MathDB

1

Part of 1997 All-Russian Olympiad

Problems(5)

P(xy)^2 \leq P(x^2)P(y^2)

Source: ARMO 1997, 9.1

4/20/2013
Let P(x)P(x) be a quadratic polynomial with nonnegative coeficients. Show that for any real numbers xx and yy, we have the inequality P(xy)2⩽P(x2)P(y2)P(xy)^2 \leqslant P(x^2)P(y^2). E. Malinnikova
quadraticsalgebrapolynomialinequalitiesalgebra proposed
x^2+bx+c = 0 and 2x^2+(b+1)x+c+1 = 0 have two integer roots

Source: ARMO 1997, 9.5

4/20/2013
Do there exist real numbers bb and cc such that each of the equations x2+bx+c=0x^2+bx+c = 0 and 2x2+(b+1)x+c+1=02x^2+(b+1)x+c+1 = 0 have two integer roots? N. Agakhanov
algebra proposedalgebra
Find integer solutions of the equation (x^2 - y^2)^2=1+16y

Source: ARMO 1997, 10.1

4/20/2013
Find all integer solutions of the equation (x2−y2)2=1+16y(x^2 - y^2)^2 = 1 + 16y. M. Sonkin
modular arithmeticnumber theory proposednumber theory
ax^2 +bx+c and (a+1)x^2 +(b + 1)x + (c + 1)

Source: ARMO 1997, 10.5

4/20/2013
Do there exist two quadratic trinomials ax2+bx+cax^2 +bx+c and (a+1)x2+(b+1)x+(c+1)(a+1)x^2 +(b + 1)x + (c + 1) with integer coeficients, both of which have two integer roots? N. Agakhanov
quadraticsmodular arithmeticalgebrapolynomialalgebra proposed
quadratic trinomials x^2 + px + q

Source: ARMO 1997, 11.5

4/20/2013
Of the quadratic trinomials x2+px+qx^2 + px + q where pp; qq are integers and 1⩽p,q⩽19971\leqslant p, q \leqslant 1997, which are there more of: those having integer roots or those not having real roots? M. Evdokimov
quadraticsalgebra proposedalgebra