Algebra
A1. \(x_1,x_2,\dots,x_n\) are positive reals with sum less than 1. Show that \[ n^{n+1} x_1x_2\cdots x_n (1 - x_1 - \dots - x_n) \le (x_1 + \dots + x_n)(1 - x_1)\cdots(1 - x_n). \] S
A2. \(x_1,x_2,\dots,x_n\) are reals \(\ge1\). Show that \[ \frac{1}{1+x_1}+\frac{1}{1+x_2}+\cdots+\frac{1}{1+x_n}\ge\frac{n}{1+(x_1x_2\cdots x_n)^{1/n}}. \] S
A3. \(x,y,z>0\) with \(xyz=1\). Show \[ \frac{x^3}{(1+y)(1+z)}+\frac{y^3}{(1+z)(1+x)}+\frac{z^3}{(1+x)(1+y)}\ge\frac{3}{4}. \] S
A4. Define \(c(n,k)\) by \(c(n,0)=c(n,n)=1\) and \(c(n+1,k)=2^k c(n,k)+c(n,k-1)\). Show \(c(n,k)=c(n-k,k)\). S
Combinatorics
C1. An \(m\times n\) array of reals has integral row and column sums. Show that each non-integral element can be changed to \(\lfloor x\rfloor\) or \(\lfloor x\rfloor+1\) keeping row/column sums unchanged. S
C2. An odd \(n\)-admissible sequence satisfies certain recursive conditions; an even one satisfies others. Show that for \(n>8\) infinitely many positive integers are not \(n\)-attainable, and all but 7 are 3-attainable. S
C3. The numbers \(1\) to \(9\) are in some order. A move reverses any block of consecutive increasing/decreasing numbers. Show at most 12 moves are enough to sort them. S
C4. If \(A\) is a permutation of \(\{1,\dots,n\}\) and \(B\subset\{1,\dots,n\}\), prove there’s a permutation that *splits* all of a family of \(n-2\) such subsets. S
C6. In the complete graph on 10 points, show 5-edge-coloring exists with certain properties, but 4-edge-coloring with analogous properties does not. S
C7. Some cards are black/white on opposite sides on an \(m\times n\) board. Determine for which \(m,n\) all cards can be removed by successive moves. S
Geometry
G2. In cyclic quadrilateral \(ABCD\), points \(E\), \(F\) vary on \(AB\), \(CD\) with \(AE/EB=CF/FD\). If \(P\) on \(EF\) satisfies \(EP/PF=AB/CD\), show \(\text{area}(APD)/\text{area}(BPC)\) is constant. S
G4. Points \(M,N\) inside triangle \(ABC\) satisfy angle equalities; show \(\frac{AM\cdot AN}{AB\cdot AC}+\frac{BM\cdot BN}{BA\cdot BC}+\frac{CM\cdot CN}{CA\cdot CB}=1\). S
G5. In triangle \(ABC\), let \(D,E,F\) be reflections of \(A,B,C\) across opposite sides. Show they are collinear iff \(OH=2R\). S
G6. In a convex hexagon \(ABCDEF\) with \(\angle B+\angle D+\angle F=360^\circ\) and \(AB\cdot CD\cdot EF=BC\cdot DE\cdot FA\), show \(BC\cdot DF\cdot EA=EF\cdot AC\cdot BD\). S
G7. In triangle \(ABC\) with \(\angle C=2\angle B\), \(D\) on \(BC\) with \(DC=2BD\), and \(E\) on line \(AD\) with \(D\) midpoint of \(AE\), show \(\angle ECB+180^\circ=2\angle EBC\). S
G8. In triangle \(ABC\) with \(\angle A=90^\circ\), tangent at \(A\) meets \(BC\) at \(D\). With \(E\), \(X\), \(Y\), \(Z\) defined as usual, show \(BD\) is tangent to circumcircle of \(ADZ\). S
Number Theory
N2. Find all real pairs \((x,y)\) such that \(x\lfloor n y\rfloor=y\lfloor n x\rfloor\) for all positive integers \(n\). S
N3. Find the smallest \(n\) such that in any set of \(n\) distinct integers there are \(a,b,c,d\) with \(a+b\equiv c+d\pmod{20}\). S
N4. A sequence \((a_n)\) is defined by certain recurrence conditions. Find \(a_{1998}\). S
N5. Find all positive integers \(n\) for which some \(m\) makes \(m^2+9\) divisible by \(2^n-1\). S
N7. Show that for any \(n>1\) there is an \(n\)-digit number, all digits non-zero, divisible by the sum of its digits. S
N8. A sequence \((a_k)\) has every non-neg integer uniquely expressible as \(a_i+2a_j+4a_k\). Find \(a_{1998}\). S
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