USAPhO 2026, Първи кръг — Задача 25
2026 Exam · 1 т.
Указания
25 QUESTIONS - 75 MINUTES
DO NOT OPEN THIS TEST UNTIL YOU ARE TOLD TO BEGIN
- Use throughout, unless otherwise specified.
- You may write in this question booklet and the scratch paper provided by the proctor.
- This test has 25 multiple choice questions. Select the best response to each question, and use a No.2 pencil to completely fill the box corresponding to your choice. If you change an answer, completely erase the previous mark. Only use the boxes numbered 1 through 25 on the answer sheet.
- All questions are equally weighted, but are not necessarily equally difficult.
- You will receive one point for each correct answer, and zero points for each incorrect or blank answer. There is no additional penalty for incorrect answers.
- You may use a hand-held calculator. Its memory must be cleared of data and programs. You may use only the basic functions found on a simple scientific calculator. Calculators may not be shared. Cell phones may not be used during the exam or while the exam papers are present. You may not use any external references, such as books or formula sheets.
- The question booklet, answer sheet and scratch paper will be collected at the end of this exam.
- To maintain exam security, do not communicate any information about the questions or their solutions until after February 13, 2026.
DO NOT OPEN THIS TEST UNTIL YOU ARE TOLD TO BEGIN
We acknowledge The US Physics Team coaches and other contributors to this year’s exams:
Tengiz Bibilashvili, Eddie Chen, Orhun Ciftcioglu, Kellan Colburn, Evan Erickson, Harry Kroft, Natalie LeBaron, Mai Qi, Elena Yudovina
Условие
The five vases below are filled with water to the same initial volume and initial height . Each vase has an identical drainage hole at the bottom.
Which vase will drain completely in the shortest amount of time?
(A) Vase a
(B) Vase b
(C) Vase c
(D) Vase d
(E) All vases drain in the same amount of time.
Отговори
- A
Решение
The rate at which water exits the container is given by Torricelli’s Law:
The rate at which the volume of water decreases is the product of the flux and the hole area :
Implicitly we can recognize that there are only one dynamic variable on each side of the equation, namely V and z and effectively reduce this to:
To get an exact solution we would need to integrate, but this is not necessarily required and we can instead use our intuition to solve it. That is, we can divide up each volume into separate slices and recognize that total time to drain the vase is the sum of the time required to drain each individual packet:
The time required to drain a specific packet is inversely proportional to the flow rate at that moment:
From this we can recognize that a slice of water drains faster if it is positioned at a high height than if it is positioned near the bottom. To minimize the total time, we must arrange the geometry of the vase such that the majority of the water volume is located where the drainage is fastest (i.e. at high ). Taking this into account we see that the answer must be a.
Оригинал в Архива: 2026_FMA_exam.pdf · официални решения: 2026_FMA_solutions.pdf