System of equations + Mathematical Olympiad - practice problems
Number of problems found: 29
- Bedrich and Adam
When Bedrich is as old as Adam today, Adam will be 14 years old. When Adam was as old as Bedrich, Bedrich was two years old today. How old are Adam and Bedrich today? - Together 70014
Matej and Anton are 44 years old together. Matej is twice as old as Anton was when Matej was half as old as Anton will be when Anton is 3 times older than Matej was when Matej was 3 times as old as Anton. - Subtract 82333
I think of three numbers; when I add them, I get 16; when I subtract the third from the sum of the first two numbers, I get 10; when I subtract the second from the sum of the first and third numbers, I get 8. Which numbers do I think? - Mathematical 40213
Sixty-two high school students took part in the Mathematical Olympiad. There were 7 fewer first-year students than sophomores, two-thirds of the number of third-year students, and 5 fourth-year students. How many students from each grade took part in the - Equations: 80499
In the field of real numbers, solve the system of equations: 2x + ⌊y⌋ = 2022, 3y + ⌊2x⌋ = 2023. (⌊a⌋ denotes the (lower) integer part of the real number a, i.e., the largest integer not greater than a., E.g., ⌊1.9⌋ = 1 and ⌊−1.1⌋ = −2.) - Circumferences 83111
Péta composed several planar shapes from mutually congruent triangles. The circumferences of the first three are 8 cm, 11.4 cm, and 14.7 cm, respectively. Determine the perimeter of the fourth shape. - Lord Ram
When lord Ram was founded, the breed of white sheep was eight more than black. White sheep are four times higher than at the beginning, and black three times more than at the beginning. White sheep is now 42 more than the black. How many white and black s - Double-digit 80970
Eva thought of two natural numbers. She first added these correctly, then subtracted them correctly. In both cases, she got a double-digit result. The product of the resulting two-digit numbers was 645. Which numbers did Eva think of? Please, what is this - Circumference 6598
Adam had three identical rectangles. He put them together and got a rectangle with a circumference of 50 cm. Then, he placed them differently and got a rectangle with a larger circumference. Calculate its perimeter. - Matemakak 9432
The cookbook by Matěj Matemakak says: The greatest common divisor of flour weight and sugar weight is 15, the greatest common divisor of sugar weight and lemon peel weight is 6, the product of sugar weight and lemon peel weight is 1800, and the smallest c - Department 82388
Last year, there were 30 more boys than girls in our scout troop. This year, the number of children in the ward increased by 10%, while the number of boys increased by 5% and the number of girls increased by 20%. How many children do we have in the depart - Intersection 5413
In the acute triangle KLM, the angle KLM is 68°. Point V is the intersection of the altitudes, and P is the foot of the altitude on the side LM. The angle P V M axis is parallel to the side KM. Compare the sizes of angles MKL and LMK. - Subtract 10001
For five whole numbers, if we add one to the first, multiply the second by the second, subtract three from the third, multiply the fourth by four, and divide the fifth by five, we get the same result each time. Find all five of the numbers that add up to - Mathematical 7136
Out of 50 pupils, 44 solved at least one of the Olympiads - MO Mathematical Olympiad and BO Biology Olympiad. Twenty pupils still need to solve the MO. Of those who dealt with both Olympiads, 1/3 of those who dealt with just one were. How many pupils solv - Different 5874
Mišo and Rišo ran back and forth on the running track. They started towards each other, each from a different end of the track. Both were still running at the same speed, each at a different speed. The first time, they met 800 m from one end of the track, - Pytagoriade
Two fifth-graders teams compete in math competitions - Mathematical Olympiad and Pytagoriade. Of the 33 students who competed in at least one of the contests, 22 students. Students who competed only in Pytagoriade were twice as many who just competed in t - Celebration 4461
Anička has 50 CZK, Anežka has 46 CZK, and they want to use all the money to buy desserts for a family celebration. They decide between cakes and pinwheels. A pinwheel is CZK 4 more expensive than a cake, and for all the money, you could buy a third more c - MO Z8-I-1 2018
Fero and David meet daily in the elevator. One morning, they found that if they multiply their current age, they get 238. If they did the same after four years, this product would be 378. Determine the sum of the current ages of Fero and David. - Something 5385
Janko got pocket money and wants to buy something good for it. If he purchased four cakes, it would increase by 0.50 euros. If he wanted to buy five cakes, he would miss 0.60 euros. He would spend all his pockets on the rest if he bought two cakes and thr - Three friends
Three friend squirrels together went to collect hazelnuts. Zrzecka found more than twice Pizizubka, and Ouska was even three times more than Pizizubka. On the way home, they talked while eating and were cracking her nuts. Pizizubka ate half of all nuts co
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