source stringclasses 1
value | problem stringlengths 53 1.53k | solution stringlengths 449 6.61k | messages listlengths 2 2 |
|---|---|---|---|
olympiads | The diagram shows two circular arcs \( CP \) and \( CQ \) in a right-angled triangle \( ABC \), where \(\angle BCA = 90^\circ\). The centers of the arcs are \( A \) and \( B \). Prove that \(\frac{1}{2} PQ^2 = AP \times BQ\). |
1. **Define Lengths and Apply Pythagoras' Theorem:**
Let the lengths of sides \(BC\), \(CA\), and \(AB\) of the right-angled triangle \(ABC\) be \(a\), \(b\), and \(c\) respectively, meaning:
\[
BC = a, \quad CA = b, \quad AB = c
\]
Given that \(\angle BCA = 90^\circ\), we apply the Pythagorean theo... | [
{
"content": "The diagram shows two circular arcs \\( CP \\) and \\( CQ \\) in a right-angled triangle \\( ABC \\), where \\(\\angle BCA = 90^\\circ\\). The centers of the arcs are \\( A \\) and \\( B \\). Prove that \\(\\frac{1}{2} PQ^2 = AP \\times BQ\\).",
"role": "user"
},
{
"content": "\n1. **D... |
olympiads | In the diagram, a newly built city's roads form a neat grid. There are $n+1$ east-west roads and $m+1$ north-south roads. A person lives at point $A$ in the southwest corner and rides a bicycle to work at point $B$ in the northeast corner every day. To sightsee along the way, this person often changes their route, with... |
To solve the problem, we need to show that the number of different routes from $A$ to $B$, denoted by $f(m, n)$, satisfies the inequality \( f(m, n) \leqslant 2^{mn} \).
1. **Identify Grid Divisions**:
The road system comprises $n+1$ east-west streets and $m+1$ north-south streets, which divide the city into \(... | [
{
"content": "In the diagram, a newly built city's roads form a neat grid. There are $n+1$ east-west roads and $m+1$ north-south roads. A person lives at point $A$ in the southwest corner and rides a bicycle to work at point $B$ in the northeast corner every day. To sightsee along the way, this person often cha... |
olympiads | The diagram shows eight circles of two different sizes. The circles are arranged in concentric pairs so that the centers form a square. Each larger circle touches one other larger circle and two smaller circles. The larger circles have a radius of 1. What is the radius of each smaller circle?
A) \(\frac{1}{3}\)
B) \(\f... |
1. **Identify the setup**:
- Let the centers of the circles be \(P, Q, R,\) and \(S\).
- The larger circles each have a radius of 1.
- Let the radius of the smaller circles be \(r\).
2. **Determine the distances**:
- The distance \(PQ\) is the sum of a larger circle's radius and the smaller circle's ... | [
{
"content": "The diagram shows eight circles of two different sizes. The circles are arranged in concentric pairs so that the centers form a square. Each larger circle touches one other larger circle and two smaller circles. The larger circles have a radius of 1. What is the radius of each smaller circle?\nA) ... |
olympiads | The diagram shows a circle and a trapezium \(ABCD\) in which \(AD\) is parallel to \(BC\) and \(AB = DC\). All four sides of \(ABCD\) are tangents to the circle. The circle has a radius of 4, and the area of \(ABCD\) is 72. What is the length of \(AB\)? |
1. **Understand the Geometry**:
We are given a trapezium \(ABCD\) with \(AD \parallel BC\) and \(AB = DC\). All sides are tangents to a circle with radius \(4\). Hence, the height of the trapezium (distance between parallel sides) is equal to the diameter of the circle, which is \(2 \times 4 = 8\).
2. **Area Calcu... | [
{
"content": "The diagram shows a circle and a trapezium \\(ABCD\\) in which \\(AD\\) is parallel to \\(BC\\) and \\(AB = DC\\). All four sides of \\(ABCD\\) are tangents to the circle. The circle has a radius of 4, and the area of \\(ABCD\\) is 72. What is the length of \\(AB\\)?",
"role": "user"
},
{
... |
olympiads | As shown in the diagram, the distance between any two adjacent points in a row is 1, and the distance between any two adjacent points in a column is also 1. Every second, a green point can turn any point that is at a distance of no more than 1 into green. Given the initial state shown in the diagram, how many red point... | ### Problem:
Given a grid where each point has a distance of 1 from its neighboring points both horizontally and vertically, suppose initially we have some green points. Every second, each green point can spread to points that are at most 1 distance away, converting them to green. Determine how many red points remain a... | [
{
"content": "As shown in the diagram, the distance between any two adjacent points in a row is 1, and the distance between any two adjacent points in a column is also 1. Every second, a green point can turn any point that is at a distance of no more than 1 into green. Given the initial state shown in the diagr... |
olympiads | In the diagram, \( T U \) is parallel to \( P S \) and points \( Q \) and \( R \) lie on \( P S \). Also, \( \angle P Q T = x^{\circ} \), \( \angle R Q T = (x-50)^{\circ} \), and \( \angle T U R = (x+25)^{\circ} \).
What is the measure of \( \angle U R S \) ?
(A) \( 115^{\circ} \)
(B) \( 140^{\circ} \)
(C) \( 135^{\c... | 1. First, we recognize that since $PQR$ forms a straight line, the sum of the angles $\angle P Q T$ and $\angle R Q T$ should give $180^\circ$. Hence,
\[
\angle P Q T + \angle R Q T = 180^\circ.
\]
2. Given the angle measures, we can write:
\[
x^\circ + (x - 50)^\circ = 180^\circ.
\]
3. Simplifyi... | [
{
"content": "In the diagram, \\( T U \\) is parallel to \\( P S \\) and points \\( Q \\) and \\( R \\) lie on \\( P S \\). Also, \\( \\angle P Q T = x^{\\circ} \\), \\( \\angle R Q T = (x-50)^{\\circ} \\), and \\( \\angle T U R = (x+25)^{\\circ} \\).\n\nWhat is the measure of \\( \\angle U R S \\) ?\n(A) \\( 1... |
olympiads | In the diagram, the \(5 \times 6\) grid is made out of thirty \(1 \times 1\) squares. What is the total length of the six solid line segments shown?
(A) 6
(B) 12
(C) 16
(D) 18
(E) 20
 | :
To determine the total length of the six solid line segments shown in the diagram, we follow these steps:
1. **Identify each individual segment length**:
- The lengths of the solid line segments from top to bottom are:
\[
5, 1, 4, 2, 3, \text{and} 3
\]
2. **Sum these lengths**:
\[
5 + 1 + 4... | [
{
"content": "In the diagram, the \\(5 \\times 6\\) grid is made out of thirty \\(1 \\times 1\\) squares. What is the total length of the six solid line segments shown?\n(A) 6\n(B) 12\n(C) 16\n(D) 18\n(E) 20\n\n grid along the dashed line such that each resulting rectangle contains exactly one number, and that number should equal the area of the rectangle. Specifically, it asks us to determine to how many different rectangles each of the 7 small squares in the 4th colum... | [
{
"content": "As shown in the diagram, divide a $7 \\times 7$ grid along the dotted lines into several rectangles such that each rectangle contains exactly one number, and this number equals the area of the rectangle. The seven small squares in the fourth column belong to different rectangles.",
"role": "us... |
olympiads | As shown in the diagram, \( AB \) is the diameter of a semicircle, \( O \) is the center of the circle, \( CD \perp AB \), and \( DE \perp OC \). If the lengths of \( AD \), \( BD \), and \( CD \) are all rational numbers, then among the following three statements:
- Statement 1: The length of \( OE \) is a rational n... | 1. We are given that $AD, BD$ and $CD$ are all rational numbers.
2. Because $O$ is the center of the semicircle and $AB$ is the diameter, we have $OA = OB = \frac{AB}{2}$. Thus, these are also rational numbers.
3. Calculate $OC$. Since $C$ is the midpoint of $AB$, $OC = OA = OB$, which is a rational number.
4. Next, $O... | [
{
"content": "As shown in the diagram, \\( AB \\) is the diameter of a semicircle, \\( O \\) is the center of the circle, \\( CD \\perp AB \\), and \\( DE \\perp OC \\). If the lengths of \\( AD \\), \\( BD \\), and \\( CD \\) are all rational numbers, then among the following three statements:\n\n- Statement 1... |
olympiads | In the diagram, $\triangle PQR$ is right-angled at $P$ and $PR = 12$. If point $S$ is on $PQ$ so that $SQ = 11$ and $SR = 13$, the perimeter of $\triangle QRS$ is
(A) 47
(B) 44
(C) 30
(D) 41
(E) 61 | 1. Let's first determine the length of \( P S \) using the Pythagorean theorem in \( \triangle P R S \):
\[
P S = \sqrt{R S^2 - P R^2} = \sqrt{13^2 - 12^2}
\]
Calculating the squares:
\[
13^2 = 169 \quad \text{and} \quad 12^2 = 144
\]
Subtracting these, we get:
\[
P S = \sqrt{169 - 144} = ... | [
{
"content": "In the diagram, $\\triangle PQR$ is right-angled at $P$ and $PR = 12$. If point $S$ is on $PQ$ so that $SQ = 11$ and $SR = 13$, the perimeter of $\\triangle QRS$ is\n(A) 47\n(B) 44\n(C) 30\n(D) 41\n(E) 61",
"role": "user"
},
{
"content": "1. Let's first determine the length of \\( P S ... |
olympiads | In the diagram, nine identical isosceles trapezoids are connected as shown to form a closed ring. (An isosceles trapezoid is a trapezoid with the properties that one pair of sides is parallel and the other two sides are equal in length.) What is the measure of $\angle ABC$? | ## Solution
1:
1. **Determine the number of sides of the inner polygon**:
The inner polygon formed by the nine trapezoids is a regular nonagon (a 9-sided polygon). This is because:
- All sides of this polygon are equal in length, which is equal to the shorter parallel side of one of the identical trapezoid... | [
{
"content": "In the diagram, nine identical isosceles trapezoids are connected as shown to form a closed ring. (An isosceles trapezoid is a trapezoid with the properties that one pair of sides is parallel and the other two sides are equal in length.) What is the measure of $\\angle ABC$?",
"role": "user"
... |
olympiads | The word "stop" starts in the position shown in the diagram to the right. It is then rotated $180^{\circ}$ clockwise about the origin, $O$, and this result is then reflected in the $x$-axis. Which of the following represents the final image?
(A)
![](https://cdn.mathpix.com/cropped/2024_04_20_6027bc27089ed4fc493cg-089.... | :
1. **Initial Position**: Let's first understand the initial position of the word "stop".
2. **Rotation by $180^\circ$**: Rotating the word $180^\circ$ clockwise abou... | [
{
"content": "The word \"stop\" starts in the position shown in the diagram to the right. It is then rotated $180^{\\circ}$ clockwise about the origin, $O$, and this result is then reflected in the $x$-axis. Which of the following represents the final image?\n(A)\n\n is the region inside the two smaller circles but outside the largest circle. Prove that the area of \( R \) is equal to the area of the triangle. |
1. **Introduce Notations and Definitions**:
Let:
- \(T\) be the area of the triangle,
- \(S_1\) be the area of the semicircle with the hypotenuse as the diameter,
- \(S_2\) and \(S_3\) be the areas of the semicircles with the shorter sides of the triangle as diameters.
2. **Set Up the Equation**:
We ca... | [
{
"content": "The diagram shows a right-angled triangle and three circles. Each side of the triangle is a diameter of one of the circles. The shaded region \\( R \\) is the region inside the two smaller circles but outside the largest circle. Prove that the area of \\( R \\) is equal to the area of the triangle... |
olympiads | The diagram shows a 16 meter by 16 meter wall. Three grey squares are painted on the wall as shown. The two smaller grey squares are equal in size and each makes an angle of \(45^\circ\) with the edge of the wall. The grey squares cover a total area of \(B\) square meters. What is the value of \(B\)? |
1. **Initialize Dimensions and Square Calculations:**
- The wall is 16 metres by 16 metres.
- There are two smaller grey squares and one larger grey square on the wall.
- The diagonal of each smaller grey square is 8 metres since they are tilted at an angle of $45^\circ$ to the edge of the wall.
2. **Usi... | [
{
"content": "The diagram shows a 16 meter by 16 meter wall. Three grey squares are painted on the wall as shown. The two smaller grey squares are equal in size and each makes an angle of \\(45^\\circ\\) with the edge of the wall. The grey squares cover a total area of \\(B\\) square meters. What is the value o... |
olympiads | In the diagram, \(D\) is on side \(AC\) of \(\triangle ABC\) so that \(BD\) is perpendicular to \(AC\). If \(AB = 29\), \(AC = 69\), and \(BD = 20\), what is the length of \(BC\)? | Given a right triangle \( \triangle ABD \) and the values \( AB = 29 \), \( AC = 69 \), and \( BD = 20 \).
1. **Apply the Pythagorean Theorem in \( \triangle ABD \):**
\[
AD^2 + BD^2 = AB^2
\]
Substitute the known values:
\[
AD^2 + 20^2 = 29^2
\]
\[
AD^2 + 400 = 841
\]
\[
AD^2 = 841... | [
{
"content": "In the diagram, \\(D\\) is on side \\(AC\\) of \\(\\triangle ABC\\) so that \\(BD\\) is perpendicular to \\(AC\\). If \\(AB = 29\\), \\(AC = 69\\), and \\(BD = 20\\), what is the length of \\(BC\\)?",
"role": "user"
},
{
"content": "Given a right triangle \\( \\triangle ABD \\) and the... |
olympiads | In the diagram, each scale shows the total mass (weight) of the shapes on that scale. What is the mass (weight) of a $\triangle$?
(A) 3
(B) 5
(C) 12
(D) 6
(E) 5.5 |
1. **Interpret the first scale**:
The first scale shows that a square \( S \) and a circle \( C \) together have a mass of 8.
\[
S + C = 8
\]
2. **Interpret the second scale**:
The second scale shows that a square \( S \) and two circles \( C \) together have a mass of 11.
\[
S + 2C = 11
\]... | [
{
"content": "In the diagram, each scale shows the total mass (weight) of the shapes on that scale. What is the mass (weight) of a $\\triangle$?\n(A) 3\n(B) 5\n(C) 12\n(D) 6\n(E) 5.5",
"role": "user"
},
{
"content": "\n1. **Interpret the first scale**:\n The first scale shows that a square \\( S \... |
olympiads | As shown in the diagram, five equilateral triangles of side length $2\sqrt{3}$ are arranged on the same side of a straight line, such that the midpoint of the base of each triangle is the vertex of the next triangle. What is the area covered by these five triangles in the plane?
(A) 10
(B) 12
(C) 15
(D) $10 \sqrt{3}$
(... |
1. **Understand the Problem:**
We have five equilateral triangles arranged in a line, with each triangle having a side length of \(2\sqrt{3}\). Every middle point of the base of one triangle is the vertex of the next triangle.
2. **Calculate the Area of a Single Equilateral Triangle:**
The area of an equila... | [
{
"content": "As shown in the diagram, five equilateral triangles of side length $2\\sqrt{3}$ are arranged on the same side of a straight line, such that the midpoint of the base of each triangle is the vertex of the next triangle. What is the area covered by these five triangles in the plane?\n(A) 10\n(B) 12\n... |
olympiads | In the cube \(A B C D - A_{1} B_{1} C_{1} D_{1}\), \(E\) is the trisection point of \(A_{1} A\), and \(F\) is the trisection point of \(C_{1} C\). Given that \(A E = 2 A_{1} E\) and \(C F = 2 C_{1} F\), a section is formed through \(B\), \(E\), and \(F\). Among the projection diagrams of this section on the respective ... |
1. **Identifying Points and Division of Line Segments**:
- In the cube \(A B C D - A_1 B_1 C_1 D_1\), point \(E\) is defined as the point that divides segment \(A_1 A\) into three equal parts. This means \(A_1E = \frac{1}{3}A_1A\) and \(AE = \frac{2}{3}A_1A\).
- Similarly, point \(F\) is defined as the point tha... | [
{
"content": "In the cube \\(A B C D - A_{1} B_{1} C_{1} D_{1}\\), \\(E\\) is the trisection point of \\(A_{1} A\\), and \\(F\\) is the trisection point of \\(C_{1} C\\). Given that \\(A E = 2 A_{1} E\\) and \\(C F = 2 C_{1} F\\), a section is formed through \\(B\\), \\(E\\), and \\(F\\). Among the projection d... |
olympiads | In the diagram, \( A M N, A P Q, Q R M \), and \( P R N \) are all straight lines. The value of \( a+b \) is:
(A) 70
(B) 55
(C) 80
(D) 90
(E) 75 | 1. **Identifying angle relationships:**
- We start by noting that $\angle PRM = 125^\circ$.
- Since $QR$ and $RM$ are straight lines, the angles around point $R$ must sum to $360^\circ$. Specifically, $\angle QRP = \angle NRM = 55^\circ$ because $PRM + QRP = 180^\circ$ and $PRM + NRM = 180^\circ$.
This gives... | [
{
"content": "In the diagram, \\( A M N, A P Q, Q R M \\), and \\( P R N \\) are all straight lines. The value of \\( a+b \\) is:\n(A) 70\n(B) 55\n(C) 80\n(D) 90\n(E) 75",
"role": "user"
},
{
"content": "1. **Identifying angle relationships:** \n - We start by noting that $\\angle PRM = 125^\\circ... |
olympiads | As shown in the diagram, quadrilateral \(ABCD\) is inscribed in a circle. The extensions of \(AB\) and \(DC\) intersect at \(E\), and the extensions of \(AD\) and \(BC\) intersect at \(F\). Point \(P\) is an arbitrary point on the circle, and \(PE\) and \(PF\) intersect the circle again at \(R\) and \(S\), respectively... |
1. Extend lines \(AB\) and \(DC\) to meet at point \(E\), and extend lines \(AD\) and \(BC\) to meet at point \(F\). Let \(P\) be any point on the circumcircle, and let \(PE\) and \(PF\) intersect the circle again at points \(R\) and \(S\), respectively. The diagonals \(AC\) and \(BD\) intersect at point \(T\).
2. We... | [
{
"content": "As shown in the diagram, quadrilateral \\(ABCD\\) is inscribed in a circle. The extensions of \\(AB\\) and \\(DC\\) intersect at \\(E\\), and the extensions of \\(AD\\) and \\(BC\\) intersect at \\(F\\). Point \\(P\\) is an arbitrary point on the circle, and \\(PE\\) and \\(PF\\) intersect the cir... |
olympiads | In the diagram, the circle is inscribed in the square. This means that the circle and the square share points \(S, T, U,\) and \(V\), and the width of the square is exactly equal to the diameter of the circle. Rounded to the nearest tenth, what percentage of line segment \(XY\) is outside the circle?
(A) 29.3
(B) 28.3
... | 1. **Assume the Side Length of the Square:**
Suppose the side length of the square is \( 2 \). This assumption doesn't affect the final percentage because the relationship between the circle and the square remains the same regardless of the actual side length.
2. **Determine the Diameter of the Circle:**
Since t... | [
{
"content": "In the diagram, the circle is inscribed in the square. This means that the circle and the square share points \\(S, T, U,\\) and \\(V\\), and the width of the square is exactly equal to the diameter of the circle. Rounded to the nearest tenth, what percentage of line segment \\(XY\\) is outside th... |
olympiads | Four squares with integer side lengths are arranged as shown in the diagram, with three vertices of square \( A \) being the centers of squares \( B \), \( C \), and \( D \). If the total area of the red parts is equal to the area of the green part, what is the minimum side length of square \( A \)? | To find the minimum edge length of the square \( \mathrm{A} \), given the conditions, let's break down the solution step-by-step.
1. **Define Variables:**
- Let the side length of square \( \mathrm{A} \) be denoted by \( a \).
- Let the side lengths of squares \( \mathrm{B}, \mathrm{C}, \) and \( \mathrm{D} \) r... | [
{
"content": "Four squares with integer side lengths are arranged as shown in the diagram, with three vertices of square \\( A \\) being the centers of squares \\( B \\), \\( C \\), and \\( D \\). If the total area of the red parts is equal to the area of the green part, what is the minimum side length of squar... |
olympiads | As shown in the diagram, from a point \( C \) on the semicircle, a perpendicular is drawn to the diameter \( AB \) with foot \( D \). Construct a circle \( \odot O_1 \) that is tangent to \(\overparen{BC}, CD\), and \(DB\) at points \(E, F, G\) respectively. Prove that \(AC = AG\). |
1. Let's denote the center of the semicircle as \( O \). So, we have \( O \), \( O_1 \), and \( E \) collinear.
2. Connect \( O_1F \). Since \( O_1F \perp CD \), it follows that \( O_1F \parallel AB \).
3. Next, connect \( EF \) and \( AE \). Observe that:
\[
\angle FEO_1 = \frac{1}{2} \angle FO_1O = \frac{... | [
{
"content": "As shown in the diagram, from a point \\( C \\) on the semicircle, a perpendicular is drawn to the diameter \\( AB \\) with foot \\( D \\). Construct a circle \\( \\odot O_1 \\) that is tangent to \\(\\overparen{BC}, CD\\), and \\(DB\\) at points \\(E, F, G\\) respectively. Prove that \\(AC = AG\\... |
olympiads | In the diagram, eight circles, each with a radius of 1, are drawn inside a rectangle. Four of the circles are tangent to two sides of the rectangle and to two other circles. The other four circles are tangent to one side of the rectangle and to three other circles. A region has been shaded, consisting of three spaces (... |
1. **Shaded Circles' Areas**:
- Each of the four shaded circles has a radius of $1$.
- The area of one circle is given by:
\[
\text{Area of one circle} = \pi \cdot 1^2 = \pi
\]
- Therefore, the total area of the four shaded circles is:
\[
4 \cdot \pi = 4\pi
\... | [
{
"content": "In the diagram, eight circles, each with a radius of 1, are drawn inside a rectangle. Four of the circles are tangent to two sides of the rectangle and to two other circles. The other four circles are tangent to one side of the rectangle and to three other circles. A region has been shaded, consis... |
olympiads | Sixteen straight segments. A sales agent set out in his car from the point indicated in the diagram, planning to cover a distance of 76 km, which consists of 16 straight segments, without traveling the same segment twice. The dots represent settlements located 1 km apart from each other, and the lines represent the rou... |
To solve this problem, we must determine a route of 76 km composed of 16 straight segments that allows the traveling agent to visit all but exactly 3 inhabited points (cities) when considering the constraints.
Let's devise the method step-by-step to ensure correctness:
1. **Total Distance and Segments Calculation**... | [
{
"content": "Sixteen straight segments. A sales agent set out in his car from the point indicated in the diagram, planning to cover a distance of 76 km, which consists of 16 straight segments, without traveling the same segment twice. The dots represent settlements located 1 km apart from each other, and the l... |
olympiads | In the diagram, \( AB \) is a chord of the circle \(\omega\). \( P \) is a point on the arc \( AB \) inside the circle. Points \( E \) and \( F \) are on the line segment \( AB \), such that \( AE = EF = FB \). Lines \( PE \) and \( PF \) are extended to intersect the circle \(\omega\) at points \( C \) and \( D \) res... |
1. **Identify Given Data and Initial Setup:**
- $AB$ is a chord of circle $\omega$.
- $P$ is a point on the chord $AB$.
- Points $E$ and $F$ lie on the segment $AB$ such that $AE = EF = FB$.
- $PE$ and $PF$, extended, intersect the circle $\omega$ again at points $C$ and $D$, respectively.
2. **Draw C... | [
{
"content": "In the diagram, \\( AB \\) is a chord of the circle \\(\\omega\\). \\( P \\) is a point on the arc \\( AB \\) inside the circle. Points \\( E \\) and \\( F \\) are on the line segment \\( AB \\), such that \\( AE = EF = FB \\). Lines \\( PE \\) and \\( PF \\) are extended to intersect the circle \... |
olympiads |
The centers \( O_1, O_2 \), and \( O_3 \) of three non-intersecting circles of the same radius are located at the vertices of a triangle. From the points \( O_1, O_2 \), and \( O_3 \), tangents are drawn to these circles as shown in the diagram. It is known that these tangents intersect to form a convex hexagon whose ... |
1. Let's denote the points of tangency as \( X_1, X_2, Y_1, Y_2, Z_1, Z_2 \), and the vertices of the hexagon as \( A, B, C, D, E, F \) (see the diagram for reference).
2. Since the radii of all the circles are equal, we have:
\[
X_1O_2 = O_1Y_2, \quad Y_1O_3 = O_2Z_2, \quad Z_1O_1 = O_3X_2
\]
3. Rewrite th... | [
{
"content": "\nThe centers \\( O_1, O_2 \\), and \\( O_3 \\) of three non-intersecting circles of the same radius are located at the vertices of a triangle. From the points \\( O_1, O_2 \\), and \\( O_3 \\), tangents are drawn to these circles as shown in the diagram. It is known that these tangents intersect ... |
olympiads | In the diagram, in the plane Cartesian coordinate system \( XOY \), the rhombus \( ABCD \) has a side length of 4, and \( |OB| = |OD| = 6 \).
1. Prove that \( |OA| \cdot |OC| \) is a constant value.
2. When point \( A \) moves on the semicircle \( (x-2)^{2}+y^{2}=4 \) with \( 2 \leq x \leq 4 \), find the trajectory of ... | ### Problem:
In the plane rectangular coordinate system \(X O Y\), the sides of the rhombus \(ABCD\) are all 4 units long, and \( |OB| = |OD| = 6 \).
1. Prove that the product \( |O A| \cdot |O C| \) is a constant value.
2. When point \( A \) moves on the semicircle \((x-2)^{2} + y^{2} = 4 \) for \(2 \leqslant x \leq... | [
{
"content": "In the diagram, in the plane Cartesian coordinate system \\( XOY \\), the rhombus \\( ABCD \\) has a side length of 4, and \\( |OB| = |OD| = 6 \\).\n1. Prove that \\( |OA| \\cdot |OC| \\) is a constant value.\n2. When point \\( A \\) moves on the semicircle \\( (x-2)^{2}+y^{2}=4 \\) with \\( 2 \\l... |
olympiads | As shown in the diagram above, the vertices of a regular decagon are colored alternately black and white. We would like to draw colored line segments between the points in such a way that:
(a) Every black point is connected to every white point by a line segment.
(b) No two line segments of the same color intersect, ... |
To solve this problem, we need to determine how many colors are required to draw line segments that connect every black point to every white point on a regular decagon, adhering to the conditions mentioned. Let's go through the steps to find the minimum number of colors required:
1. **Identify the constraints**:
... | [
{
"content": "As shown in the diagram above, the vertices of a regular decagon are colored alternately black and white. We would like to draw colored line segments between the points in such a way that:\n\n(a) Every black point is connected to every white point by a line segment.\n\n(b) No two line segments of ... |
olympiads | Using the four 4-square tiles in the bottom left diagram, we created the shape shown in the bottom right diagram. If we fill each of the 16 squares in the right diagram with the numbers 1, 3, 5, and 7 (each number is used once per square), such that each row and each column contains all different numbers, and the four ... |
1. **Understand the Problem Statement:**
- We are given a \(4 \times 4\) grid with 16 squares.
- We need to fill each square with the numbers 1, 3, 5, and 7.
- Each number must be placed such that:
- No row or column contains repeated numbers.
- No 2x2 subgrid (which are the individual pieces of the... | [
{
"content": "Using the four 4-square tiles in the bottom left diagram, we created the shape shown in the bottom right diagram. If we fill each of the 16 squares in the right diagram with the numbers 1, 3, 5, and 7 (each number is used once per square), such that each row and each column contains all different ... |
olympiads | Given three circles of the same radius on a plane, prove that if all of them intersect at a single point (as shown in the diagram), the sum of the marked arcs $AK$, $CK$, and $EK$ is equal to $180^\circ$. |
1. Let's denote the centers of the given circles as \( O_1, O_2, \) and \( O_3 \).
2. Define the angles as follows:
\[
\angle C O_1 K = \alpha, \quad \angle A O_2 K = \beta, \quad \angle E O_3 K = \gamma.
\]
3. Our aim is to prove that \( \alpha + \beta + \gamma = 180^\circ \).
4. Note that the quadrilatera... | [
{
"content": "Given three circles of the same radius on a plane, prove that if all of them intersect at a single point (as shown in the diagram), the sum of the marked arcs $AK$, $CK$, and $EK$ is equal to $180^\\circ$.",
"role": "user"
},
{
"content": "\n1. Let's denote the centers of the given cir... |
olympiads | The diagram shows a rectangle inside an isosceles triangle. The base of the triangle is \( n \) times the base of the rectangle, for some integer \( n \) greater than 1. Prove that the rectangle occupies a fraction \( \frac{2}{n} - \frac{2}{n^{2}} \) of the total area. |
We are given an isosceles triangle with a rectangle inscribed in such a way that the base of the triangle is \( n \) times the base of the rectangle, where \( n \) is an integer greater than 1. We need to find the fraction of the area of the triangle that is occupied by the rectangle.
Let:
- \( AB = b \) be the base ... | [
{
"content": "The diagram shows a rectangle inside an isosceles triangle. The base of the triangle is \\( n \\) times the base of the rectangle, for some integer \\( n \\) greater than 1. Prove that the rectangle occupies a fraction \\( \\frac{2}{n} - \\frac{2}{n^{2}} \\) of the total area.",
"role": "user"... |
olympiads | In the diagram, $\mathrm{ABCD}$ is a right trapezoid with $\angle \mathrm{DAB} = \angle \mathrm{ABC} = 90^\circ$. A rectangle $\mathrm{ADEF}$ is constructed externally along $\mathrm{AD}$, with an area of 6.36 square centimeters. Line $\mathrm{BE}$ intersects $\mathrm{AD}$ at point $\mathrm{P}$, and line $\mathrm{PC}$ ... |
1. **Setup and Given Information:**
- We are given a right-angled trapezoid \( \mathrm{ABCD} \) such that \( \angle \mathrm{DAB} = \angle \mathrm{ABC} = 90^\circ \).
- Extend \( \mathrm{AD} \) to create a rectangle \( \mathrm{ADEF} \) with area \( 6.36 \) square meters.
- Connect \( \mathrm{BE} \), intersecti... | [
{
"content": "In the diagram, $\\mathrm{ABCD}$ is a right trapezoid with $\\angle \\mathrm{DAB} = \\angle \\mathrm{ABC} = 90^\\circ$. A rectangle $\\mathrm{ADEF}$ is constructed externally along $\\mathrm{AD}$, with an area of 6.36 square centimeters. Line $\\mathrm{BE}$ intersects $\\mathrm{AD}$ at point $\\ma... |
olympiads | As shown in the diagram, circles \(\odot \mathrm{O}_{1}, \odot \mathrm{O}_{2}\), and \(\odot \mathrm{O}_{3}\) intersect at point \(P\). A straight line intersects \(\odot \mathrm{O}_{1}\) and \(\odot \mathrm{O}_{2}\) at points \(B\) and \(C\), respectively. The extension of \(\mathrm{AP}\) intersects \(\odot \mathrm{O}... | 1. **Identify Key Points and Relations**
Let $\odot \mathrm{O}_{1}$, $\odot \mathrm{O}_{2}$, and $\odot \mathrm{O}_{3}$ be circles intersecting at point $P$. A line intersects $\odot \mathrm{O}_{1}$ and $\odot \mathrm{O}_{2}$ at points $B$ and $C$ respectively. The extension of $AP$ intersects $\odot \mathrm{O}_{3}$... | [
{
"content": "As shown in the diagram, circles \\(\\odot \\mathrm{O}_{1}, \\odot \\mathrm{O}_{2}\\), and \\(\\odot \\mathrm{O}_{3}\\) intersect at point \\(P\\). A straight line intersects \\(\\odot \\mathrm{O}_{1}\\) and \\(\\odot \\mathrm{O}_{2}\\) at points \\(B\\) and \\(C\\), respectively. The extension of... |
olympiads | Eight paper squares of size $2 \times 2$ were sequentially placed on a table to form a larger square of size $4 \times 4$. The last square placed was square $E$. The diagram shows how the squares are visible: square $E$ is fully visible, while the other squares are partially visible. Which square was placed on the tabl... |
1. **Understanding the Problem:**
We start with the description of the squares and how they were placed. Given the constraints that the square \( E \) was the last to be placed on the table, and all other squares are partially visible, we proceed to determine the order they were laid.
2. **Identification of Squar... | [
{
"content": "Eight paper squares of size $2 \\times 2$ were sequentially placed on a table to form a larger square of size $4 \\times 4$. The last square placed was square $E$. The diagram shows how the squares are visible: square $E$ is fully visible, while the other squares are partially visible. Which squar... |
olympiads | In the given diagram, in the cube \( A B C D - A_{1} B_{1} C_{1} D_{1} \), what is the measure of the dihedral angle \( B - A_{1} C - D \)? |
1. Let's denote the side length of the given cube $ ABCD-A_1B_1C_1D_1 $ as $1$.
2. As shown in Fig. 4, we draw a perpendicular from $B$ to the plane defined by $A_1C$ at point $E$ and connect $D$ to $E$.
3. Due to the properties of the cube, we know that:
\[
\triangle A_{1} D C \cong \triangle A_{1} B C
\]... | [
{
"content": "In the given diagram, in the cube \\( A B C D - A_{1} B_{1} C_{1} D_{1} \\), what is the measure of the dihedral angle \\( B - A_{1} C - D \\)?",
"role": "user"
},
{
"content": "\n1. Let's denote the side length of the given cube $ ABCD-A_1B_1C_1D_1 $ as $1$. \n\n2. As shown in Fig. 4,... |
olympiads | In the diagram, square \(ABCD\) has a side length of 4. What is the total area of the shaded regions?
(A) 4
(B) 8
(C) 9
(D) 12
(E) 16 |
1. **Understand the problem**:
We need to find the total area of the shaded regions in a square \(ABCD\) with a side length of 4 units. The square is divided into four rectangles, and each rectangle is split into two identical triangles, with some triangles being shaded.
2. **Calculate the area of the square**:
... | [
{
"content": "In the diagram, square \\(ABCD\\) has a side length of 4. What is the total area of the shaded regions?\n(A) 4\n(B) 8\n(C) 9\n(D) 12\n(E) 16",
"role": "user"
},
{
"content": "\n1. **Understand the problem**:\n We need to find the total area of the shaded regions in a square \\(ABCD\\... |
olympiads | As shown in the diagram, \( A \) and \( B \) are fixed points on two fixed lines \( AX \) and \( BY \), respectively. Points \( P \) and \( R \) are on ray \( AX \), and points \( Q \) and \( S \) are on ray \( BY \). Given \(\frac{AP}{BQ} = \frac{AR}{BS}\), let \( M \), \( N \), and \( T \) be points on segments \( AB... |
We need to show that the points \(M\), \(N\), and \(T\) are collinear given the conditions:
1. \(\frac{AP}{BQ}=\frac{AR}{BS}\).
2. \(\frac{AM}{MB}=\frac{PN}{NQ}=\frac{RT}{TS}\).
Let's start by using vectors and setting up the given conditions in vector form. Let:
- \(\overrightarrow{AB} = \boldsymbol{a}\)
- \(\overr... | [
{
"content": "As shown in the diagram, \\( A \\) and \\( B \\) are fixed points on two fixed lines \\( AX \\) and \\( BY \\), respectively. Points \\( P \\) and \\( R \\) are on ray \\( AX \\), and points \\( Q \\) and \\( S \\) are on ray \\( BY \\). Given \\(\\frac{AP}{BQ} = \\frac{AR}{BS}\\), let \\( M \\), ... |
olympiads | In the diagram, $\triangle PQR$ is right-angled at $P$ and $\angle PRQ=\theta$. A circle with center $P$ is drawn passing through $Q$. The circle intersects $PR$ at $S$ and $QR$ at $T$. If $QT=8$ and $TR=10$, determine the value of $\cos \theta$. |
1. Note the given information:
- Triangle $PQR$ is right-angled at $P$.
- $\angle PRQ = \theta$.
- A circle with center $P$ passes through $Q$.
- The circle intersects $PR$ at $S$ and $QR$ at $T$.
- $QT = 8$ and $TR = 10$.
- Therefore, $QR = QT + TR = 8 + 10 = 18$.
2. Let the radius of the circle be... | [
{
"content": "In the diagram, $\\triangle PQR$ is right-angled at $P$ and $\\angle PRQ=\\theta$. A circle with center $P$ is drawn passing through $Q$. The circle intersects $PR$ at $S$ and $QR$ at $T$. If $QT=8$ and $TR=10$, determine the value of $\\cos \\theta$.",
"role": "user"
},
{
"content": "... |
olympiads | Thirteen identical metallic rods are connected as shown in the diagram. It is known that the resistance of the entire construction is $R=8$ Ω. Determine the resistance of one rod $R_{0}$, if this construction is connected to a current source at points $A$ and $B$. |
1. **Identify the Problem and Equivalent Resistance**:
We are given a network of 13 identical metal rods connected in a specific manner, and the total resistance of the configuration between points \( A \) and \( B \) is \( R = 8 \Omega \). We need to find the resistance of a single rod, denoted as \( R_0 \).
2. ... | [
{
"content": "Thirteen identical metallic rods are connected as shown in the diagram. It is known that the resistance of the entire construction is $R=8$ Ω. Determine the resistance of one rod $R_{0}$, if this construction is connected to a current source at points $A$ and $B$.",
"role": "user"
},
{
... |
olympiads | In the diagram, \(\triangle PQR\) has \(PQ = QR = RP = 30\). Points \(S\) and \(T\) are on \(PQ\) and \(PR\), respectively, so that \(ST\) is parallel to \(QR\). Points \(V\) and \(U\) are on \(QR\) so that \(TU\) is parallel to \(PQ\) and \(SV\) is parallel to \(PR\). If \(VS + ST + TU = 35\), the length of \(VU\) is:... |
To find the length of $V U$, we will leverage the properties of the equilateral triangles formed within $\triangle PQR$ and the given condition that $VS + ST + TU = 35$.
1. **Identify the type of triangles within $\triangle PQR$:**
Since $P Q = Q R = R P = 30$, $\triangle PQR$ is an equilateral triangle, and each ... | [
{
"content": "In the diagram, \\(\\triangle PQR\\) has \\(PQ = QR = RP = 30\\). Points \\(S\\) and \\(T\\) are on \\(PQ\\) and \\(PR\\), respectively, so that \\(ST\\) is parallel to \\(QR\\). Points \\(V\\) and \\(U\\) are on \\(QR\\) so that \\(TU\\) is parallel to \\(PQ\\) and \\(SV\\) is parallel to \\(PR\\... |
olympiads | There are 16 halls in the museum, arranged as shown in the diagram. Half of these halls exhibit paintings, while the other half exhibit sculptures. From any hall, one can move to any adjacent hall (one with a common wall). During any tour of the museum, the halls alternate: a hall with paintings, a hall with sculptures... |
1. We start by recognizing that there are 16 halls in the museum. Half of them contain pictures, and the other half contain sculptures. This implies:
\[
\text{Number of halls with pictures} = \frac{16}{2} = 8
\]
and
\[
\text{Number of halls with sculptures} = \frac{16}{2} = 8
\]
2. Given that any... | [
{
"content": "There are 16 halls in the museum, arranged as shown in the diagram. Half of these halls exhibit paintings, while the other half exhibit sculptures. From any hall, one can move to any adjacent hall (one with a common wall). During any tour of the museum, the halls alternate: a hall with paintings, ... |
olympiads | As shown in the figure, there are four small squares with a side length of 1. The diagram contains a total of 9 grid points (grid points are the vertices of the small squares). If we use these grid points as the vertices, how many isosceles triangles can be formed in total? | ```markdown
### Problem:
如图是由四个边长为 \(1\) 的小正方形组成的图形, 图中共有 9 个格点 (格点即为小正方形的顶点). 如果以这些格点为顶点, 那么一共可组成多少个等腰三角形.
为了计算一共有多少个等腰三角形, 我们可以按照等脚边的长度进行分类.
1. **第一种**：
考虑同一个小正方形内的三角形. 每个小正方形有两个等腰三角形, 其中两条等长的边分别为小正方形的边. 由于图中有 4 个小正方形:
\[
4 \text{ 个正方形} \times 4 \text{ 个等腰三角形} = 16 \text{ 个}
\]
2. **第二种**：
考虑跨越两个相... | [
{
"content": "As shown in the figure, there are four small squares with a side length of 1. The diagram contains a total of 9 grid points (grid points are the vertices of the small squares). If we use these grid points as the vertices, how many isosceles triangles can be formed in total?",
"role": "user"
... |
olympiads | As shown in the diagram, a square pond with a side length of 12 meters is surrounded by grassland. There are wooden stakes at points $A$, $B$, $C$, and $D$ along the edge of the pond, and the distance between each pair of adjacent stakes $A B$, $B C$, and $C D$ is 3 meters. A goat is to be tied with a 4-meter rope to o... |
1. Given the problem is to find which point (A, B, C, or D) on a square pool's side, a goat should be tied to maximize the grazing area.
2. The square pool has a side length of 12 meters, and the goat is tied with a 4-meter rope.
3. Key distances: \( AB = BC = CD = 3 \) meters, meaning each side of the pool is divide... | [
{
"content": "As shown in the diagram, a square pond with a side length of 12 meters is surrounded by grassland. There are wooden stakes at points $A$, $B$, $C$, and $D$ along the edge of the pond, and the distance between each pair of adjacent stakes $A B$, $B C$, and $C D$ is 3 meters. A goat is to be tied wi... |
olympiads | Denis divided a triangle into nine smaller triangles, as shown in the diagram, and placed numbers in them. The numbers in the white triangles are equal to the sums of the numbers in the adjacent gray triangles. After this, Lyosha erased the numbers $1, 2, 3, 4, 5,$ and $6$ and replaced them with the letters $A, B, C, D... |
1. **Identifying Unique Sum for 6:**
- The largest number, 6, can only be the sum of three different numbers from the set \(\{1, 2, 3, 4, 5, 6\}\). The only possible combination is \(6 = 1 + 2 + 3\).
2. **Analyzing the Condition for $B, D, E$:**
- Recall the problem's condition: the sum of $D$ and $E$, along wi... | [
{
"content": "Denis divided a triangle into nine smaller triangles, as shown in the diagram, and placed numbers in them. The numbers in the white triangles are equal to the sums of the numbers in the adjacent gray triangles. After this, Lyosha erased the numbers $1, 2, 3, 4, 5,$ and $6$ and replaced them with t... |
olympiads | The king promised a reward to anyone who could plant a beautiful fruit orchard in a rocky wasteland. Two brothers heard about this. The elder brother managed to dig 18 holes (see the left diagram). He could not dig more and broke all the shovels. The king became angry and imprisoned him. Then the younger brother, Ivan,... |
To save his brother, earn money, and potentially gain half a kingdom, Ivan needs to plant six different types of trees in the vertices of six equilateral triangles in such a way that they do not overlap, with one tree in each vertex of a triangle. Here is a detailed step-by-step solution:
1. **Understanding the Probl... | [
{
"content": "The king promised a reward to anyone who could plant a beautiful fruit orchard in a rocky wasteland. Two brothers heard about this. The elder brother managed to dig 18 holes (see the left diagram). He could not dig more and broke all the shovels. The king became angry and imprisoned him. Then the ... |
olympiads | In the diagram, \(PQRS\) is a trapezoid with an area of 12. \(RS\) is twice the length of \(PQ\). The area of \(\triangle PQS\) is
(A) 3
(B) 4
(C) 5
(D) 6
(E) 8 |
By considering the properties of the trapezoid \(PQRS\), let's solve the problem step-by-step.
#### Step 1: Analyzing the trapezoid \(PQRS\)
Given:
- \(PQRS\) is a trapezoid with an area of 12.
- \(RS\) is twice the length of \(PQ\).
Since \(PQ && SR\):
#### Step 2: Assign variables
Let the length of \(PQ\) be \(a\... | [
{
"content": "In the diagram, \\(PQRS\\) is a trapezoid with an area of 12. \\(RS\\) is twice the length of \\(PQ\\). The area of \\(\\triangle PQS\\) is\n(A) 3\n(B) 4\n(C) 5\n(D) 6\n(E) 8",
"role": "user"
},
{
"content": "\nBy considering the properties of the trapezoid \\(PQRS\\), let's solve the ... |
olympiads | As shown in the diagram, it is known that $AOGE$, $BDHF$, $MKSI$, and $MLTJ$ are all squares, and $M$ is exactly the midpoint of $AB$. Points $E$, $I$, $J$, and $F$ lie on a straight line, as do points $C$, $K$, $L$, and $D$. Prove that $EI^2 + FJ^2 = CK^2 + DL^2$. |
1. **Construct Additional Geometry**:
- Construct $C'$ such that $C'I \perp AB$ and $C'I = CK$.
- Construct $D'$ such that $D'J \perp AB$ and $D'J = DL$.
- Draw perpendiculars: $EP \perp C'I$ at $P$ and $FQ \perp D'J$ at $Q$.
- Passing through $M$, draw a parallel line to $CD$ and denote the intersecti... | [
{
"content": "As shown in the diagram, it is known that $AOGE$, $BDHF$, $MKSI$, and $MLTJ$ are all squares, and $M$ is exactly the midpoint of $AB$. Points $E$, $I$, $J$, and $F$ lie on a straight line, as do points $C$, $K$, $L$, and $D$. Prove that $EI^2 + FJ^2 = CK^2 + DL^2$.",
"role": "user"
},
{
... |
olympiads | The diagram shows a solid with six triangular faces and five vertices. Andrew wants to write an integer at each of the vertices so that the sum of the numbers at the three vertices of each face is the same. He has already written the numbers 1 and 5 as shown. What is the sum of the other three numbers he will write? |
1. **Define Variables**:
- Let the three unknown integers be represented as \( x, y, z \).
2. **Analyze the Face Sums**:
- Consider the 'top' three triangular faces of the solid. Since the sum of the numbers at the vertices of each face must be the same, we set up the following equations:
\[
1 + 5 + x ... | [
{
"content": "The diagram shows a solid with six triangular faces and five vertices. Andrew wants to write an integer at each of the vertices so that the sum of the numbers at the three vertices of each face is the same. He has already written the numbers 1 and 5 as shown. What is the sum of the other three num... |
olympiads |
An electric field is created in space, which depends on the coordinates as shown in the diagram.
This field moved a particle with a mass of 1 mg and a charge of 1 mC from the point with the coordinate 5 cm to the point with the coordinate 15 cm. Determine the speed of the charge at the final point, assuming it was at... | 1. Given the electric field strength at the point with coordinate \(10 \, \text{cm}\) is \(200 \, \text{V/m}\).
2. The potential difference between the final and initial points is equal to the area under the graph. We calculate this area in steps:
- For the first region from \(5 \, \text{cm}\) to \(10 \, \text{c... | [
{
"content": "\nAn electric field is created in space, which depends on the coordinates as shown in the diagram.\n\nThis field moved a particle with a mass of 1 mg and a charge of 1 mC from the point with the coordinate 5 cm to the point with the coordinate 15 cm. Determine the speed of the charge at the final ... |
olympiads | In the diagram, \( DA = CB \). What is the measure of \(\angle DAC\)?
(A) \(70^\circ\)
(B) \(100^\circ\)
(C) \(95^\circ\)
(D) \(125^\circ\)
(E) \(110^\circ\) |
Given the problem, we need to find the measure of $\angle D A C$. We know the given conditions:
- $DA = CB$
- $\angle BAC = 70^\circ$
- $\angle ABC = 55^\circ$
Let's proceed step by step.
1. First, let's find $\angle BCA$ in $\triangle ABC$:
\[
\angle BAC + \angle ABC + \angle BCA = 180^\circ \quad \text{(Sum... | [
{
"content": "In the diagram, \\( DA = CB \\). What is the measure of \\(\\angle DAC\\)?\n(A) \\(70^\\circ\\)\n(B) \\(100^\\circ\\)\n(C) \\(95^\\circ\\)\n(D) \\(125^\\circ\\)\n(E) \\(110^\\circ\\)",
"role": "user"
},
{
"content": "\nGiven the problem, we need to find the measure of $\\angle D A C$. ... |
olympiads | As shown in the diagram, a rectangular grassland is divided into three parts. It is known that the second part is 10 meters wider than the first part. The area of the second part is 1000 square meters, and the third part is 4 meters narrower than the first part. The area of the third part is 650 square meters. What is ... |
1. **Identify Variables**:
Let the width of the first piece of land be \( a \).
2. **Width of Other Pieces**:
The width of the second piece of land is \( a + 10 \) meters.
The width of the third piece of land is \( a - 4 \) meters.
3. **Given Areas**:
The area of the second piece of land is 1000 square m... | [
{
"content": "As shown in the diagram, a rectangular grassland is divided into three parts. It is known that the second part is 10 meters wider than the first part. The area of the second part is 1000 square meters, and the third part is 4 meters narrower than the first part. The area of the third part is 650 s... |
olympiads | As shown in the diagram, given that \( BD \) is the angle bisector of \( \angle ABC \), \( DE \perp BC \) at \( E \), \( S_{ABC} = 24 \), \( AB = 8 \), and \( BC = 12 \), find the length of \( DE \). |
Given: \( BD \) is the angle bisector of \( \angle ABC \), \( DE \perp BC \) at \( E \), and the area of \( \triangle ABC \) is \( S_{ABC} = 24 \), with \( AB = 8 \) and \( BC = 12 \).
We need to find the length \( DE \).
1. **Construct Perpendicular \( DF \)**:
Draw \( DF \perp AB \) at \( F \).
2. **Identify E... | [
{
"content": "As shown in the diagram, given that \\( BD \\) is the angle bisector of \\( \\angle ABC \\), \\( DE \\perp BC \\) at \\( E \\), \\( S_{ABC} = 24 \\), \\( AB = 8 \\), and \\( BC = 12 \\), find the length of \\( DE \\).",
"role": "user"
},
{
"content": "\nGiven: \\( BD \\) is the angle b... |
olympiads | The diagram shows five congruent right-angled isosceles triangles. What is the total area of the triangles?
A) \( 25 \, \text{cm}^2 \)
B) \( 30 \, \text{cm}^2 \)
C) \( 35 \, \text{cm}^2 \)
D) \( 45 \, \text{cm}^2 \)
E) \( 60 \, \text{cm}^2 \) |
1. **Understand the given information and the problem**: We need to find the total area of five congruent right-angled isosceles triangles.
2. **Identify the individual dimensions of each triangle**:
- Each triangle is a right-angled isosceles triangle.
- Since each section of the diagram is a part of the whole... | [
{
"content": "The diagram shows five congruent right-angled isosceles triangles. What is the total area of the triangles?\n\nA) \\( 25 \\, \\text{cm}^2 \\)\n\nB) \\( 30 \\, \\text{cm}^2 \\)\n\nC) \\( 35 \\, \\text{cm}^2 \\)\n\nD) \\( 45 \\, \\text{cm}^2 \\)\n\nE) \\( 60 \\, \\text{cm}^2 \\)",
"role": "user"... |
olympiads | Find the area of the triangle (see the diagram) on graph paper. (Each side of a square is 1 unit.) | :
1. **Identify the areas of the two parts of the triangle:**
The given triangle can be divided into two right-angled triangles. Let's denote them as triangle \( \triangle A \) and triangle \( \triangle B \).
2. **Calculate the area of triangle \( \triangle A \):**
- The base of \( \triangle A \) is 1 uni... | [
{
"content": "Find the area of the triangle (see the diagram) on graph paper. (Each side of a square is 1 unit.)",
"role": "user"
},
{
"content": ": \n\n1. **Identify the areas of the two parts of the triangle:**\n\n The given triangle can be divided into two right-angled triangles. Let's denote t... |
olympiads | In the diagram, the parabola has \( x \)-intercepts \(-1\) and \( 4 \), and \( y \)-intercept \( 8 \). If the parabola passes through the point \((3, w)\), what is the value of \( w \)?
(A) 4
(B) 5
(C) 6
(D) 7
(E) 8 |
We are given a parabola with \( x \)-intercepts at \(-1\) and \(4\), and a \( y \)-intercept at \(8\). We are asked to find the value of \( w \) when the parabola passes through the point \((3, w)\). We will solve this problem using two different methods:
#### Method 1: Using Symmetry
1. **Find the axis of symmetry:... | [
{
"content": "In the diagram, the parabola has \\( x \\)-intercepts \\(-1\\) and \\( 4 \\), and \\( y \\)-intercept \\( 8 \\). If the parabola passes through the point \\((3, w)\\), what is the value of \\( w \\)? \n(A) 4 \n(B) 5 \n(C) 6 \n(D) 7 \n(E) 8",
"role": "user"
},
{
"content": "\nWe are giv... |
olympiads | In the given diagram, each of the 10 circles represents a digit. Fill in the diagram with a valid equality by placing one of the digits from 0 to 9 in each circle, using each digit exactly once.
## $0 \times O O \times O O O=0 O O$ |
Given the problem, we are tasked with filling in a mathematical equality using the numbers 0 to 9 to create a valid multiplication that fits the specified form:
$$0 \times O O \times O O O = 0 O O$$
Let's define variables \( a, b, c, \ldots, j \) to represent each of the digits 0 to 9. By filling in the diagram appr... | [
{
"content": "In the given diagram, each of the 10 circles represents a digit. Fill in the diagram with a valid equality by placing one of the digits from 0 to 9 in each circle, using each digit exactly once.\n\n## $0 \\times O O \\times O O O=0 O O$",
"role": "user"
},
{
"content": "\nGiven the pro... |
olympiads | In the following diagram, \(ABCD\) is a square with \(PA = a, PB = 2a\) and \(PC = 3a\). Find \(\angle APB\).
(A) \(120^\circ\)
(B) \(130^\circ\)
(C) \(135^\circ\)
(D) \(140^\circ\)
(E) \(145^\circ\) |
1. **Let \(AB = x\)**. We need to find \(\angle APB\).
2. **Cosine Rule:**
Using the Law of Cosines on \(\triangle PAB\), we get:
\[
x^2 + a^2 - 2ax \cos \alpha = (2a)^2
\]
Simplifying, we obtain:
\[
x^2 + a^2 - 2ax \cos \alpha = 4a^2 \implies x^2 + a^2 - 4a^2 = 2ax \cos \alpha
\]
Hence,
... | [
{
"content": "In the following diagram, \\(ABCD\\) is a square with \\(PA = a, PB = 2a\\) and \\(PC = 3a\\). Find \\(\\angle APB\\).\n(A) \\(120^\\circ\\)\n(B) \\(130^\\circ\\)\n(C) \\(135^\\circ\\)\n(D) \\(140^\\circ\\)\n(E) \\(145^\\circ\\)",
"role": "user"
},
{
"content": "\n1. **Let \\(AB = x\\)... |
olympiads | Given the diagram in Figure 5, circles $\odot O_{1}$ and $\odot O_{2}$ intersect at points $A$ and $B$. Line segment $AD$ is a chord of $\odot O_{2}$ and line segment $AC$ is a chord of $\odot O_{1}$, with both chords being tangents to the other circle. The angle bisector of $\angle CAD$ intersects $\odot O_{1}$ and $\... | 1. To begin, consider the intersections and the relevant angles. Given the problem, intersections \(A, B\) are on the circles \(\odot O_1\) and \(\odot O_2\); \(F\) and \(E\) intersect on the perpendicular bisector at \(C\) and \(D\). Define angles at points accordingly.
2. Let us denote and analyze the various angles ... | [
{
"content": "Given the diagram in Figure 5, circles $\\odot O_{1}$ and $\\odot O_{2}$ intersect at points $A$ and $B$. Line segment $AD$ is a chord of $\\odot O_{2}$ and line segment $AC$ is a chord of $\\odot O_{1}$, with both chords being tangents to the other circle. The angle bisector of $\\angle CAD$ inte... |
olympiads | In the diagram, \(PQRS\) is a square, \(PST\) is an equilateral triangle, and \(SRUVW\) is a regular pentagon. What is the size of angle \(WTS\)?
A) \(35^{\circ}\)
B) \(36^{\circ}\)
C) \(37^{\circ}\)
D) \(38^{\circ}\)
E) \(39^{\circ}\) |
1. First, we need to understand the geometry of the diagram given in the problem:
- \( PQRS \) is a square.
- \( PST \) is an equilateral triangle.
- \( SRUVW \) is a regular pentagon.
2. Since \( PQRS \) is a square, each internal angle of the square is \( 90^\circ \).
3. Since \( PST \) is an equilateral ... | [
{
"content": "In the diagram, \\(PQRS\\) is a square, \\(PST\\) is an equilateral triangle, and \\(SRUVW\\) is a regular pentagon. What is the size of angle \\(WTS\\)?\n\nA) \\(35^{\\circ}\\)\nB) \\(36^{\\circ}\\)\nC) \\(37^{\\circ}\\)\nD) \\(38^{\\circ}\\)\nE) \\(39^{\\circ}\\)",
"role": "user"
},
{
... |
olympiads | Which of the following nets can be used to build the partial cube shown in the diagram? |
To solve this problem, we need to determine which net can be folded to form the partial cube shown in the diagram. Let's analyze each net in detail:
1. **Net A**:
- Net A would produce a cube with holes on two edges of the same face. This does not match the configuration of the partial cube given in the diagram.
... | [
{
"content": "Which of the following nets can be used to build the partial cube shown in the diagram?",
"role": "user"
},
{
"content": "\nTo solve this problem, we need to determine which net can be folded to form the partial cube shown in the diagram. Let's analyze each net in detail:\n\n1. **Net A... |
olympiads | The diagram shows three examples of readings from functional electronic clocks. How many segments can stop working such that the time can always be determined unambiguously? |
1. **Consider the first position:**
- We need to distinguish three numbers: \(0\), \(1\), and \(2\).
- We can distinguish these numbers with two sticks, for example, the top and middle sticks:
- If both the top and middle sticks are lit, it represents \(2\).
- If only the top stick is lit, it ... | [
{
"content": "The diagram shows three examples of readings from functional electronic clocks. How many segments can stop working such that the time can always be determined unambiguously?",
"role": "user"
},
{
"content": "\n1. **Consider the first position:** \n - We need to distinguish three num... |
olympiads | As shown in the diagram, it is known that $SP$ is the tangent to circle $\odot O$ at point $P$. $PQ$ is perpendicular to $OS$ at point $Q$. The secant $SAB$ intersects $PQ$ at point $T$. Prove that $\angle AQT = \angle BQT$. |
Given:
- \( SP \) is a secant line of circle \( \odot O \).
- \( P \) is the point of intersection.
- \( PQ \perp OS \) at \( Q \).
- The secant line \( SAB \) intersects \( PQ \) at \( T \).
We need to prove that \( \angle AQT = \angle BQT \).
1. **Identify Key Elements:**
- Connect \( OP \) and \( OB \).
2. **... | [
{
"content": "As shown in the diagram, it is known that $SP$ is the tangent to circle $\\odot O$ at point $P$. $PQ$ is perpendicular to $OS$ at point $Q$. The secant $SAB$ intersects $PQ$ at point $T$. Prove that $\\angle AQT = \\angle BQT$.",
"role": "user"
},
{
"content": "\nGiven:\n- \\( SP \\) i... |
olympiads | In the diagram, \(ABCD\) is a rectangle. If the area of triangle \(ABP\) is 40, then the area of the shaded region is
(A) 20
(B) 40
(C) 60
(D) 50
(E) 80 |
1. **Construct Perpendicular from Point $P$ to Line $AB$**:
Let's draw a perpendicular line from point $P$ to line $AB$. Let the intersection point be $H$.
2. **Area of Triangle $\triangle ABP$ Calculation**:
We know the area of $\triangle ABP$ is 40. Thus, using the area formula for a triangle, we have:
... | [
{
"content": "In the diagram, \\(ABCD\\) is a rectangle. If the area of triangle \\(ABP\\) is 40, then the area of the shaded region is\n(A) 20\n(B) 40\n(C) 60\n(D) 50\n(E) 80",
"role": "user"
},
{
"content": "\n1. **Construct Perpendicular from Point $P$ to Line $AB$**: \n Let's draw a perpendic... |
olympiads | The diagram shows a weaver's design for a rihlèlò, a winnowing tray from Mozambique. How many lines of symmetry does the design have?
A 0
B 1
C 2
D 4
E 8 | 1. **Understanding Lines of Symmetry**:
- A line of symmetry divides an image or a figure into two identical parts such that one part is the mirror image of the other.
- For a shape or design, identifying lines of symmetry involves finding axes where folding along those lines results in matching parts on both s... | [
{
"content": "The diagram shows a weaver's design for a rihlèlò, a winnowing tray from Mozambique. How many lines of symmetry does the design have?\nA 0\nB 1\nC 2\nD 4\nE 8",
"role": "user"
},
{
"content": "1. **Understanding Lines of Symmetry**:\n - A line of symmetry divides an image or a figur... |
olympiads | In the diagram, a $4 \times 9$ grid $PQTV$ is formed from thirty-six $1 \times 1$ squares. Lines $PR$ and $US$ are drawn with $R$ and $S$ on $QT$. What is the ratio of the shaded area to the unshaded area?
(A) $5:9$
(B) $9:8$
(C) $4:5$
(D) $9:5$
(E) $5:4$ | 1. **Calculate the Area of the Entire Grid:**
The grid is a $4 \times 9$ rectangle.
\[
\text{Area of the grid} = 4 \times 9 = 36.
\]
2. **Calculate the Area of \(\triangle PQR\):**
\(\triangle PQR\) is a right triangle with base \(QR\) of 3 units and height \(PQ\) of 4 units. Using the formula for the a... | [
{
"content": "In the diagram, a $4 \\times 9$ grid $PQTV$ is formed from thirty-six $1 \\times 1$ squares. Lines $PR$ and $US$ are drawn with $R$ and $S$ on $QT$. What is the ratio of the shaded area to the unshaded area?\n(A) $5:9$\n(B) $9:8$\n(C) $4:5$\n(D) $9:5$\n(E) $5:4$",
"role": "user"
},
{
"... |
olympiads | In the addition problem shown in the diagram, the same Chinese characters represent the same digits, and different Chinese characters represent different digits. What is the three-digit number represented by "春蕾杯"? |
1. **Understanding the Problem**:
- We have a cryptarithm involving Chinese characters and we want to determine the three-digit number represented by "春萧杯".
- The rule given is that identical characters represent identical digits and different characters represent different digits.
2. **Analyzing the Given Prob... | [
{
"content": "In the addition problem shown in the diagram, the same Chinese characters represent the same digits, and different Chinese characters represent different digits. What is the three-digit number represented by \"春蕾杯\"?",
"role": "user"
},
{
"content": "\n1. **Understanding the Problem**:... |
olympiads | A person lives in town $A$ and is planning to drive to his workplace at location $B$. If traffic congestion incidents on all road sections are independent and each road section can experience traffic congestion at most once, and given the probability of traffic congestion as shown in the diagram: Please choose a route ... | 1. Let’s denote \( P(AC) \) as the probability of congestion on segment \( AC \). Similarly, \( P(\overline{AC}) \) represents the probability of no congestion on segment \( AC \).
2. For each segment, we are given the probabilities of no congestion:
- \( P(\overline{AC}) = \frac{9}{10} \)
- \( P(\overline{CD})... | [
{
"content": "A person lives in town $A$ and is planning to drive to his workplace at location $B$. If traffic congestion incidents on all road sections are independent and each road section can experience traffic congestion at most once, and given the probability of traffic congestion as shown in the diagram: ... |
olympiads | As shown in the diagram, \(ABCD\) is a square with an area of 1, and \(\triangle BPC\) is an equilateral triangle. Find the area of \(\triangle BPD\).
(A) \(\frac{\sqrt{3}-1}{2}\)
(B) \(\frac{2\sqrt{3}-1}{8}\)
(C) \(\frac{\sqrt{3}}{4}\)
(D) \(\frac{\sqrt{3}-1}{4}\)
(E) \(\frac{1}{4}\) | ### Problem:
如图，$ABCD$ 是面積为 $1$ 的正方形， $\triangle BPC$ 是正三角形，则 $\triangle BPD$ 的面積为:
\begin{itemize}
\item (A) $\frac{\sqrt{3}-1}{2}$
\item (B) $\frac{2\sqrt{3}-1}{8}$
\item (C) $\frac{\sqrt{3}}{4}$
\item (D) $\frac{\sqrt{3}-1}{4}$
\item (E) $\frac{1}{4}$
\end{itemize}
(中国北京市初中数学竞赛, 1984年)
1. **Identify Geome... | [
{
"content": "As shown in the diagram, \\(ABCD\\) is a square with an area of 1, and \\(\\triangle BPC\\) is an equilateral triangle. Find the area of \\(\\triangle BPD\\).\n\n(A) \\(\\frac{\\sqrt{3}-1}{2}\\)\n\n(B) \\(\\frac{2\\sqrt{3}-1}{8}\\)\n\n(C) \\(\\frac{\\sqrt{3}}{4}\\)\n\n(D) \\(\\frac{\\sqrt{3}-1}{4}... |
olympiads | The diagram shows a square \(PQRS\) with sides of length 2. The point \(T\) is the midpoint of \(RS\), and \(U\) lies on \(QR\) so that \(\angle SPT = \angle TPU\). What is the length of \(UR\)? |
**Method 1**
1. **Setup the problem:**
- Given a square \( PQRS \) with sides of length 2.
- Point \( T \) is the midpoint of \( RS \).
- Point \( U \) lies on \( QR \) such that \(\angle SPT = \angle TPU\).
2. **Define Point \( F \):**
- Let \( F \) be the point on \( PU \) so that \( \angle TFP = 90^\ci... | [
{
"content": "The diagram shows a square \\(PQRS\\) with sides of length 2. The point \\(T\\) is the midpoint of \\(RS\\), and \\(U\\) lies on \\(QR\\) so that \\(\\angle SPT = \\angle TPU\\). What is the length of \\(UR\\)?",
"role": "user"
},
{
"content": "\n**Method 1**\n1. **Setup the problem:**... |
olympiads | In the given diagram, the distances between the homes of five friends are shown. The shortest distance along the roads from Asya to Galya is 12 km, from Galya to Borya is 10 km, from Asya to Borya is 8 km, from Dasha to Galya is 15 km, and from Vasya to Galya is 17 km. What is the shortest distance along the roads from... |
Given:
- The shortest distance from Ася to Галя (between Asya and Galya) is 12 km.
- The shortest distance from Галя to Боря (between Galya and Borya) is 10 km.
- The shortest distance from Ася to Боря (between Asya and Borya) is 8 km.
- The shortest distance from Даша to Галя (between Dasha and Galya) is 15 km.
- The... | [
{
"content": "In the given diagram, the distances between the homes of five friends are shown. The shortest distance along the roads from Asya to Galya is 12 km, from Galya to Borya is 10 km, from Asya to Borya is 8 km, from Dasha to Galya is 15 km, and from Vasya to Galya is 17 km. What is the shortest distanc... |
olympiads | Let $n$ lines $l_{1}, l_{2}, \ldots, l_{n}$ and two points $A$ and $B$ be given in a plane. In which direction should a ball be shot from point $A$ so that, after reflecting sequentially off the lines $l_{1}, l_{2}, \ldots, l_{n}$, it passes through point $B$? (See diagram 38 for the case $n=3$).
b) Let $n=4$ and the ... |
**Part (a)**
1. Suppose the problem is solved and consider the sequence of lines $l_{1}, l_{2}, \ldots, l_{n}$. Let $A$ and $B$ be the starting and ending points, respectively.
2. Reflect point $B$ successively across each of the lines $l_{1}, l_{2}, \ldots, l_{n}$. Denote the reflections of $B$ as $B_{1}$, $B_{2}, \... | [
{
"content": "Let $n$ lines $l_{1}, l_{2}, \\ldots, l_{n}$ and two points $A$ and $B$ be given in a plane. In which direction should a ball be shot from point $A$ so that, after reflecting sequentially off the lines $l_{1}, l_{2}, \\ldots, l_{n}$, it passes through point $B$? (See diagram 38 for the case $n=3$)... |
olympiads | Place 26 matches along the lines so that they divide the entire diagram into two parts of equal size and identical shape, with two stars in one part and two crosses in the other. |
1. **Understand the task requirements:**
- We need to place **26 matchsticks** to divide a large square into two smaller parts.
- The two parts must be the **same size and shape**, with one part containing **two stars** and the other containing **two crosses**.
2. **Analyze the initial setup:**
- The init... | [
{
"content": "Place 26 matches along the lines so that they divide the entire diagram into two parts of equal size and identical shape, with two stars in one part and two crosses in the other.",
"role": "user"
},
{
"content": "\n1. **Understand the task requirements:**\n - We need to place **26 m... |
olympiads | The diagram shows the grid points of a square grid, with two points labeled $A$ and $B$. Let $C$ be one of the remaining grid points. Find all possible positions of point $C$ such that triangle $ABC$ has an area of 4.5 square units. |
To find all possible positions of point \(C\) so that the triangle \(ABC\) has an area of \(4.5\) square units, we need to discuss three qualitatively different cases: when triangle \(ABC\) is right-angled at \(A\) or \(B\), acute-angled, or obtuse-angled.
1. **Right-angled Triangle at \(A\) or \(B\)**
Let's co... | [
{
"content": "The diagram shows the grid points of a square grid, with two points labeled $A$ and $B$. Let $C$ be one of the remaining grid points. Find all possible positions of point $C$ such that triangle $ABC$ has an area of 4.5 square units.",
"role": "user"
},
{
"content": "\nTo find all possi... |
olympiads | A city's plan follows the scheme shown in the diagram.
All streets have one-way traffic: you can only travel "right" or "up".
How many different routes are there from point $A$ to point $B$?
| ```markdown
# Problem
План города имеет схему, изображенную на рисунке.
На всех улицах введено односторонне движение: можно ехать только "вправо" или "вверх". Сколько есть разных м... | [
{
"content": "A city's plan follows the scheme shown in the diagram. \n\nAll streets have one-way traffic: you can only travel \"right\" or \"up\".\n\nHow many different routes are there from point $A$ to point $B$?\n",
"role": "user"
},
{
"content": "```markdown\n# Problem\nПлан города имеет схему,... |
olympiads | In the diagram, rectangle $\mathrm{ABCD}$ has a length of 30 centimeters and a width of 20 centimeters. The area of $\triangle \mathrm{CEF}$ is 210 square centimeters. What is the length of $\mathrm{OG}$ in centimeters? |
1. We start by analyzing the given problem:
- The length of rectangle \( \mathrm{ABCD} \) is 30 meters.
- The width of rectangle \( \mathrm{ABCD} \) is 20 meters.
- The area of \( \triangle \mathrm{CEF} \) is 210 square meters.
2. We draw the rectangle \( \mathrm{ABCD} \), and within it, we locate point \( \... | [
{
"content": "In the diagram, rectangle $\\mathrm{ABCD}$ has a length of 30 centimeters and a width of 20 centimeters. The area of $\\triangle \\mathrm{CEF}$ is 210 square centimeters. What is the length of $\\mathrm{OG}$ in centimeters?",
"role": "user"
},
{
"content": "\n1. We start by analyzing t... |
olympiads | There were several people standing in a square, each facing one of four objects arranged as shown in the diagram.
Each person recorded which object was in front of them, which was to their left, and which was to their right. As a result, "house" was mentioned 5 times, "fountain" - 6 times, "bench" - 7 times, "tree" - ... | To determine the number of people standing in the square and how many were facing each object, follow these steps:
1. **Calculate the Total Number of Object Mentions**:
- The problem states the number of mentions for each object:
- "дом" (house): 5 times
- "фонтан" (fountain): 6 times
- "скамейка... | [
{
"content": "There were several people standing in a square, each facing one of four objects arranged as shown in the diagram.\n\nEach person recorded which object was in front of them, which was to their left, and which was to their right. As a result, \"house\" was mentioned 5 times, \"fountain\" - 6 times, ... |
olympiads |
In each diagram shown in this problem, the number on the line connecting two circles is the sum of the two numbers in these two circles.
(a) What is the value of \( x \)?
(b) With justification, determine the value of \( y \).
(c) With justification, determine the values of \( p, q \), and \( r \). |
We will break down the problem step-by-step for each part of the question.
1. **Finding the value of \( x \)**
- We are given that the sum of the numbers along the top line is 13. Let's denote the left number as \( a \) and the right number as 4:
\[
a + 4 = 13
\]
- Solving for \( a \), we sub... | [
{
"content": "\nIn each diagram shown in this problem, the number on the line connecting two circles is the sum of the two numbers in these two circles.\n\n(a) What is the value of \\( x \\)?\n\n(b) With justification, determine the value of \\( y \\).\n\n(c) With justification, determine the values of \\( p, q... |
olympiads | As shown in the diagram, in the right trapezoid $ABCD$, $AB = AD = \frac{1}{2} DC = 1$. Triangle $\triangle ABD$ is folded along $BD$ so that plane $ABD$ is perpendicular to plane $BCD$. Let $G$ be the midpoint of $AD$, $AH = 2HC$, and the midpoint of $BD$ is $O$.
(1) Prove that $AO$ is perpendicular to plane $BCD$;
(2... |
#### 1. Proof that $A O \perp$ plane $BCD$
Given:
- $AB = AD = \frac{1}{2} DC = 1$
- $O$ is the midpoint of $BD$
- $G$ is the midpoint of $AD$
- Plane $ABD$ is perpendicularly folded along $BD$ to make it perpendicular to plane $BCD$
We need to prove $A O \perp$ plane $BCD$.
1. Since $AB = AD$, by the properties of... | [
{
"content": "As shown in the diagram, in the right trapezoid $ABCD$, $AB = AD = \\frac{1}{2} DC = 1$. Triangle $\\triangle ABD$ is folded along $BD$ so that plane $ABD$ is perpendicular to plane $BCD$. Let $G$ be the midpoint of $AD$, $AH = 2HC$, and the midpoint of $BD$ is $O$.\n(1) Prove that $AO$ is perpend... |
olympiads | Bully Dima arranged 38 wooden toothpicks in the shape of a $3 \times 5$ rectangle. Then he simultaneously ignited two adjacent corners of this rectangle, as marked in the diagram.
It is known that one toothpick burns in 10 seconds. How many seconds will it take for the entire structure to burn down?
(The fire spreads... | 1. **Observing the Configuration**: The construction is represented as a rectangle of \( 3 \times 5 \) made from wooden toothpicks. There are 38 toothpicks in total, arranged to form this rectangular shape.
2. **Understanding the Fire Spread**: The fire spreads from any burning toothpick to all adjacent unburnt toothp... | [
{
"content": "Bully Dima arranged 38 wooden toothpicks in the shape of a $3 \\times 5$ rectangle. Then he simultaneously ignited two adjacent corners of this rectangle, as marked in the diagram.\n\nIt is known that one toothpick burns in 10 seconds. How many seconds will it take for the entire structure to burn... |
olympiads | The diagram shows a semicircle with centre \( O \). Two of the angles are given. What is the value of \( x \)?
Options:
- A) 9
- B) 11
- C) 16
- D) 17.5
- E) 18 |
1. Recognize that triangle $OPQ$ is an isosceles triangle because $OP$ and $OQ$ are both radii of the semicircle. This gives us:
\[
\angle OQP = \angle OPQ = 67^\circ
\]
2. Since we have a semicircle with center $O$ and angle $PQS$ is an angle inscribed in a semicircle, $\angle PQS$ is a right angle. Therefo... | [
{
"content": "The diagram shows a semicircle with centre \\( O \\). Two of the angles are given. What is the value of \\( x \\)? \nOptions:\n- A) 9\n- B) 11\n- C) 16\n- D) 17.5\n- E) 18",
"role": "user"
},
{
"content": "\n1. Recognize that triangle $OPQ$ is an isosceles triangle because $OP$ and $OQ... |
olympiads |
In the diagram, two circles and a square lie between a pair of parallel lines that are a distance of 400 units apart. The square has a side length of 279 units and one of its sides lies along the lower line. The circles are tangent to each other, and each circle is tangent to one of the lines. Each circle also touches... |
1. **Labeling and Basic Steps:**
Let \( r \) be the radius of the lower circle. We label the square as \( ABCD \), the center of the upper circle as \( U \), and the center of the lower circle as \( L \). The key points are as follows:
- \( E \) where the upper circle touches the top line.
- \( G \) where th... | [
{
"content": "\nIn the diagram, two circles and a square lie between a pair of parallel lines that are a distance of 400 units apart. The square has a side length of 279 units and one of its sides lies along the lower line. The circles are tangent to each other, and each circle is tangent to one of the lines. E... |
olympiads | In the diagram, \(AB\) is a diameter of a circle with center \(O\). \(C\) and \(D\) are points on the circle. \(OD\) intersects \(AC\) at \(P\), \(OC\) intersects \(BD\) at \(Q\), and \(AC\) intersects \(BD\) at \(R\). If \(\angle BOQ = 60^{\circ}\) and \(\angle APO = 100^{\circ}\), calculate the measure of \(\angle BQ... | :
1. **Identify Relationships:**
Since \( O \) is the center of the circle, it is given that \( OA = OC \) and \( OB = OD \). This implies that triangles \( \triangle OAC \) and \( \triangle OBD \) are isosceles.
2. **Isosceles Triangle Properties:**
- In \( \triangle OAC \), we have \( \angle OAC = \angle OCA... | [
{
"content": "In the diagram, \\(AB\\) is a diameter of a circle with center \\(O\\). \\(C\\) and \\(D\\) are points on the circle. \\(OD\\) intersects \\(AC\\) at \\(P\\), \\(OC\\) intersects \\(BD\\) at \\(Q\\), and \\(AC\\) intersects \\(BD\\) at \\(R\\). If \\(\\angle BOQ = 60^{\\circ}\\) and \\(\\angle APO... |
olympiads | A determined spider wants to reach a fly caught in its web (see the diagram). It can only crawl upwards and to the right along the threads of the web. How many different ways are there for the spider to reach its goal? |
1. **Understanding the problem**: The spider needs to reach the fly by only moving up or to the right on a web made of threads forming a grid.
2. **Objective**: Find the number of distinct paths from the bottom-left corner to the top-right corner of the grid. Specifically, the spider must make 5 steps upwards and 6 s... | [
{
"content": "A determined spider wants to reach a fly caught in its web (see the diagram). It can only crawl upwards and to the right along the threads of the web. How many different ways are there for the spider to reach its goal?",
"role": "user"
},
{
"content": "\n1. **Understanding the problem*... |
olympiads | In the diagram, what is the value of \( x \)?
(A) 65
(B) 75
(C) 85
(D) 95
(E) 105
 | 1. **Identify the sum of the angles in a triangle**:
The sum of the interior angles in any triangle is always \(180^\circ\).
2. **Set up the equation for the given triangle**:
We are given two angles in the triangle, \(50^\circ\) and \(45^\circ\), along with \(x^\circ\) which we need to find.
\[
50^\... | [
{
"content": "In the diagram, what is the value of \\( x \\)?\n(A) 65\n(B) 75\n(C) 85\n(D) 95\n(E) 105\n\n",
"role": "user"
},
{
"content": "1. **Identify the sum of the a... |
olympiads |
In the given diagram, an ellipse $C: \frac{x^{2}}{4}+\frac{y^{2}}{3}=1$ has its left and right vertices at points $A$ and $B$, respectively. Choose any point $P$ on the ellipse $C$ other than $A$ or $B$. The lines $PA$ and $PB$ intersect the line $x=3$ at points $M$ and $N$, respectively. The line $MB$ intersects the ... |
#### Part (1): Finding the value of $\overrightarrow{F M} \cdot \overrightarrow{F N}$
1. **Identify Coordinates of Point \( P \)**:
- Let \( P \left( x_0, y_0 \right) \) be any point on the ellipse defined by the equation:
\[
\frac{x_0^2}{4} + \frac{y_0^2}{3} = 1
\]
- Rearranging this gives:
\... | [
{
"content": "\nIn the given diagram, an ellipse $C: \\frac{x^{2}}{4}+\\frac{y^{2}}{3}=1$ has its left and right vertices at points $A$ and $B$, respectively. Choose any point $P$ on the ellipse $C$ other than $A$ or $B$. The lines $PA$ and $PB$ intersect the line $x=3$ at points $M$ and $N$, respectively. The ... |
olympiads | In the given diagram, \( A, B, C, D \) are the midpoints of the sides of a regular hexagon, and the area of the hexagon is 16. Find the area of the shaded region. | :
Given a regular hexagon with an area of 16 square units, and points \(A, B, C, D\) as the midpoints of the hexagon's sides, we are asked to find the area of the shaded region.
1. **Identify the Structure Inside the Hexagon:**
- Consider the hexagon as made up of six equilateral triangles.
- Each equilateral ... | [
{
"content": "In the given diagram, \\( A, B, C, D \\) are the midpoints of the sides of a regular hexagon, and the area of the hexagon is 16. Find the area of the shaded region.",
"role": "user"
},
{
"content": ": \nGiven a regular hexagon with an area of 16 square units, and points \\(A, B, C, D\\... |
olympiads | As shown in the diagram, in a plane Cartesian coordinate system, there is a circle \( M \) with the equation \( x^2 + y^2 + Dx + Ey + F = 0 \). The inscribed quadrilateral \( ABCD \) has diagonals \( AC \) and \( BD \) that are perpendicular to each other, and \( AC \) and \( BD \) lie on the \( x \)-axis and \( y \)-a... | 1. **Step 1: Proof that \( F < 0 \)**
- **Method One:** From the given problem, point \( O \) (the origin) must lie within the circle \( M \). If point \( O \) lies inside the circle, substituting the coordinates \( (0, 0) \) into the circle's equation:
\[
x^2 + y^2 + Dx + Ey + F = 0
\]
and evalua... | [
{
"content": "As shown in the diagram, in a plane Cartesian coordinate system, there is a circle \\( M \\) with the equation \\( x^2 + y^2 + Dx + Ey + F = 0 \\). The inscribed quadrilateral \\( ABCD \\) has diagonals \\( AC \\) and \\( BD \\) that are perpendicular to each other, and \\( AC \\) and \\( BD \\) l... |
olympiads | As shown in the diagram, given that $PA$ and $PB$ are two tangents drawn from an external point $P$ to the circle centered at $O$, and $M$ and $N$ are the midpoints of segments $AP$ and $AB$ respectively, the line $MN$ intersects the circle at points $C$ and $E$. Point $N$ lies between $M$ and $C$. The line $PC$ inters... | :
1. **Connect $O$, $P$, $A$, $C$, $E$:**
\[
O, P, N \text{ are collinear}, \text{ and } OP \perp AB
\]
2. **Midpoints and Parallel Lines:**
- $M$ is the midpoint of $PA$,
- $N$ is the midpoint of $AB$,
Hence,
\[
MN = MP = MA \quad \text{and} \quad MN \parallel PB
\]
3. **Power of... | [
{
"content": "As shown in the diagram, given that $PA$ and $PB$ are two tangents drawn from an external point $P$ to the circle centered at $O$, and $M$ and $N$ are the midpoints of segments $AP$ and $AB$ respectively, the line $MN$ intersects the circle at points $C$ and $E$. Point $N$ lies between $M$ and $C$... |
olympiads | In each box of the diagram below, we need to write a four-digit natural number such that all the indicated arithmetic operations are correct. In how many different ways can we fill the diagram?
Hint: Mark the number in some box as unknown and use it to fill the entire diagram. |
1. **Let the unknown be \( a \):**
Consider the unknown number in the first box as \( a \). Express the numbers in all the other boxes using \( a \). Here are the calculated values as per the given schema:
| Box | Expression in terms of \( a \) |
|:----:|:--------------------------------:|
| 1 | \( a \) |
... | [
{
"content": "In each box of the diagram below, we need to write a four-digit natural number such that all the indicated arithmetic operations are correct. In how many different ways can we fill the diagram?\n\nHint: Mark the number in some box as unknown and use it to fill the entire diagram.",
"role": "us... |
olympiads | The subway of city $\mathrm{N}$ consists of three lines, and its diagram is shown in the figure. Each train travels one segment between stations per minute. Upon reaching an end station, a train immediately turns around and starts moving in the opposite direction, and at each transfer station, it must change its line o... |
To solve this problem, we'll use the concept of bipartite graphs and parity arguments. Let's rewrite the detailed step-by-step proof for clarity.
1. **Graph Representation of the Metro System:**
Let's consider the stations and the connections between them as a graph. Each station is a vertex, and each direct route... | [
{
"content": "The subway of city $\\mathrm{N}$ consists of three lines, and its diagram is shown in the figure. Each train travels one segment between stations per minute. Upon reaching an end station, a train immediately turns around and starts moving in the opposite direction, and at each transfer station, it... |
olympiads | In the diagram, a city’s road system is shown. This city has 8 straight streets, and 11 intersections labeled with the letters $A, B, C, \ldots, J, K$.
You need to place a police officer at three different intersections such that on each of the 8 streets there is at least one police officer. Which three intersections ... |
1. **Identify the constraint:**
There are 8 streets in the city and 11 intersections named $A, B, C, \ldots, J, K$. We need to place policemen at 3 intersections such that each of the 8 streets has at least one policeman.
2. **Determine the arrangement:**
Since there are three vertical streets and each must hav... | [
{
"content": "In the diagram, a city’s road system is shown. This city has 8 straight streets, and 11 intersections labeled with the letters $A, B, C, \\ldots, J, K$.\n\nYou need to place a police officer at three different intersections such that on each of the 8 streets there is at least one police officer. W... |
olympiads | In the diagram, there is a layout of 8 cities connected by 12 roads. How many ways are there to close 5 roads simultaneously for repairs, such that it is still possible to travel from any city to any other city? | To solve the problem of determining the number of ways to close 5 out of 12 roads such that it is still possible to travel between any two cities, we need to look at the problem of maintaining the connectivity of an 8-vertex graph.
1. **Understanding the Graph Structure**:
The given graph has 8 vertices (cities) an... | [
{
"content": "In the diagram, there is a layout of 8 cities connected by 12 roads. How many ways are there to close 5 roads simultaneously for repairs, such that it is still possible to travel from any city to any other city?",
"role": "user"
},
{
"content": "To solve the problem of determining the ... |
olympiads | In the diagram, point \( P \) is outside of circle \( \odot O \), with \( PA \) and \( PB \) tangential to \( \odot O \) at points \( A \) and \( B \) respectively. Line \( PCD \) is a secant of \( \odot O \) intersecting the circle at points \( C \) and \( D \). Line \( CO \) intersects \( \odot O \) at another point ... |
Let's rewrite and detail the solution step-by-step with reference to the given problem and solutions.
#### Problem:
Given $\odot O$, point $P$ is outside of $\odot O$. $\mathrm{PA}$ and $\mathrm{PB}$ respectively intersect $\odot O$ at points $\mathrm{A}$ and $\mathrm{B}$. $\mathrm{PCD}$ is a secant line to $\odot O$... | [
{
"content": "In the diagram, point \\( P \\) is outside of circle \\( \\odot O \\), with \\( PA \\) and \\( PB \\) tangential to \\( \\odot O \\) at points \\( A \\) and \\( B \\) respectively. Line \\( PCD \\) is a secant of \\( \\odot O \\) intersecting the circle at points \\( C \\) and \\( D \\). Line \\( ... |
olympiads | A square is divided into two equal parts (see the diagram). Find different ways of dividing the given square into two equal parts that result in different shapes. |
The problem asks to find different ways to divide the square into two equal parts, distinct in the shape of the resulting figures. Below are detailed steps:
1. **Vertical Division:**
Divide the square through the middle with a vertical line from the top side to the bottom side.
If the side length of the squar... | [
{
"content": "A square is divided into two equal parts (see the diagram). Find different ways of dividing the given square into two equal parts that result in different shapes.",
"role": "user"
},
{
"content": "\nThe problem asks to find different ways to divide the square into two equal parts, dist... |
olympiads | The angle bisector $AM$ of triangle $ABC$ is extended to intersect the circumcircle at point $K$. Find three triangles in the diagram that are all similar to each other. |
To identify three similar triangles within the given construction, we need to analyze their angles and sides. We start by acknowledging some properties and theorems in geometry related to circles and triangle similarity.
1. **Identify the Key Elements:**
- Triangle \( \triangle ABC \) is inscribed in a circle.
... | [
{
"content": "The angle bisector $AM$ of triangle $ABC$ is extended to intersect the circumcircle at point $K$. Find three triangles in the diagram that are all similar to each other.",
"role": "user"
},
{
"content": "\nTo identify three similar triangles within the given construction, we need to an... |
olympiads | The diagram gives the lengths of six lines meeting at a point. Eric wishes to redraw this diagram without lifting his pencil from the paper. He can choose to start his drawing anywhere. What is the shortest distance he can draw to reproduce the figure?
A) 14 cm
B) 15 cm
C) 16 cm
D) 17 cm
E) 18 cm |
To determine the shortest distance Eric can draw without lifting his pencil, we need to analyze how to trace all lines meeting at a point in the most efficient manner.
1. **Understanding the Problem:**
- There are six lines of various lengths.
- Eric must draw all lines without lifting his pencil, starting and ... | [
{
"content": "The diagram gives the lengths of six lines meeting at a point. Eric wishes to redraw this diagram without lifting his pencil from the paper. He can choose to start his drawing anywhere. What is the shortest distance he can draw to reproduce the figure?\nA) 14 cm\nB) 15 cm\nC) 16 cm\nD) 17 cm\nE) 1... |
olympiads | In the diagram, \( OAB \) is a quarter circle, \( OE \) and \( CD \) are parallel, \( BC \) and \( EA \) are parallel, and \( \angle BCD = 4 \times \angle OBC \). What is the size of \( \angle OBC \)?
Given:
- Quadrilateral \( FCGE \) must be a parallelogram since \( OE \) is parallel to \( CD \) and \( BC \) is paral... |
1. Let us start by letting \(\angle OBC = x^\circ\).
2. Given that \(O A B\) is a quarter circle, this implies \(\angle BOA = 90^\circ\) since the angle subtended by a quarter circle is a right angle.
3. It is given that \(\angle BCD = 4 \times \angle OBC\). Therefore,
\[
\angle BCD = 4x^\circ
\]
4. We kno... | [
{
"content": "In the diagram, \\( OAB \\) is a quarter circle, \\( OE \\) and \\( CD \\) are parallel, \\( BC \\) and \\( EA \\) are parallel, and \\( \\angle BCD = 4 \\times \\angle OBC \\). What is the size of \\( \\angle OBC \\)?\n\nGiven:\n- Quadrilateral \\( FCGE \\) must be a parallelogram since \\( OE \\... |
olympiads | The diagram shows an equilateral triangle \(ADE\) inside a square \(ABCD\). What is the value of \(\frac{\text{area of } \triangle ADE}{\text{area of } \triangle DEC}\)?
(A) \(\frac{\sqrt{3}}{4}\);
(B) \(\frac{1}{4}\);
(C) \(\frac{\sqrt{3}}{2}\);
(D) \(\sqrt{3}\);
(E) 2. |
1. **Define the length of the side of the square and necessary angles:**
Let the side length of square \(ABCD\) be \(a\). Since \(A D E\) is an equilateral triangle, each internal angle of \(\triangle ADE\) is \(60^\circ\).
2. **Calculate the area of \(\triangle ADE\):**
The area of \(\triangle ADE\) can be d... | [
{
"content": "The diagram shows an equilateral triangle \\(ADE\\) inside a square \\(ABCD\\). What is the value of \\(\\frac{\\text{area of } \\triangle ADE}{\\text{area of } \\triangle DEC}\\)?\n\n(A) \\(\\frac{\\sqrt{3}}{4}\\);\n(B) \\(\\frac{1}{4}\\);\n(C) \\(\\frac{\\sqrt{3}}{2}\\);\n(D) \\(\\sqrt{3}\\);\n(... |
olympiads | Show how to cut the figure shown in the top diagram into three equal parts and then assemble these parts into the regular hexagon shown in the bottom diagram. It is not allowed to leave gaps or overlap parts. |
To solve this problem, we need to decompose the given figure into three equal parts and then rearrange those parts to form a regular hexagon. Here are the steps to achieve this:
1. **Understand the Composition of the Figures**:
The initial figure is made up of several equilateral triangles. By counting the nu... | [
{
"content": "Show how to cut the figure shown in the top diagram into three equal parts and then assemble these parts into the regular hexagon shown in the bottom diagram. It is not allowed to leave gaps or overlap parts.",
"role": "user"
},
{
"content": "\nTo solve this problem, we need to decompo... |
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