . R$ and $V$ are fixed, but $r$ can vary. Show the maximum power dissipation across $r$ is
$$
P_{\max }=\frac{V^{2}}{4 R} .
\tag{7}
$$
Give the optimal value of $r$ in terms of $R$ and $V$. | null | false | null | null | null | TP_TO_physics_en_COMP | |
980 | Electromagnetism | 2. Johnson-Nyquist noise
In this problem we study thermal noise in electrical circuits. The goal is to derive the JohnsonNyquist spectral (per-frequency, $f$ ) density of noise produced by a resistor, $R$ :
$$
\frac{d\left\langle V^{2}\right\rangle}{d f}=4 k T R
\tag{2}
$$
Here, \langle\rangle denotes an average ove... | (b) If the average power per frequency interval delivered to the resistor $r$ is $\frac{d\left\langle P_{\max }\right\rangle}{d f}=$ $\frac{d E}{d f}=k T$, show that $\frac{d\left\langle V^{2}\right\rangle}{d f}=4 k T R$. | null | false | null | null | null | TP_TO_physics_en_COMP |
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