id stringlengths 6 18 | image imagewidth (px) 70 4.26k | question stringlengths 10 747 | reference_answer stringlengths 1 336 |
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understand0521-14 | Identify the colored node on the second level (directly below the root "Disruptive Technologies") that has exactly two child nodes, where one child is a leaf node (having no subsequent children) and the other child acts as a parent to exactly two leaf nodes. | Artificial Intelligence | |
understand0521-8 | Calculate the absolute numerical difference between the 'Before' and 'After' values for each of the three metrics presented in the bar charts (Energy Consumption, Waste Generation, and Recycling Rate). Identify the metric that has the smallest absolute difference. According to the blue text sections in the upper half o... | 0.4 | |
understand0521-11 | Consider the thick, dark outline strokes in the image as a network of boundaries. How many distinct, completely enclosed areas (or 'cells') are formed entirely by these dark outline strokes?
(Rules: Count only the smallest non-overlapping enclosed areas. For example, the interior of the pink mouth and the rest of the ... | 6 | |
understand0521-15 | Puzzle Content: Observe the four colored rectangular boxes at the second tier of the diagram (Internet, Artificial Intelligence, Blockchain, Internet of Things). We will refer to them as "Root Branches". Now, let's define a concept—"Generational Balance": If all the direct child nodes branching downwards from a "Root B... | Artificial Intelligence. | |
understand0521-23 | In this diagram, nodes are categorized by background color as pink or yellow-green. Please find the ONLY node that simultaneously meets both of the following conditions:
It is a yellow-green node itself.
It is connected EXACTLY to 3 pink nodes and 1 yellow-green node." | Physiotherapist | |
understand0521-12 | In the provided flowchart, consider all distinct geometric shapes containing text (circles, ovals, diamonds, and rounded rectangles) as individual nodes. Do not count the large, colored background rectangular regions as nodes.
A "simple path" is defined as a continuous sequence of nodes connected by directed arrows (f... | 7 | |
understand0521-13 | The training schedule is visually divided into two sections: "Week 1" (the top 7 rows, denoted by a light purple background) and "Week 2" (the bottom 7 rows, with a white background). Calculate the total scheduled training distance for each week by summing all explicit numerical values listed in parentheses alongside t... | 1 | |
understand0521-0 | Examine the subset of trajectories that form perfectly vertical lines (parallel to the y-axis). How many perfectly vertical trajectories are present in the entire chart, and what is the exact longitude (in degrees) of the rightmost trajectory in this subset? | There are exactly 8 perfectly vertical trajectories, and the rightmost one is located at 100 degrees longitude. | |
understand0518-23 | Se disponen tres bloques para crear un bloque más grande como se muestra en la figura.
El ancho de uno de ellos es 6 y las áreas de algunas de sus caras son 14, 21, 16, 30, como se muestra. ¿Cuál es el área de la cara $x$? | 24 | |
understand0518-17 | Назовём лодочкой трапецию
с основаниями $1$ и $3$, являющуюся прилепленным к противоположным сторонам единичного квадратика двух треугольничков (полуклеток). В квадрате $100 \times 100$ расположена невидимая лодочка (её можно поворачивать, она не выходит за границы квадрата, её средняя клетка целиком лежит на одной из ... | $4000$ выстрелов. | |
understand0518-50 | Os pontos da rede quadriculada abaixo são numerados a partir do vértice inferior esquerdo seguindo o caminho poligonal sugerido no desenho. Considere o ponto correspondente ao número 2001. Quais são os números dos pontos situados imediatamente abaixo e imediatamente à esquerda dele?
[IMAGE: ] | 1874 | |
understand0518-53 | Na figura, os triângulos $ABC$ e $EGF$ são equiláteros. O perímetro do triângulo $ABC$ é 132cm e, além disso,
$$AE = EC \\ BD = DC \\ EF = FC \\ DG = GE$$
(1)、Qual o perímetro da área sombreada?
(2)、Que fração da área do triângulo ABC representa a área sombreada? | a) 12
b) $S' = \frac{13}{16}S$ | |
understand0518-60 | Quantos triângulos existem cujos lados estão sobre alguns dos segmentos traçados na figura ao lado? | 17 | |
understand0518-105 | Find the sum of the shaded angles. | 360 | |
understand0518-70 | Four squares are drawn below, with the area of the bottom left square being 4 units².
Determine the area of the shaded triangle. | $8 \text{ units}^2$ | |
understand0518-188 | All the telephone numbers in Georgetown have six digits and each of them begins with the digits 81. Kate finds the scrap of paper shown, with part of Jenny's telephone number on it.
How many different possibilities are there for Jenny's telephone number? | 102 | |
understand0518-133 | The figure shows a dark shaded cube partially enclosed by smaller light shaded cubes. There are no cubes missing on the faces not visible in the picture. How many extra light shaded cubes are needed to enclose the dark shaded cube and complete the large four by four cube? | 22 | |
understand0518-173 | A structure is built with identical cubes. The top view, the front view and the side view are shown below.
What is the least number of cubes required to build this structure? | 8 | |
understand0518-150 | Duas fitas de largura 1 cm, uma preta e outra cinza, foram cortadas em retângulos menores, que foram colados para formar a faixa decorativa abaixo. Os retângulos encaixaram perfeitamente, sem espaços vazios nem superposições e as duas fitas foram utilizadas completamente. Se a faixa decorativa tem 100 cm de comprimento... | 300 | |
understand0518-165 | $ABCD$ in the figure is a rectangle with inscribed semicircle on $AD$. Semicircles on $AE$ and $DE$ touch each other in point E on $AD$, and rectangle $ABCD$ respectively in point $A$ and point $D$. Both BE and CE are straight line segments. If $CD = 2$ and $AE = DE$, calculate the total area of all shaded regions. | 6 | |
understand0518-172 | Grasp the two loose ends of each rope firmly in your mind. Then imagine yourself pulling them until you have a straight piece of rope – either with a knot or without one.
Which of these four ropes will give you a knot? | A will form a knot. | |
understand0518-217 | A square with side length 6 has a circle with radius 2 inside of it, with the centers of the square and circle vertically aligned. Aarush is standing 4 units directly above the center of the circle, at point $P$. What is the area of the region inside the square that he can see? (Assume that Aarush can only see parts of... | $13\sqrt{3} - \frac{4\pi}{3}$ | |
understand0518-210 | Na figura a seguir, o pentágono regular ABCDE e o triângulo EFG estão inscritos na circunferência $C_o$, e M é ponto médio de BC. Para qual valor de $\alpha$, em graus, os triângulos EFG e HIG são semelhantes? | 36 | |
understand0518-206 | No quadriculado a seguir, cada quadradinho tem 1 cm² de área.
a) Qual é a área e o perímetro da figura formada pelos quadradinhos pintados de cinza?
b) Pintando outros quadradinhos, podemos aumentar a área dessa figura, sem mudar o seu perímetro. Qual é o valor máximo da área que podemos obter dessa maneira? | a)$20 \text{ cm}^2$;$34 \text{ cm}$
b)$72 \text{ cm}^2$ | |
understand0518-256 | Three isosceles right triangles are erected from the larger side of a rectangle into the interior of the rectangle, as shown on the right, where M is the midpoint of that side. Five circles are inscribed tangent to some of the sides and to one another as shown. One of the circles touches the vertex of the largest trian... | $$a:a:b:c:c = 1:1:1:2:2.$$ | |
understand0518-252 | Catherine has a plate containing 300 circular crumbling mooncakes, arranged as follows:
(This continues for 100 total columns). She wants to pick some of the mooncakes to eat, however whenever she takes a mooncake all adjacent mooncakes will be destroyed and cannot be eaten. Let $M$ be the maximal number of mooncakes s... | $(100, F_{104} + 1223)$ | |
understand0518-236 | Given a triangle $ABC$, let $D$ be a point on segment $BC$. Construct the circumcircle $\omega$ of triangle $ABD$ and point $E$ on $\omega$ such that $CE$ is tangent to $\omega$ and $A, E$ are on opposite sides of $BC$ (as shown in diagram). If $\angle CAD = \angle ECD$ and $AC = 12, AB = 7$, find $AE$. | ${2\sqrt{21}}$ | |
understand0518-271 | (a) If $n = 49 997$ how many times does it go through each loop and what is the final output?
(b) Some positive even numbers go through both loops at least once. Describe these numbers.
(c) Some positive odd numbers less than 150 go through both loops at least once. Describe these numbers. | (a)2;10
(b)any even number not ending on a 0 will pass through both loops at least once
(c)
$$1, 3, 5, 7, 9, 21, 23, 25, 27, 41, 43, 45, 61$$
$$63, 81, 111, 113, 115, 117, 131, 133, 135.$$
The positive odd numbers less than 150 that go through both loops at least once are all the odd numbers less than 150 not in the ... | |
understand0518-232 | Points W, X, Y, and Z are chosen inside a regular octagon so that four congruent rhombuses are formed, as shown in the diagram below. If the side length of the octagon is 1, compute the area of quadrilateral WXYZ. | $2 - \sqrt{2}$ | |
understand0518-260 | The diagram shows four identical white triangles symmetrically placed on a grey square. Each triangle is isosceles and right-angled.
The total area of the visible grey regions is equal to the total area of the white triangles.
What is the smallest angle in each of the (identical) grey triangles in the diagram? | $15^\circ$ | |
understand0518-264 | Triangle $ABC$ is an obtuse isosceles triangle with the property that three squares of equal size can be inscribed in it as shown on the right. The ratio $AC/AB$ is an irrational number that is the root of a cubic polynomial. Determine that polynomial. | $2x^3 - 2x^2 - 3x + 2$ | |
understand0518-309 | 试把 $1, 2, 3, ..., 9$ 等九个数字不重复地填入 $3×3$ 的方格表中, 使得下列六个图中打星号位置上的四个数字之和全部都相等。 | $$
\begin{array}{|c|c|c|}
\hline
9 & 2 & 7 \\
\hline
4 & 5 & 6 \\
\hline
3 & 8 & 1 \\
\hline
\end{array}
$$ | |
understand0518-294 | The unfinished cube is being built from identical blocks each of size 1 cm by 1 cm by 2 cm, as shown. How many more blocks are needed to complete the cube? | 48 | |
understand0518-212 | Consider 27 unit-cubes assembled into one $3 × 3 × 3$ cube. Let $A$ and $B$ be two opposite corners of this large cube. Remove the one unit-cube not visible from the exterior, along with all six unit-cubes in the center of each face. Compute the minimum distance an ant has to walk along the surface of the modified cube... | $\sqrt{41}$ | |
understand0518-321 | 正方形 $PQRS$在正九边形的内部且点$P$是正九边形的一个顶点，如下图所示。若正九边形的边长为 48 cm，请问正方形$PQRS$的面积是多少cm²？ | 3456 cm² | |
understand0518-364 | The large equilateral triangle shown consists of 36 smaller equilateral triangles. Each of the smaller equilateral triangles has area $10 \text{ cm}^2$.
The area of the shaded triangle is $K \text{ cm}^2$. Find $K$. | $110$ | |
understand0518-358 | The diagram shows a semicircle with diameter $PQ$ inscribed in a rhombus $ABCD$. The rhombus is tangent to the arc of the semicircle in two places. Points $P$ and $Q$ lie on sides $BC$ and $CD$ of the rhombus respectively. The line of symmetry of the semicircle is coincident with the diagonal $AC$ of the rhombus. It is... | 603 | |
understand0518-350 | An integer is to be written in each circle of the network shown. The integers must be written so that the sum of the numbers at the end of each line segment is the same. Two of the integers have already been written. What is the total of all the integers in the completed diagram? | 132 | |
understand0518-341 | Four straight lines intersect as shown.
What is the value of $p + q + r + s$? | 540 | |
understand0518-403 | Im Parallelogramm $ABCD$ schneidet eine Gerade durch den Punkt $C$ die Seite $AB$ im Punkt $E$ so, dass $\overline{EB} = \frac{1}{5} \overline{AE}$ erfüllt ist. Die Strecke $CE$ schneidet die Diagonale $BD$ im Punkt $F$. Bestimme das Verhältnis $\overline{BF} : \overline{BD}$. | 01:07:00 | |
understand0518-373 | In the hexagonal grid below, some edges are bolded, dividing the grid into six regions. Each of the nine cells is to be filled with a distinct single digit integer, such that
* no consecutive numbers share a **bolded** edge;
* the one or two digit number formed by each region is a prime number.
What is the product of t... | 672 | |
understand0518-400 | Die Star-Architektin Pauline will sich in ihrem rechteckigen Grundstück mit den Seitenlängen 35 m und 25 m ein ganz modernes fünfeckiges Haus bauen. Die Grundfläche des Hauses passt sie in das Grundstück ein wie in der Abbildung zu sehen ist:
Dabei markieren die Punkte am Rand jeweils einen Abstand von 5 m. Welchen Ant... | $\frac{41}{70}$ | |
understand0514-9 | 图中的化学式表示什么物质 | 英文名:Methyl ethanesulfonate 中文名: 乙烷磺酸甲酯 | |
understand0514-20 | 如图, 1mol 单原子理想气体构成的系统分别经历循环过程 $abcda$ 和 $abc'a$。已知理想气体在任一缓慢变化过程中, 压强 $p$ 和体积 $V$ 满足函数关系 $p = f(V)$。
计算系统经 $bc'$ 直线变化过程中的摩尔热容是多少？ | $C_{\pi} = \frac{8V - 35V_1}{4V - 14V_1} R$ | |
understand0514-7 | 图中的化学式表示什么物质 | 英文名:2-Chloropyridine-4-boronic acid pinacol ester
中文名:2-氯吡啶-4-硼酸频哪醇酯 | |
understand0514-27 | Three isosceles right triangles are erected from the larger side of a rectangle into the interior of the rectangle, as shown on the right, where M is the midpoint of that side. Five circles are inscribed tangent to some of the sides and to one another as shown. One of the circles touches the vertex of the largest trian... | $$a:a:b:c:c = 1:1:1:2:2.$$ | |
understand0514-17 | 图中有多少个三角形？ | 56 | |
understand0514-43 | 下图是一个被挖空的长方体，每个洞都是贯穿的。如果把它丢进染缸里，4个面染色的小正方体比3个面染色的小正方体多几个？ | 12 | |
understand0514-28 | The figure shows the map of Squareville, where each city block is of the same length. Two friends, Alexandra and Brianna, live at the corners marked by A and B, respectively. They start walking toward each other's house, leaving at the same time, walking with the same speed, and independently choosing a path to the oth... | $\frac{245}{841}$ | |
understand0514-90 | 如图所示，将一个三角形纸片ABC折叠，使得点C落在三角形ABC所在平面上，折痕为DE。已知$\angle ABE = 74^\circ$，$\angle DAB = 70^\circ$，$\angle CEB = 20^\circ$，那么$\angle CDA$等于多少 | 92° | |
understand0514-75 | Four straight lines intersect as shown.
What is the value of $p + q + r + s$? | 540 | |
understand0514-102 | 四个正方形如图摆放, 如果较小的两个正方形面积分别为 15 和 60, 那么较大的两个正方形面积差为多少 | 27 | |
understand0514-105 | 如图，已知正六边形ABCDEF的面积是314，那么阴影部分面积总和是多少 | 628 | |
understand0514-147 | "如图.$正六边形ABCDEF$的边长为1,作$正方形GHMN$使得点G在AB上、 点M在ED上.则$正方形GHMN$的面积的最大值是______.
希望杯（1-8年级） 46套 Word版 真题 【更新PDF 在本目录赠送文件夹】/【02】 -【赠送】希望数学100题2017-2022 全年级培训题+答案PDF/4- 希望数学2017-2019资料100题/images/050aa2dc6d4ab485c1028e320d434a21.png( 希望杯（1-8年级） 46套 Word版 真题 【更新PDF 在本目录赠送文件夹】/【02】 -【赠送】希望数学100题2017-2022 全年级培训题+答案PDF/4- ... | 2 | |
understand0514-146 | "如图，在正方形网格中有一个三角形，问图中含有三角形的正方形有几个？
希望杯（1-8年级） 46套 Word版 真题 【更新PDF 在本目录赠送文件夹】/【02】 -【赠送】希望数学100题2017-2022 全年级培训题+答案PDF/4- 希望数学2017-2019资料100题/images/1f787a9a26dc9859d62a3c5b2ba758de.png( 希望杯（1-8年级） 46套 Word版 真题 【更新PDF 在本目录赠送文件夹】/【02】 -【赠送】希望数学100题2017-2022 全年级培训题+答案PDF/4- 希望数学2017-2019资料100题/images/1f787a9a26dc9... | 14个 | |
understand0514-164 | "如图，在直角三角形ABC中，D是斜边AB上一点，正方形CEDF的边长为4。若蓝色部分的面积为12，则红色部分的面积为______。
希望杯（1-8年级） 46套 Word版 真题 【更新PDF 在本目录赠送文件夹】/【05】 -2021 冬令营 1-8年级 希望杯 思维挑战真题PDF/images/8dcef2288f32943733e96a8d548c14a9.png( 希望杯（1-8年级） 46套 Word版 真题 【更新PDF 在本目录赠送文件夹】/【05】 -2021 冬令营 1-8年级 希望杯 思维挑战真题PDF/images/8dcef2288f32943733e96a8d548c14a9.png)" | 2 | |
understand0514-16 | 图中是一个环状算式, 需要将数字1、2、3、4、5、7、8、15填入, 使算式成立，每个数字只能用1次。输出从环形的左上角开始，按照顺时针顺序依次填入数字的顺序。 | 从环形的左上角开始，按照顺时针顺序依次填入 8,7,15,3,5,1,4,2 | |
understand0514-13 | 图中分子式的相对分子质量是多少？ | 254 | |
understand0514-92 | 右图中, $\triangle ABC$的面积为100平方厘米, $\triangle ABD$的面积为72平方厘米. M为CD边的中点, $\angle MHB = 90^\circ$, 已知$AB = 20$厘米, 则MH的长度为多少 cm | 8.6 cm | |
understand0514-78 | The figure shows a quadrilateral ABCD in which $AD = DC$ and $∠ADC = ∠ABC = 90°$. The point E is the foot of the perpendicular from D to AB. The length DE is 25. What is the area of quadrilateral ABCD? | 625 | |
understand0514-63 | Given a triangle ABC with the property that $|AB| + |AC| = 3|BC|$. Let T be the point on line segment AC such that $|AC| = 4|AT|$. Let K and L be points on the interior of line segments AB and AC respectively such that, firstly, $KL is parallel to BC$ and, secondly, KL is tangent to the inscribed circle of $\triangle A... | $\frac{2}{3}$ | |
understand0514-53 | 某弹簧振子在水平方向上做简谐运动,其振动图像如图所示,则振子在0~60 s 内的路程为多少米？ | 15 m | |
understand0514-41 | An aquarium is made of glass with a thickness of 2 cm. From the outside, the length of the aquarium is 24 cm, the width is 20 cm and the height is 17 cm. What volume of water do we need, in cm³, to completely fill the aquarium? | $4800 cm³$ | |
understand0514-157 | "如图, 以等腰直角三角形ABC的三边为边, 分别向外作正方形, 已知三角形ABC的面积是4, 则得到的正方形的顶点构成的六边形EFGHIJ的面积是____.
希望杯（1-8年级） 46套 Word版 真题 【更新PDF 在本目录赠送文件夹】/【03】 -2019 冬令营4-8年级 希望杯 思维挑战真题PDF/images/22b09ccc8b8cded790f4eeb334cc799b.png( 希望杯（1-8年级） 46套 Word版 真题 【更新PDF 在本目录赠送文件夹】/【03】 -2019 冬令营4-8年级 希望杯 思维挑战真题PDF/images/22b09ccc8b8cded790f4eeb334... | 48 | |
understand0514-37 | Let triangle ABC be inscribed in a circle with center O. Let D be the perpendicular line from point C, and M be the midpoint of triangle BC. Given that OM = OD = 7, find the radius of the circle. | 14 | |
understand0514-171 | "如图, 矩形ABCD的对角线BD经过坐标原点O, 矩形的边分别平行于坐标轴, 反比例函数 $y=\frac{k}{x}$ 的图象分别与BC, CD交于点M, N. 已知点 $A(-3, -3)$, 且$\triangle OMN$的面积为3, 则$k^2=$______.
希望杯（1-8年级） 46套 Word版 真题 【更新PDF 在本目录赠送文件夹】/【03】 -2019 冬令营4-8年级 希望杯 思维挑战真题PDF/images/03810873f42ec107c5effe75db50dd48.png( 希望杯（1-8年级） 46套 Word版 真题 【更新PDF 在本目录赠送文件夹】/【03】 -2019... | 27 | |
understand0514-121 | 如图，在直角 $△ABC$ 的两个直角边AC，BC上分别作正方形ACDE和CBFG. 若$AC=14$，$BC=28$. 则 $△BEG$ 的面积等于____.
华杯赛（2-6年级）103套 Word版 真题【更新PDF 在本目录赠送文件夹】/2020华杯赛汇总/images/2458119db2080d24ebcc4cf76d4d8876.png( 华杯赛（2-6年级）103套 Word版 真题【更新PDF 在本目录赠送文件夹】/2020华杯赛汇总/images/2458119db2080d24ebcc4cf76d4d8876.png) | 784 | |
understand0514-19 | 在太阳内部存在两个主要的核聚变反应过程：碳循环和质子-质子循环；其中碳循环是贝蒂在1938年提出的，碳循环反应过程如图所示。图中p、e⁺和vₑ分别表示质子、正电子和电子型中微子；粗箭头表示循环反应进行的先后次序。当从循环图顶端开始，质子p与¹²C核发生反应生成¹³N核，反应按粗箭头所示的次序进行，直到完成一个循环后，重新开始下一个循环。已知e⁺、p和He核的质量分别为0.511 MeV/c²、1.0078 u和4.0026 u ($1u≈931.494 MeV/c²$)，电子型中微子vₑ的质量可以忽略。写出图中X和Y代表的核素； | $^{15}\text{O}$；$^{13}\text{C}$ | |
understand0514-2 | 带电粒子绕着带电量为$+Q$的源电荷做轨迹为椭圆的曲线运动, 源电荷固定在椭圆左焦点$F$上, 带电粒子电量为$-q$; 已知椭圆焦距为$c$, 半长轴为$a$, 电势计算公式为$\varphi = \frac{kQ}{r}$, 带点粒子速度的平方与其到电荷的距离的倒数满足如图关系。
在椭圆轨道半短轴顶点$B$的电势是多少？ | $\varphi_B = \frac{kQ}{a}$ | |
understand0514-32 | Two squares $ABCD$ and $BEFG$ share the vertex B, with E on the side BC and G on the side AB, as shown.The length of CG is 9 cm and the area of the shaded region is 47 cm².Calculate the perimeter of the shaded region. | 32 cm | |
understand0514-69 | 圆$\omega_1$的半径为6, 圆心为A, 与半径为15的圆$\omega_2$在点B处内切. 点C和D位于圆$\omega_2$上, 使得BC是圆$\omega_2$的直径, 并且$BC \perp AD$. 如图所示, 矩形EFGH内接于圆$\omega_1$, 使得$EF \perp BC$, C更靠近GH, 而不是EF, 并且D更靠近FG, 而不是EH. 三角形$\triangle DGF$和三角形$\triangle CHG$的面积相等. 矩形EFGH的面积为$\frac{m}{n}$, 其中m和n是互质的正整数. 求$m+n$的值. | 293 | |
understand0514-77 | 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 $UR$? | $\frac{1}{2}$ | |
understand0514-122 | 如图, $△ABC$ 是一个等腰直角三角形, DEFG 是其内接正方形, H 是正方形的对角线交点; 那么, 由图中的线段所构成的三角形中, 有______对相互全等的三角形.
希望杯（1-8年级） 46套 Word版 真题 【更新PDF 在本目录赠送文件夹】/2024希望杯1-8年级汇总/2024年希望数学冬令营训练真题及答案（3-8年级 6份卷）/2024培训题答案版/images/6b0b0518bc4c69426aa6181e0f1bff82.png( 希望杯（1-8年级） 46套 Word版 真题 【更新PDF 在本目录赠送文件夹】/2024希望杯1-8年级汇总/2024年希望数学冬令营训练真题及答案（3-... | 26 | |
understand0514-161 | "如图, 若①②③④⑤五个平行四边形拼成一个含30°内角的菱形 EFGH (不重复、无缝隙). 已知①②③④四个平行四边形面积的和为14, 四边形ABCD的面积为11, 则菱形EFGH的周长为______.
希望杯（1-8年级） 46套 Word版 真题 【更新PDF 在本目录赠送文件夹】/【02】 -【赠送】希望数学100题2017-2022 全年级培训题+答案PDF/2- 希望数学2021资料100题/images/8392b42f3233369daa1e5460e440d02b.png( 希望杯（1-8年级） 46套 Word版 真题 【更新PDF 在本目录赠送文件夹】/【02】 -【赠送】希望数学100题20... | 24 | |
understand0514-160 | "正方形 ABCD 每条边上的点均为三等分点, 如图摆放两个相同的长方形 EFGH、MNPQ, 如果 $NE=12$, 那么正方形 EPGM 与长方形 EFGH 的面积差是______。
希望杯（1-8年级） 46套 Word版 真题 【更新PDF 在本目录赠送文件夹】/2024希望杯1-8年级汇总/2024年希望数学冬令营训练真题及答案（3-8年级 6份卷）/2024培训题答案版/images/efd1ffe48fff285d98cc5415dea44012.png( 希望杯（1-8年级） 46套 Word版 真题 【更新PDF 在本目录赠送文件夹】/2024希望杯1-8年级汇总/2024年希望数学冬令营训练真题及... | 144 | |
understand0514-50 | 古希腊数学家阿波罗尼奥斯在研究圆锥曲线时发现了椭圆的光学性质：从椭圆的一个焦点射出的光线，经椭圆反射，其反射光线必经过椭圆的另一个焦点。设椭圆$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>b>0)$的左、右焦点分别为$F_1, F_2$，若从椭圆右焦点$F_2$发出的光线经过椭圆上的点$A$和点$B$反射后，满足$AB \perp AD$，且$\cos\angle ABC = \frac{3}{5}$，则该椭圆的离心率为多少？ | $\frac{\sqrt{5}}{3}$ | |
understand0514-175 | 函数 $y=1+x+\frac{\sin x}{x^2}$ 的部分图象大致为图中的 ABCD 中的哪一个 | D | |
understand0514-119 | 如图所示, 在梯形ABCD中, E、F、G、H分别是BA、CB、DC、AD上的三等分点, 则阴影部分占梯形ABCD面积的比值是多少?
华杯赛（2-6年级）103套 Word版 真题【更新PDF 在本目录赠送文件夹】/2020华杯赛汇总/images/93c5e0507624c4d2016a12c27300881b.png( 华杯赛（2-6年级）103套 Word版 真题【更新PDF 在本目录赠送文件夹】/2020华杯赛汇总/images/93c5e0507624c4d2016a12c27300881b.png) | $\frac{5}{9}$ | |
understand0514-70 | 在$\triangle ABC$中, $\angle BAC = 84^\circ$, $\angle ABC = 60^\circ$, $\angle ACB = 36^\circ$. D, E, F 分别是边BC, AC, AB的中点. 三角形DEF的外接圆与BD, AE, AF在点G, H, J处相交. 如图所示, 点G, D, E, H, J, F将三角形DEF的外接圆分成六段小圆弧. 求$DE + 2 \cdot HJ + 3 \cdot FG$的度数. | $336^\circ$ | |
understand0514-129 | 如图, 直角三角形ABC的面积是2022, $\angle CAB=90^\circ$, $AC:AB=2:3$, 分别以三角形ABC的三边AB、AC、BC向外作正方形ABED、ACGH、BCKF, 则六边形DEFKGH的面积是 ______。
希望杯（1-8年级） 46套 Word版 真题 【更新PDF 在本目录赠送文件夹】/2023希望杯1-8年级汇总/2023年希望数学冬令营真题及答案（1-8年级 8份卷）/images/f9dac847ed3677cd0559e4a373f0629b.png( 希望杯（1-8年级） 46套 Word版 真题 【更新PDF 在本目录赠送文件夹】/2023希望杯1-8年级汇总/20... | 25612 | |
understand0514-85 | 如图，甲、乙、丙、丁四个长方形拼成正方形EFGH，中间阴影为正方形。已知，甲乙丙丁四个长方形面积和是32cm²，四边形ABCD的面积是20cm²。求甲、乙、丙、丁四个长方形周长的总和是多少厘米。 | 48厘米 | |
understand0514-0 | 如图所示, 长度为 $d$ 的水平传送带 $M$ 顺时针匀速运动。质量为 $m$ 的小物块 $A$ 在传送带左端 $M$ 由静止释放。$A$ 还未与传送带达到相同速度时就从右端 $N$ 平滑地进入光滑水平面 $NO$, 与向右运动的小物块 $B$ 发生碰撞(碰撞时间极短)。碰后 $A$、$B$ 均向右运动, 从 $O$ 点进入粗糙水平地面。设 $A$ 与传送带间的动摩擦因数和 $A$、$B$ 与地面间的动摩擦因数均为 $\mu$, 重力加速度为 $g$。
$A$ 在传送带上的加速度大小及离开传送带时的速度大小分别是？ | $a=\mu g$；$\sqrt{2\mu gd}$ | |
understand0514-47 | 卫星导航系统中, 地球静止同步轨道卫星的轨道位于地球赤道所在平面,轨道高度为36 000 km(轨道高度指卫星到地球表面的最短距离),把地球看成一个球心为O,半径r为$6 400^④ km$的球,其上点A的纬度是指OA与赤道所在$平面^③$所成角的度数, 地球表面能直接观测到的一颗地球静止同步轨道卫星的点的纬度的最大值记为$α^②$,该卫星信号覆盖的地球表面积$S=2\pi r^2 \cdot (1-\cos \alpha)$(单位:km²),则S占地球表面积的百分$比^①$约为多少？ | 0.42 | |
understand0514-166 | "如图, 在平面直角坐标系$xOy$中, 多边形$OABCDE$的顶点坐标分别是$O(0, 0)$, $A(0, 6)$, $B(4, 6)$, $C(4, 4)$, $D(6, 4)$, $E(6, 0)$. 若直线$l$经过点$M(2, 3)$, 且将多边形$OABCDE$分割成面积相等的两部分, 则直线$l$的函数表达式是______.
希望杯（1-8年级） 46套 Word版 真题 【更新PDF 在本目录赠送文件夹】/【02】 -【赠送】希望数学100题2017-2022 全年级培训题+答案PDF/2- 希望数学2021资料100题/images/5670dd6a5371b51c72c06529b46d162a.p... | $y=-\frac{1}{3}x+\frac{11}{3}$ | |
understand0514-104 | 图中两圆的半径分别为16和12，两个圆心的距离是24，过两圆交点之一的直线被两圆截出相等的弦 $QP$ 和 $QR$，则 $PQ^2 等于多少 | 520 | |
understand0514-151 | "如图, 平行四边形ABCD中, 在边AB、AD上分别取点E、F, AB与AD长度的乘积等于AE与AF长度乘积的5.8倍. DE与BF相交于点O, 四边形CBOD的面积是四边形AEOF的面积的______倍.
希望杯（1-8年级） 46套 Word版 真题 【更新PDF 在本目录赠送文件夹】/2023希望杯1-8年级汇总/2023年希望数学冬令营真题及答案（1-8年级 8份卷）/images/a9752ece72bb6266570306026e0644d8.png( 希望杯（1-8年级） 46套 Word版 真题 【更新PDF 在本目录赠送文件夹】/2023希望杯1-8年级汇总/2023年希望数学冬令营真题及答案（1-... | 5.8 | |
understand0514-52 | 如图所示是某细胞的几种质膜转运体,鱼藤酮是一种线粒体抑制剂,若使用鱼藤酮处理该细胞，是否可能出现细胞摄取葡萄糖的能力加强的现象？ | 不可能 | |
understand0514-172 | "在平面直角坐标系中, 四边形ABCD的顶点和各边中点的坐标如图所示. $m+n+p+q+r+s+u+v=$ ______.
希望杯（1-8年级） 46套 Word版 真题 【更新PDF 在本目录赠送文件夹】/【06】 -2022 夏令营1-8年级 希望杯 个人PDF/2022HMTC个人战/images/479dc28e34c0cedddb9184694ab97f22.png( 希望杯（1-8年级） 46套 Word版 真题 【更新PDF 在本目录赠送文件夹】/【06】 -2022 夏令营1-8年级 希望杯 个人PDF/2022HMTC个人战/images/479dc28e34c0cedddb9184694ab97... | 30 | |
understand0514-94 | 右图中，$\angle A+\angle B+\angle C+\angle D+\angle F+\angle G$ 等于多少度 | 360 | |
understand0514-139 | 如图5，由若干个小等边三角形构成，其中每个三角形的顶点都被称为格点，则以图中的格点为顶点的等边三角形有多少个？
希望杯（1-8年级） 46套 Word版 真题 【更新PDF 在本目录赠送文件夹】/【02】 -【赠送】希望数学100题2017-2022 全年级培训题+答案PDF/5- 2016希望数学100题/2016希望杯考前100题试题/images/34ab94572268a56bce95d05aa0b72166.png( 希望杯（1-8年级） 46套 Word版 真题 【更新PDF 在本目录赠送文件夹】/【02】 -【赠送】希望数学100题2017-2022 全年级培训题+答案PDF/5- 2016希望数学10... | 66个 | |
understand0514-144 | "如图, 将一个正方形分割成两个相同的L形, 若分成的两个L形可以组成一个周长是26的长方形, 求这个正方形的面积.
希望杯（1-8年级） 46套 Word版 真题 【更新PDF 在本目录赠送文件夹】/【02】 -【赠送】希望数学100题2017-2022 全年级培训题+答案PDF/4- 希望数学2017-2019资料100题/images/bab141ee6c4840ead194b4e60a15b545.png( 希望杯（1-8年级） 46套 Word版 真题 【更新PDF 在本目录赠送文件夹】/【02】 -【赠送】希望数学100题2017-2022 全年级培训题+答案PDF/4- 希望数学2017-2019资料1... | 36 | |
understand0514-109 | 如图,P是等边三角形ABC内一点,△ABC的面积是2019.三个阴影三角形的面积和是多少 | 1009.5 | |
understand0514-15 | 图中“风车”（阴影部分）的面积是多少平方厘米？ | $2 \times 2 = 4 \text{ (平方厘米)}$ | |
understand0514-42 | 有一个正方体木块，每个面上分别写上了 1、2、3、4、5、6，并且相对两面上的数的和是 7。这个木块按下图放置后，按照图中箭头所示方向翻动。翻动到最后一格时，木块朝上一面的数是？ | 6 | |
understand0514-149 | "如图所示，大圆的直径是小圆的5倍，大圆内的“S”形曲线（图中虚线）由两段半圆弧组成。如果已知阴影部分的面积等于4，那么图中空白部分的面积等于______。
希望杯（1-8年级） 46套 Word版 真题 【更新PDF 在本目录赠送文件夹】/2024希望杯1-8年级汇总/2024年希望数学冬令营训练真题及答案（3-8年级 6份卷）/2024培训题答案版/images/7676da1d52adfedeeeb679499566536e.png( 希望杯（1-8年级） 46套 Word版 真题 【更新PDF 在本目录赠送文件夹】/2024希望杯1-8年级汇总/2024年希望数学冬令营训练真题及答案（3-8年级 6份卷）/2... | 21 | |
understand0514-60 | 函数$y=(2^x-2^{-x})\sin x$在区间$[-\pi,\pi]$上的图象大致为图中的哪个选项对应的图？ | B | |
understand0514-44 | 已知函数 $f(x) = 2\sin(\omega x + \varphi)$ ($\omega > 0, 0 < \varphi < \frac{\pi}{2}$) 的部分图象如图, $f(x_1) = f(x_2) = -\frac{2}{3}$, 则 $\cos[\frac{\pi}{6}(x_2 - x_1)] 等于多少？ | $\frac{1}{3}$ | |
understand0514-24 | As shown in the picture,in $\triangle ACF$, $B$ is the midpoint of $\overline{AC}$, $D$ and $E$ divide side $\overline{CF}$ into three equal parts, while $G$, $H$, and $I$ divide side $\overline{FA}$ into four equal parts.
Seventeen segments are drawn to connect these six
points to one another and to the opposite verti... | $(12r + 6s, 6t)$ and $(30r + 10s, 10t)$ | |
understand0514-114 | 在下图所示的各边相等的正五角星中, $\angle A=\alpha$, $\angle CGH = \beta$, $\angle HIJ = \gamma$, 则 $\alpha : \beta : \gamma$ 等于多少 | $1:2:3$ | |
understand0514-14 | 如右图所示,从长、宽、高为15,5,4的长方体中切走一块长、宽、高为y,5,x的长方体(x,y为整数),余下部分的体积为120,求x和y. | x = 3, y = 12 |
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