구 위 점과 고정점들로 정의된 벡터 내적의 최댓값

문제

좌표공간에 원점 $\mathrm{O}(0,0,0)$과 두 점 $\mathrm{A}(2,2,0)$, $\mathrm{B}(0,0,2\sqrt{2})$가 있다. 구 $S: x^2+y^2+z^2=16$ 위를 움직이는 점 $\mathrm{P}$에 대하여, 선분 $\mathrm{AP}$의 중점을 $\mathrm{M}$이라 하자. 점 $\mathrm{Q}$는 선분 $\mathrm{OM}$ 위에 있으며 $2|\vec{\mathrm{OQ}}| = |\vec{\mathrm{OM}}|$을 만족시킨다. 이때, $\vec{\mathrm{BQ}} \cdot \vec{\mathrm{BM}}$의 최댓값은?

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        <path d="M0,0 L0,6 L6,3 z" fill="#1F2937" />
    </marker>
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<style>
    .line {
        stroke:#1F2937;
        stroke-width:2;
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        font-size: 14px;
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        font-family: 'Noto Sans KR', sans-serif;
        font-size: 14px;
        fill: #1F2937;
        text-anchor: middle;
        dominant-baseline: middle;
    }
    .dot {
        fill: #1F2937;
        stroke: none;
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    Projection function used for points:
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    unit_scale = 20
    center_x = 200, center_y = 200
    px = center_x + x * unit_scale - y * unit_scale * 0.5
    py = center_y - z * unit_scale - y * unit_scale * 0.5
-->

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<line x1="200" y1="200" x2="300" y2="200" class="line" marker-end="url(#arrow)" />
<!-- Y-axis -->
<line x1="200" y1="200" x2="150" y2="150" class="line" marker-end="url(#arrow)" />
<!-- Z-axis -->
<line x1="200" y1="200" x2="200" y2="100" class="line" marker-end="url(#arrow)" />

<!-- Axis labels -->
<text x="310" y="200" class="axis-label">X</text>
<text x="140" y="165" class="axis-label">Y</text>
<text x="200" y="90" class="axis-label">Z</text>

<!-- Sphere S -->
<ellipse cx="200" cy="200" rx="80" ry="56" class="line" stroke-dasharray="4 4" />
<text x="200" y="265" class="point-label">S</text> <!-- Label for sphere -->

<!-- Origin O(0,0,0) -->
<circle cx="200" cy="200" r="3" class="dot" />
<text x="190" y="210" class="point-label">O</text>

<!-- Point A(2,2,0) -->
<circle cx="220" cy="180" r="3" class="dot" />
<text x="230" y="180" class="point-label">A</text>

<!-- Point B(0,0,2√2) -->
<circle cx="200" cy="143.43" r="3" class="dot" />
<text x="190" y="143.43" class="point-label">B</text>

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<circle cx="200" cy="120" r="3" class="dot" />
<text x="190" y="120" class="point-label">P</text>

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<line x1="220" y1="180" x2="200" y2="120" class="line" />

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<circle cx="210" cy="150" r="3" class="dot" />
<text x="220" y="150" class="point-label">M</text>

<!-- Segment OM -->
<line x1="200" y1="200" x2="210" y2="150" class="line" />

<!-- Point Q on OM such that 2|OQ| = |OM| (Q is midpoint of OM)
     For M(1,1,2), Q is (0.5,0.5,1) -->
<circle cx="205" cy="175" r="3" class="dot" />
<text x="215" y="175" class="point-label">Q</text>

<!-- Vector BQ -->
<line x1="200" y1="143.43" x2="205" y2="175" class="line" marker-end="url(#arrow)" />

<!-- Vector BM -->
<line x1="200" y1="143.43" x2="210" y2="150" class="line" marker-end="url(#arrow)" />