2025 AMC 10A Problem 8

You cannot tell whether one statement is true on its own, because each one talks about all four. But every statement depends on just one number: how many of the four are true. Call that number $T$. There are only five possible values, $0$ to $4$, so try each.

1. Rewrite each statement in terms of $T$. If $T$ statements are true, then $4 - T$ are false.
- Statement 1 ("at least one true") is true when $T$ is $1$ or more.
- Statement 2 ("at least two true") is true when $T$ is $2$ or more.
- Statement 3 ("at least two false") is true when $4 - T$ is $2$ or more, which means $T$ is $2$ or less.
- Statement 4 ("at least one false") is true when $4 - T$ is $1$ or more, which means $T$ is $3$ or less.

1. Test each value of $T$. For each guess, count how many statements come out true. The guess only works if that count equals the guess.

1. Read off the answer. Only $T = 3$ agrees with itself. Statements 1, 2 and 4 are true, and statement 3 is false.

1. Double-check. With exactly one false statement, "at least two are false" really is false, and the other three really are true. Everything fits.

1. Answer the actual question. It asks for the number of false statements: $4 - 3 = 1$.

The answer is $\boxed{\textbf{(B)}\ 1}$.

When statements talk about themselves, you cannot judge them one at a time. Instead, guess the one number they all depend on and check whether the guess produces itself. Here that number is "how many are true", and with only five possibilities a small table settles it. Any puzzle of the form "at least $k$ of these are true/false" can be handled the same way.

Let $T$ be the number of true statements, so $4 - T$ are false. Every statement speaks only about $T$:

• statement 1 is true exactly when $T \ge 1$;
• statement 2 is true exactly when $T \ge 2$;
• statement 3 is true exactly when $4 - T \ge 2$, that is $T \le 2$;
• statement 4 is true exactly when $4 - T \ge 1$, that is $T \le 3$.

A value of $T$ is possible only if exactly $T$ of these four conditions hold. Test all five values.

• $T = 0$: conditions 3 and 4 hold, so $2$ statements are true, not $0$.
• $T = 1$: conditions 1, 3, 4 hold, so $3$ are true, not $1$.
• $T = 2$: all four conditions hold, so $4$ are true, not $2$.
• $T = 3$: conditions 1, 2, 4 hold and condition 3 fails, so $3$ are true. Consistent.
• $T = 4$: conditions 1, 2, 4 hold, so $3$ are true, not $4$.

Only $T = 3$ survives: statements 1, 2 and 4 are true and statement 3 is false. Reading it back confirms the arrangement, since "at least two of these statements are false" is indeed false when exactly one is false. The number of false statements is $4 - 3 = 1$.

The answer is $\boxed{\textbf{(B)}\ 1}$.