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9.3

Part of VI Soros Olympiad 1999 - 2000 (Russia)

Problems(4)

(x+\sqrt{x^2 +1} )(y+\sqrt{y^2 +1} - VI Soros Olympiad 1999-00 Round 1 9.3

Source:

5/21/2024
Find x+yx + y if (x+x2+1)(y+y2+1)=1.(x+\sqrt{x^2 +1} )(y+\sqrt{y^2 +1} ) = 1.
algebra
x^2 + bx + c has 2 roots in (2,3) (VI Soros Olympiad 1990-00 R1 9.3)

Source:

5/27/2024
The quadratic trinomial x2+bx+cx^2 + bx + c has two roots belonging to the interval (2,3)(2, 3). Prove that 5b+2c+12<05b+2c+12 < 0.
algebrainequalities
angle bisector wanted, statring with isosceles

Source: VI Soros Olympiad 1990-00 R2 9.3 https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics

5/28/2024
On the sides BCBC and ACAC of the isosceles triangle ABCABC (AB=BCAB = BC), points EE and DD are marked, respectively, so that DE∥ABDE \parallel AB. On the extendsion of side CBCB beyond the point BB, point KK was arbitrarily marked. Let PP be the intersection point of the lines ABAB and KDKD. Let QQ be the intersection point of the lines AKAK and DEDE. Prove that CACA is the bisector of angle ∠PCQ\angle PCQ.
geometryangle bisector
y=x^2 and 1998 points

Source: VI Soros Olympiad 1990-00 R3 9.3 https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics

5/28/2024
On the coordinate plane, the parabola y=x2y = x^2 and the points A(x1,x12)A(x_1, x_1^2), B(x2,x22)B(x_2, x_2^2) are set such that x1=−998x_1=-998, x2=1999x_2 =1999 The segments BX1BX_1, AX2AX_2, BX3BX_3, AX4AX_4,..., BX1997BX_{1997}, AX1998AX_{1998} and XkX_k are constructed succesively with (xk,0)(x_k,0), 1≤k≤19981 \le k \le 1998 and x3x_3, x4x_4,..., x1998x_{1998} are abscissas of the points of intersection of the parabola with segments BX1BX_1, AX2AX_2, BX3BX_3, AX4AX_4,..., BX1997BX_{1997}, AX1998AX_{1998}. Find the value 1x1999+1x2000\frac{1}{x_{1999}}+\frac{1}{x_{2000}}
analytic geometryparabolaalgebra