A car and a bus left City A at the same time and traveled along the same road to City B which was 160 km away. They ran at different but constant speeds which were integral numbers of km per hour. The difference in their speeds was a prime number of km per hour. After traveling for two hours, the distance the car had covered was five times the distance the bus still had to cover. What is the speed, in km per hour, of the bus?
The sum of five distinct positive integers is equal to 2024. We denote by $d$ the greatest common divisor of the 5 integers. What is the largest possible value that $d$ can take?
What is the smallest positive integer $m$, such that there exists a positive integer $n$ for which the first four digits after the decimal point in the decimal representation of the fraction $\frac{n}{m}$ are 2, 0, 2, 4 in that order?
In triangle $ABC$, points $D$ and $E$ are on the sides $AC$ and $AB$, respectively, and point $F$ is the intersection of $BD$ and $CE$, as shown in the diagram below. If the areas of triangles $DFC$, $BCF$ and $BEF$ are equal to $4\text{ cm}^2$, $8\text{ cm}^2$ and $10\text{ cm}^2$, respectively, what is the area, in $\text{cm}^2$, of quadrilateral $ADFE$?
It is known that $2^{84}$ is a 26-digit number so that 1 is its first digit. Among $2^1,2^2,2^3,2^4,\ldots,2^{83}$ and $2^{84}$, how many numbers are there so that 4 is their first digit?
In triangle $ABC$, $\angle BAC=30^\circ$ and $\angle ABC=105^\circ$. Let $P$ be a point inside the triangle such that $\angle PAB=15^\circ$ and $\angle ABP=75^\circ$, as shown in the diagram below. What is the measure, in degrees, of $\angle PCB$?
There are one semicircle and three shaded circles, $C_1,C_2,C_3$, where the radii of the two smaller shaded circles, $C_2$ and $C_3$, are equal. Each of $C_1,C_2,C_3$ is tangent to the semicircle and $C_1$ is tangent to each of $C_2,C_3$. Each shaded circle is also tangent to the diameter of the semicircle, with $C_1$ tangent to that diameter at the centre of the semicircle, as shown. If the total shaded area is $777\text{ cm}^2$, what is the unshaded area, in $\text{cm}^2$? (Take $\pi=\frac{22}{7}$.)
Consider 5 points located on the same plane, where any three of these points are not collinear. What is the maximum possible measure, in degrees, of the smallest angle among all the interior angles of a triangle with three of these points as its vertices?
Let $n$ be a positive integer so that the least common multiple of $n$ and 49 is a perfect square number. How many possible values of $n$ are there if $n\le2024$?
There is a road from town A to town B with a line of villages. Adjacent villages are 10 km apart; the first village is 20 km from A and the last is 20 km from B. Each village has some statues. Arthur travels from A to B and must travel at least 20 km per day. If he stops at a village, he stays there that day and resumes his journey toward B the next day, so he misses some villages. He wants to visit as many statues as possible. For example:
- A 4 3 4 3 B: he stops at villages 1 and 3.
- A 2 4 5 4 B: he stops at villages 2 and 4.
- A 5 4 2 3 B: he stops at villages 1 and 4.
In each case he visits 8 statues. For the following line, what is the maximum number of statues he can visit?
Two children, Ann and Bob, play a game in which an ideal coin is tossed repeatedly. The probability of heads or tails on each toss is $\frac12$. Ann throws the coin 4 times and Bob throws it 6 times, independently. Let $\frac pq$ be the probability that they obtain the same number of heads, with the fraction irreducible. What is $p+q$?
Some squares of a $7\times5$ board are colored gray, as shown in the diagram below. What is the number of ways to choose five gray squares so that any two squares chosen are on different rows and different columns? Give a detailed solution.
Triangle $ABC$ is isosceles with $AB=AC$. Let $E$ be the midpoint of $AC$ and $K$ the intersection other than $A$ of the bisector of $\angle CAB$ and the circumcircle of triangle $ABE$, as shown. Show that the diameter of the circumcircle of triangle $ABC$ is equal to $\frac43 AK$.
