Question 1
Given a parallelogram $ABCD$, let $E$ lie on side $BC$ and $F$ on side $CD$ so that triangles $ABE$ and $BCF$ have equal area. The diagonal $BD$ meets $AE$ at $M$ and $AF$ at $N$.
(a) Prove that a triangle exists with side lengths $BM,MN,ND$.
(b) As $E,F$ vary so that $MN$ decreases, prove that the circumradius of that triangle also decreases.
Editorial status: the literal statement allows endpoints and is false there. The source proofs establish the conditional version with $E,F$ strictly inside their sides, equivalently $0\lt BE/BC=CF/CD\lt1$. This additional interiority condition is a substantive repair, not implied by the two given area triangles. The original statement above is retained for review.
The supplied source has no question point values. Each problem uses 1 editorial point for import compatibility; this is not an official scoring scale.
The supplied PDF does not state a time limit. This edition uses the user-approved 240 minutes per session, 480 minutes total for both sessions; this is an editorial convention.
The printed dates are 18–19 May 1994; grouping Q1–3 and Q4–6 into sessions is editorial, since no separate day boundary is printed. Import scoring multipliers are correct 1, incorrect 0, blank 0. All six essays require manual assessment; the rubrics are editorial, not official step marks.
(a) Prove that a triangle exists with side lengths $BM,MN,ND$.
(b) As $E,F$ vary so that $MN$ decreases, prove that the circumradius of that triangle also decreases.
Editorial status: the literal statement allows endpoints and is false there. The source proofs establish the conditional version with $E,F$ strictly inside their sides, equivalently $0\lt BE/BC=CF/CD\lt1$. This additional interiority condition is a substantive repair, not implied by the two given area triangles. The original statement above is retained for review.
The supplied source has no question point values. Each problem uses 1 editorial point for import compatibility; this is not an official scoring scale.
The supplied PDF does not state a time limit. This edition uses the user-approved 240 minutes per session, 480 minutes total for both sessions; this is an editorial convention.
The printed dates are 18–19 May 1994; grouping Q1–3 and Q4–6 into sessions is editorial, since no separate day boundary is printed. Import scoring multipliers are correct 1, incorrect 0, blank 0. All six essays require manual assessment; the rubrics are editorial, not official step marks.