5. It is known that among 100 balls, exactly 51 are radioactive. There is a device into which two balls can be placed, and if both are radioactive, a light will turn on (if at least one of the two balls is not radioactive, the light will not turn on). Can all the radioactive balls be found using the device no more than 145 times?
Answer. Yes. Solution. Let's divide the balls into 50 pairs and test them. Consider two possible cases.

1) Exactly one of these tests revealed two radioactive balls. Then in each of the remaining 49 pairs, there is exactly one radioactive ball. Testing one of the found radioactive balls with one ball from each of the remaining pairs, we can identify all 98 remaining balls. In total, we conducted $50+49=99$ tests.
2) At least two tests revealed two radioactive balls each. Then we have already found 4 radioactive balls. Testing one of the found radioactive balls with 95 balls from the remaining 48 pairs, we will know about 99 balls whether they are radioactive or not. If there are 50 such balls, the remaining ball is radioactive, and if there are 51, it is not. We have found all the radioactive balls in $50+95=145$ tests.

Remark. More subtle reasoning can improve the estimate of 145.