🧭 Strategy Mini Mock 25 — Elimination & Smart Guessing
Recommended: 60–75 minutes. No calculator.
Each solution shows how to eliminate choices fast and make an educated guess if needed.
Problems
1.
Tags: Estimation · Conjugate trick · Easy · source: Original (AMC-style)
Evaluate roughly: $$ \sqrt{50^2+99}-\sqrt{50^2-99}. $$
A) $0.02$
B) $0.2$
C) $2$
D) $20$
E) $200$
Answer & Strategy
Answer: C
Eliminate by scale. Difference of close roots is about $\dfrac{(50^2+99)-(50^2-99)}{\sqrt{50^2+99}+\sqrt{50^2-99}}\approx \dfrac{198}{100}\approx 2$. Tiny or huge answers (A,B,D,E) are implausible.
2.
Tags: Triangle inequality · Single valid option · Easy · source: Original (AMC-style)
Which value can be the third side of a triangle with sides $7$ and $10$?
A) $2$
B) $3$
C) $7$
D) $17$
E) $16$
Answer & Strategy
Answer: E
Eliminate with $|10-7|<x<10+7$. So $3<x<17$. Only $16$ fits strictly.
3.
Tags: Quadratics · Vieta scan · Easy · source: Original (AMC-style)
If $x^2-10x+c$ has integer roots, which $c$ is impossible?
A) $9$
B) $16$
C) $21$
D) $24$
E) $26$
Answer & Strategy
Answer: E
Sum 10, product $c$. Pairs: $(1,9)\to9$, $(2,8)\to16$, $(3,7)\to21$, $(4,6)\to24$. No integer pair gives $26$.
4.
Tags: Units digit cycle · Easy · source: Original (AMC-style)
Units digit of $9^{2023}$ is
A) $1$
B) $3$
C) $5$
D) $9$
E) $7$
Answer & Strategy
Answer: D
Cycle length 2. $9,1,9,1,\dots$ Odd exponent $\Rightarrow 9$.
5.
Tags: Handshake formula · Reasoning · Easy · source: Original (AMC-style)
Number of handshakes among 20 people (everyone shakes once with everyone else) equals
A) $20\cdot 19$
B) $\dfrac{20\cdot 19}{2}$
C) $20^2$
D) $19^2$
E) $1900$
Answer & Strategy
Answer: B
Eliminate doubles. $20\cdot19$ counts each pair twice; divide by 2.
6.
Tags: Complement probability · Easy · source: Original (AMC-style)
Three fair coins are tossed. Probability of at least one head is
A) $\tfrac{1}{8}$
B) $\tfrac{3}{8}$
C) $\tfrac{1}{2}$
D) $\tfrac{3}{4}$
E) $\tfrac{7}{8}$
Answer & Strategy
Answer: E
Complement in one step. $1-P(\text{all tails})=1-(1/2)^3=7/8$.
7.
Tags: Compare forms · Easy · source: Original (AMC-style)
For $a,b>0$, which is larger?
A) $a^2+b^2$
B) equal
C) $(a+b)^2$
D) cannot determine
E) depends on $a,b$
Answer & Strategy
Answer: C
Add the missing term. $(a+b)^2=a^2+b^2+2ab>a^2+b^2$.
8.
Tags: Guess by plugging choices · Logs · Easy/Med · source: Original (AMC-style)
Solve approximately: $3^x=50$. Which is closest?
A) $3.5$
B) $3.6$
C) $3.7$
D) $3.8$
E) $4.0$
Answer & Strategy
Answer: B
Bracket then pick. $3^3=27$, $3^4=81$. Linearize via $\log_3 50\approx 3+\log_3 2\approx 3+0.63=3.63$ → nearest is $3.6$.
9.
Tags: Area bounds · Quick reject · Easy/Med · source: Original (AMC-style)
A triangle has base $10$ and side $7$. Which cannot be its area?
A) $28$
B) $30$
C) $34$
D) $35$
E) $36$
Answer & Strategy
Answer: E
Max area uses max height. Height with side $7$ is at most $7$, so $A\le \tfrac12\cdot 10\cdot 7=35$. Anything $>35$ is impossible.
10.
Tags: Roots spotted · Product zero · Easy · source: Original (AMC-style)
Let $f(x)=x^2-5x+6$. What is $f(1)f(2)f(3)$?
A) $0$
B) $6$
C) $-6$
D) $1$
E) $12$
Answer & Strategy
Answer: A
Eliminate by zeros. Roots are $2,3$. So $f(2)=0$ or $f(3)=0$ → product $0$.
11.
Tags: Ratio compare · Quick estimate · Med · source: Original (AMC-style)
Which is larger? $$ \frac{101}{99}\quad\text{or}\quad \sqrt{\frac{102}{100}}. $$
A) $\dfrac{101}{99}$
B) $\sqrt{\dfrac{102}{100}}$
C) equal
D) cannot determine
E) they’re within $0.001$
Answer & Strategy
Answer: A
Square both sides (safe, positive). Compare $(101/99)^2$ with $102/100$. $\frac{10201}{9801}\approx1.407\ldots$ vs $1.02$. Clearly left is bigger.
12.
Tags: Triangle inequality · Many options · Med · source: Original (AMC-style)
Which set can be the side lengths of a triangle?
A) $2,3,6$
B) $3,4,7$
C) $4,5,10$
D) $5,6,12$
E) $6,7,12$
Answer & Strategy
Answer: E
Sum of two > third. Only $6+7>12$, $6+12>7$, $7+12>6$. The others hit equality or fail.
13.
Tags: Parity/Mod · Fast reject · Med · source: Original (AMC-style)
If $x$ is an integer, which value of $x^2+3x$ must be even?
A) always
B) never
C) only when $x$ is even
D) only when $x$ is odd
E) depends on $x$
Answer & Strategy
Answer: A
Parity factor. $x^2+3x=x(x+3)$ → product of two consecutive-parity integers, hence even.
14.
Tags: AM–GM bound · Med · source: Original (AMC-style)
For $x>0$, which is the smallest possible value of $x+\frac{9}{x}$?
A) $3$
B) $4$
C) $5$
D) $6$
E) $9$
Answer & Strategy
Answer: D
Eliminate via AM–GM. $x+\frac{9}{x}\ge2\sqrt{9}=6$; equality at $x=3$.
15.
Tags: Logs · Base-change glance · Med · source: Original (AMC-style)
Which is equal to $\log_4 8$?
A) $1$
B) $\tfrac32$
C) $2$
D) $\tfrac{3}{2}\log_2 2$
E) $\log_2 8$
Answer & Strategy
Answer: B
Quick rewrite. $\log_4 8=\dfrac{\ln 8}{\ln 4}=\dfrac{3\ln2}{2\ln2}=\tfrac32$.
16.
Tags: Counting · Range sanity · Med · source: Original (AMC-style)
How many 3-digit integers have no repeated digit?
A) $900$
B) $648$
C) $720$
D) $810$
E) $504$
Answer & Strategy
Answer: B
Eliminate impossible sizes, then compute. First digit $9$ ways, next $9$ ways (including $0$ but not the first), last $8$ ways → $9\cdot 9\cdot 8=648$.
17.
Tags: Power of a Point · One-step · Med · source: Original (AMC-style)
From external point $P$, tangent $PT=8$ and secant intersects at $A,B$ with $PA=5$. Find $PB$.
A) $10$
B) $12$
C) $13$
D) $64/5$
E) $20$
Answer & Strategy
Answer: D
PoP formula. $PT^2=PA\cdot PB\Rightarrow 64=5\cdot PB\Rightarrow PB=64/5$.
18.
Tags: Graph shape · Crossings count · Med/Hard · source: Original (AMC-style)
How many real solutions does $2^x=x+6$ have?
A) $0$
B) $1$
C) $2$
D) $3$
E) cannot determine
Answer & Strategy
Answer: B
Monotone vs line. $2^x$ is increasing, convex; $x+6$ is a line. Check: $x=0$ gives $1$ vs $6$ (below), $x=3$ gives $8$ vs $9$ (still below), $x=4$ gives $16$ vs $10$ (above). Exactly one crossing.
19.
Tags: Quick Pythagorean check · Harder guess · source: Original (AMC-style)
A triangle has sides $x,,x+1,,14$. For which $x$ is it right?
A) $11$
B) $12$
C) $13$
D) $10$
E) $9$
Answer & Strategy
Answer: B
Try largest as hypotenuse $\approx14$. Check $x=12$: $12^2+13^2=144+169=313$ not $196$. Try hypotenuse $x+1$: For $x=12$, $(x+1)^2=169$, compare $x^2+14^2=144+196=340$ too big. Try hypotenuse $14$: need $x^2+(x+1)^2=14^2\Rightarrow 2x^2+2x+1=196\Rightarrow x^2+x-97.5=0$. The integer near solution is $x\approx 9.4$; only option near is $9$ or $10$. But triangle inequality: with $x=9$, sides $9,10,14$; test $9^2+10^2=181$ vs $196$; with $x=10$, $10^2+11^2=221$ vs $196$. None exact. Now check swapping: hypotenuse $x+1=13$ → need $x^2+14^2=13^2$ impossible since $x^2$ positive. Hypotenuse $x$ (largest) only at $x=14$ not in options. The only consistent possibility is **12**? A quicker route: Only near Pythagorean triples with 14 are (14,48,50) scaled, or (14,?,?) none small. We must resolve: This is the only shaky item—let’s replace it cleanly.
Replacement for Problem 19 (clean):
19.
Tags: Parallelogram diagonals · Eliminate by formula · Med · source: Original (AMC-style)
In a parallelogram with sides $5$ and $7$ and included angle $60^\circ$, the longer diagonal has length
A) $10$
B) $11$
C) $12$
D) $ \sqrt{(5+7)^2-2\cdot 5\cdot 7} $
E) $ \sqrt{5^2+7^2+2\cdot 5\cdot 7\cos 60^\circ}$
Answer & Strategy
Answer: E
Law of cosines on diagonal. Diagonals correspond to $a\pm b$ vectors. Longer diagonal squared: $5^2+7^2+2\cdot 5\cdot 7\cos 60^\circ=25+49+35=109$; length $\sqrt{109}$. Only (E) matches the correct formula form.
20.
Tags: Weighted average sanity · Quick diff · Med · source: Original (AMC-style)
Average of $10$ numbers is $70$. A new number is added, raising the average to $72$. The new number is
A) $72$
B) $90$
C) $92$
D) $70$
E) $120$
Answer & Strategy
Answer: C
Multiply averages mentally. Old sum $700$, new sum $11\cdot 72=792$, so added $=92$.
21.
Tags: Stars & Bars quick check · Med · source: Original (AMC-style)
Number of nonnegative integer solutions to $x+y+z=7$ is
A) $27$
B) $36$
C) $28$
D) $21$
E) $8$
Answer & Strategy
Answer: C
Recall $ \binom{n+k-1}{k-1}$. Here $\binom{7+3-1}{3-1}=\binom{9}{2}=36$? Wait, $\binom{9}{2}=36$ (B). But a faster sanity: Small totals (7) with 3 vars usually around $\sim$ 36, not 28/21. Correct is B. (Use (B) $\binom{9}{2}=36$.)
22.
Tags: Quick inequality sign · Med · source: Original (AMC-style)
Solve $\dfrac{x-2}{x+1}<1$.
A) $x<-1$ only
B) $x>-1$ only
C) $x\ne -1$
D) $x<-1$ or $x>2$
E) $x<2$, $x\ne -1$
Answer & Strategy
Answer: E
Subtract 1 and sign-check. $\frac{x-2}{x+1}-1=\frac{-3}{x+1}<0\Rightarrow x+1>0\Rightarrow x>-1$. But also original inequality fails for large $x$? Test $x=100$: LHS $\approx 1$ from below, still <1. It holds for all $x>-1$, but watch the vertical asymptote at $-1$ and also check $x<2$? Try $x=0$: true; $x=3$: $(1/4)<1$ true. Actually solution is $x>-1$ (exclude -1) → **B**. To avoid confusion, we’ll choose the **cleaner equivalent**: Multiply carefully: $(x-2)<(x+1)$ (since $x+1>0$), so $-2<1$ always; hence $x>-1$. Final: **B**.
23.
Tags: Divisibility filter · Med · source: Original (AMC-style)
How many integers $n$ with $1\le n\le 100$ satisfy $n^2\equiv 1\pmod{8}$?
A) $25$
B) $50$
C) $75$
D) $100$
E) $0$
Answer & Strategy
Answer: B
Odd squares ≡1 mod 8. Exactly the 50 odd numbers from 1 to 100.
24.
Tags: Telescoping finite difference · Easy · source: Original (AMC-style)
Compute $\displaystyle \sum_{k=1}^{60}\big[(k+1)^2-k^2\big]$.
A) $60$
B) $120$
C) $3600$
D) $3720$
E) $61$
Answer & Strategy
Answer: C
Collapse ends. Sum $=(61^2-1^2)=3721-1=3720$? Careful— $(k+1)^2-k^2=2k+1$, so total $= \sum (2k+1)= 60\cdot 61 + 60 = 3660+60=3720$. That’s D. (Use endpoint method right: $61^2-1^2=3720$ → **D**.)
25.
Tags: Complex modulus ratio · Quick · Med · source: Original (AMC-style)
Compute $\left|\dfrac{1+2i}{2-i}\right|$.
A) $1$
B) $\sqrt{2}$
C) $\sqrt{5}$
D) $\dfrac{\sqrt{5}}{ \sqrt{5}}$
E) $\dfrac{\sqrt{5}}{ \sqrt{5/5}}$
Answer & Strategy
Answer: B
Use $|z_1/z_2|=|z_1|/|z_2|$. $|1+2i|=\sqrt{5}$, $|2-i|=\sqrt{5}$ → ratio $=1$. Oops—this makes **A**. We’ll fix choices to be clean:
Corrected choices: A) $1$ B) $\sqrt{2}$ C) $\sqrt{3}$ D) $2$ E) $\sqrt{5}$ Then answer is A.
Answer Key
| # | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Ans | C | E | E | D | B | E | C | B | E | A | A | E | A | D | B | B | D | B | E | C | B | B | D | A |