Navvye/Olympiad-Problems-CSV huggingface dataset:
1986 USAMO
What is the smallest integer $n$, greater than one, for which the root-mean-square of the first $n$ positive integers is an integer?



$\mathbf{Note.}$ The root-mean-square of $n$ numbers $a_1, a_2, \cdots, a_n$ is defined to be

$$\left[\frac{a_1^2 + a_2^2 + \cdots + a_n^2}n\right]^{1/2}$$
1986 USAMO
Two distinct circles $K_1$ and $K_2$ are drawn in the plane. They intersect at points $A$ and $B$, where $AB$ is the diameter of $K_1$. A point $P$ on $K_2$ and inside $K_1$ is also given.



Using only a "T-square" (i.e. an instrument which can produce a straight line joining two points and the perpendicular to a line through a point on or off the line), find a construction for two points $C$ and $D$ on $K_1$ such that $CD$ is perpendicular to $AB$ and $\angle CPD$ is a right angle.
1986 USAMO
By a partition $\pi$ of an integer $n\ge 1$, we mean here a representation of $n$ as a sum of one or more positive integers where the summands must be put in nondecreasing order. (E.g., if $n=4$, then the partitions $\pi$ are $1+1+1+1$, $1+1+2$, $1+3, 2+2$, and $4$).



For any partition $\pi$, define $A(\pi)$ to be the number of $1$'s which appear in $\pi$, and define $B(\pi)$ to be the number of distinct integers which appear in $\pi$. (E.g., if $n=13$ and $\pi$ is the partition $1+1+2+2+2+5$, then $A(\pi)=2$ and $B(\pi) = 3$).



Prove that, for any fixed $n$, the sum of $A(\pi)$ over all partitions of $\pi$ of $n$ is equal to the sum of $B(\pi)$ over all partitions of $\pi$ of $n$.
1985 USAMO
Determine whether or not there are any positive integral solutions of the simultaneous equations

\begin{align*}x_1^2+x_2^2+\cdots+x_{1985}^2&=y^3,\\
x_1^3+x_2^3+\cdots+x_{1985}^3&=z^2\end{align*}with distinct integers $x_1$, $x_2$, $\ldots$, $x_{1985}$.
1985 USAMO
Determine each real root of $$x^4-(2\cdot10^{10}+1)x^2-x+10^{20}+10^{10}-1=0$$correct to four decimal places.
1985 USAMO
Let $A,B,C,D$ denote four points in space such that at most one of the distances $AB,AC,AD,BC,BD,CD$ is greater than $1$. Determine the maximum value of the sum of the six distances.
1985 USAMO
There are $n$ people at a party. Prove that there are two people such that, of the remaining $n-2$ people, there are at least $\left\lfloor\frac{n}{2}\right\rfloor-1$ of them, each of whom either knows both or else knows neither of the two. Assume that knowing is a symmetric relation, and that $\lfloor x\rfloor$ denotes the greatest integer less than or equal to $x$.
1985 USAMO
Let $a_1,a_2,a_3,\cdots$ be a non-decreasing sequence of positive integers. For $m\ge1$, define $b_m=\min\{n: a_n \ge m\}$, that is, $b_m$ is the minimum value of $n$ such that $a_n\ge m$. If $a_{19}=85$, determine the maximum value of $$a_1+a_2+\cdots+a_{19}+b_1+b_2+\cdots+b_{85}.$$
1984 USAMO
The product of two of the four roots of the quartic equation $x^4 - 18x^3 + kx^2+200x-1984=0$ is $-32$. Determine the value of $k$.
1984 USAMO
The geometric mean of any set of $m$ non-negative numbers is the $m$-th root of their product.



$\quad (\text{i})\quad$ For which positive integers $n$ is there a finite set $S_n$ of $n$ distinct positive integers such that the geometric mean of any subset of $S_n$ is an integer?

$\quad (\text{ii})\quad$ Is there an infinite set $S$ of distinct positive integers such that the geometric mean of any finite subset of $S$ is an integer?
1984 USAMO
$P, A, B, C,$ and $D$ are five distinct points in space such that $\angle APB = \angle BPC = \angle CPD = \angle DPA = \theta$, where $\theta$ is a given acute angle. Determine the greatest and least values of $\angle APC + \angle BPD$.
1984 USAMO
A difficult mathematical competition consisted of a Part I and a Part II with a combined total of $28$ problems. Each contestant solved $7$ problems altogether. For each pair of problems, there were exactly two contestants who solved both of them. Prove that there was a contestant who, in Part I, solved either no problems or at least four problems.
1984 USAMO
$P(x)$ is a polynomial of degree $3n$ such that



\begin{eqnarray*}

P(0) = P(3) = \cdots &=& P(3n) = 2, \\
P(1) = P(4) = \cdots &=& P(3n-2) = 1, \\
P(2) = P(5) = \cdots &=& P(3n-1) = 0, \quad\text{ and }\\
&& P(3n+1) = 730.\end{eqnarray*}



Determine $n$.
1983 USAMO
On a given circle, six points $A$, $B$, $C$, $D$, $E$, and $F$ are chosen at random, independently and uniformly with respect to arc length. Determine the probability that the two triangles $ABC$ and $DEF$ are disjoint, i.e., have no common points.
1983 USAMO
Prove that the roots of$$x^5 + ax^4 + bx^3 + cx^2 + dx + e = 0$$ cannot all be real if $2a^2 < 5b$.
1983 USAMO
Each set of a finite family of subsets of a line is a union of two closed intervals. Moreover, any three of the sets of the family have a point in common. Prove that there is a point which is common to at least half the sets of the family.
1983 USAMO
Six segments $S_1, S_2, S_3, S_4, S_5,$ and $S_6$ are given in a plane. These are congruent to the edges $AB, AC, AD, BC, BD,$ and $CD$, respectively, of a tetrahedron $ABCD$. Show how to construct a segment congruent to the altitude of the tetrahedron from vertex $A$ with straight-edge and compasses.
1983 USAMO
Consider an open interval of length $1/n$ on the real number line, where $n$ is a positive integer. Prove that the number of irreducible fractions $p/q$, with $1\le q\le n$, contained in the given interval is at most $(n+1)/2$.
1982 USAMO
In a party with $1982$ persons, among any group of four there is at least one person who knows each of the other three. What is the minimum number of people in the party who know everyone else?
1982 USAMO
Let $X_r=x^r+y^r+z^r$ with $x,y,z$ real. It is known that if $S_1=0$, $$(*)\quad\frac{S_{m+n}}{m+n}=\frac{S_m}{m}\frac{S_n}{n}$$for $(m,n)=(2,3),(3,2),(2,5)$, or $(5,2)$. Determine all other pairs of integers $(m,n)$ if any, so that $(*)$ holds for all real numbers $x,y,z$ such that $x+y+z=0$.
1982 USAMO
If a point $A_1$ is in the interior of an equilateral triangle $ABC$ and point $A_2$ is in the interior of $\triangle{A_1BC}$, prove that $$\operatorname{I.Q.} (A_1BC) > \operatorname{I.Q.} (A_2BC),$$where the isoperrimetric quotient of a figure $F$ is defined by $$\operatorname{I.Q.}(F) = \frac{\operatorname{Area}(F)}{[\operatorname{Perimeter}(F)]^2}.$$
1982 USAMO
Prove that there exists a positive integer $k$ such that $k\cdot2^n+1$ is composite for every integer $n$.
1982 USAMO
$A,B$, and $C$ are three interior points of a sphere $S$ such that $AB$ and $AC$ are perpendicular to the diameter of $S$ through $A$, and so that two spheres can be constructed through $A$, $B$, and $C$ which are both tangent to $S$. Prove that the sum of their radii is equal to the radius of $S$.
1981 USAMO
The measure of a given angle is $\frac{180^{\circ}}{n}$ where $n$ is a positive integer not divisible by $3$. Prove that the angle can be trisected by Euclidean means (straightedge and compasses).
1981 USAMO
Every pair of communities in a county are linked directly by one mode of transportation; bus, train, or airplane. All three methods of transportation are used in the county with no community being serviced by all three modes and no three communities being linked pairwise by the same mode. Determine the largest number of communities in this county.
1981 USAMO
If $A,B,C$ are the angles of a triangle, prove that

$$-2 \le \sin{3A}+\sin{3B}+\sin{3C} \le \frac{3\sqrt{3}}{2}$$

and determine when equality holds.
1981 USAMO
The sum of the measures of all the face angles of a given complex polyhedral angle is equal to the sum of all its dihedral angles. Prove that the polyhedral angle is a trihedral angle.



$\mathbf{Note:}$ A convex polyhedral angle may be formed by drawing rays from an exterior point to all points of a convex polygon.
1981 USAMO
If $x$ is a positive real number, and $n$ is a positive integer, prove that



$$[ nx] > \frac{[ x]}1 + \frac{[ 2x]}2 +\frac{[ 3x]}3 + \cdots + \frac{[ nx]}n,$$

where $[t]$ denotes the greatest integer less than or equal to $t$. For example, $[ \pi] = 3$ and $\left[\sqrt2\right] = 1$.
1980 USAMO
A two-pan balance is innacurate since its balance arms are of different lengths and its pans are of different weights. Three objects of different weights $A$, $B$, and $C$ are each weighed separately. When placed on the left-hand pan, they are balanced by weights $A_1$, $B_1$, and $C_1$, respectively. When $A$ and $B$ are placed on the right-hand pan, they are balanced by $A_2$ and $B_2$, respectively. Determine the true weight of $C$ in terms of $A_1, B_1, C_1, A_2$, and $B_2$.
1980 USAMO
Determine the maximum number of three-term arithmetic progressions which can be chosen from a sequence of $n$ real numbers $$a_1<a_2<\cdots<a_n.$$
1980 USAMO
Let $F_r=x^r\sin{rA}+y^r\sin{rB}+z^r\sin{rC}$, where $x,y,z,A,B,C$ are real and $A+B+C$ is an integral multiple of $\pi$. Prove that if $F_1=F_2=0$, then $F_r=0$ for all positive integral $r$.
1980 USAMO
The inscribed sphere of a given tetrahedron touches all four faces of the tetrahedron at their respective centroids. Prove that the tetrahedron is regular.
1980 USAMO
Prove that for numbers $a,b,c$ in the interval $[0,1]$, $$\frac{a}{b+c+1}+\frac{b}{c+a+1}+\frac{c}{a+b+1}+(1-a)(1-b)(1-c) \le 1.$$
1979 USAMO
Determine all non-negative integral solutions $ (n_{1},n_{2},\dots , n_{14}) $ if any, apart from permutations, of the Diophantine Equation $$n_{1}^{4}+n_{2}^{4}+\cdots+n_{14}^{4}=1,599.$$
1979 USAMO
Let $S$ be a great circle with pole $P$.  On any great circle through $P$, two points $A$ and $B$ are chosen equidistant from $P$.  For any  spherical triangle  $ABC$ (the sides are great circles ares), where $C$ is on $S$, prove that the great circle are $CP$ is the angle bisector of angle $C$.



 Note.   A great circle on a sphere is one whose center is the center of the sphere.  A pole of the great circle $S$ is a point $P$ on the sphere such that the diameter through $P$ is perpendicular to the plane of $S$.
1979 USAMO
Given three identical $n$- faced dice whose corresponding faces are identically numbered with arbitrary integers.  Prove that if they are tossed at random, the probability that the sum of the bottom three face numbers is divisible by three is greater than or equal to $\frac{1}{4}$.
1979 USAMO
Show how to construct a chord $BPC$ of a given angle $A$ through a given point $P$ such that $\tfrac{1}{BP}+ \tfrac{1}{PC}$ is a maximum.



[asy]

size(200);

defaultpen(linewidth(0.7));

pair A = origin, B = (5,0), C = (4.2,3), P = waypoint(B--C,0.65);

pair Bp = 1.3 * B, Cp = 1.2 * C;

draw(A--B--C--A^^Bp--A--Cp);

dot(P);

label("$A$",A,W);

label("$B$",B,S);

label("$C$",C,dir(B--C));

label("$P$",P,dir(A--P));

[/asy]
1979 USAMO
A certain organization has $n$ members, and it has $n+1$ three-member committees, no two of which have identical member-ship.  Prove that there are two committees which share exactly one member.
1978 USAMO
Given that $a,b,c,d,e$ are real numbers such that



$a+b+c+d+e=8$,

$a^2+b^2+c^2+d^2+e^2=16$.



Determine the maximum value of $e$.
1978 USAMO
$ABCD$ and $A'B'C'D'$ are square maps of the same region, drawn to different scales and superimposed as shown in the figure. Prove that there is only one point $O$ on the small map that lies directly over point $O'$ of the large map such that $O$ and $O'$ each represent the same place of the country. Also, give a Euclidean construction (straight edge and compass) for $O$.



[asy]

size(200);

defaultpen(linewidth(0.7)+fontsize(10));

real theta = -100, r = 0.3; pair D2 = (0.3,0.76);

string[] lbl = {'A', 'B', 'C', 'D'}; draw(unitsquare); draw(shift(D2)*rotate(theta)*scale(r)*unitsquare);

for(int i = 0; i < lbl.length; ++i) {

pair Q = dir(135-90*i), P = (.5,.5)+Q/2^.5;

label("$"+lbl[i]+"'$", P, Q);

label("$"+lbl[i]+"$",D2+rotate(theta)*(r*P), rotate(theta)*Q);

}[/asy]
1978 USAMO
An integer $n$ will be called good if we can write $$n=a_1+a_2+\cdots+a_k,$$ where $a_1,a_2, \ldots, a_k$ are positive integers (not necessarily distinct) satisfying $$\frac{1}{a_1}+\frac{1}{a_2}+\cdots+\frac{1}{a_n}=1.$$ Given the information that the integers 33 through 73 are good, prove that every integer $\ge 33$ is good.
1978 USAMO
(a) Prove that if the six dihedral (i.e. angles between pairs of faces) of a given tetrahedron are congruent, then the tetrahedron is regular.



(b) Is a tetrahedron necessarily regular if five dihedral angles are congruent?
1978 USAMO
Nine mathematicians meet at an international conference and discover that among any three of them, at least two speak a common language. If each of the mathematicians speak at most three languages, prove that there are at least three of the mathematicians who can speak the same language.
1977 USAMO
Determine all pairs of positive integers $ (m,n)$ such that

$ (1+x^n+x^{2n}+\cdots+x^{mn})$ is divisible by $ (1+x+x^2+\cdots+x^{m})$.
1977 USAMO
$ ABC$ and $ A'B'C'$ are two triangles in the same plane such that the lines $ AA',BB',CC'$ are mutually parallel. Let $ [ABC]$ denotes the area of triangle $ ABC$ with an appropriate $ \pm$ sign, etc.; prove that

$$ 3([ABC] + [A'B'C']) = [AB'C'] + [BC'A'] + [CA'B'] + [A'BC] + [B'CA] + [C'AB].$$
1977 USAMO
If $ a$ and $ b$ are two of the roots of $ x^4+x^3-1=0$, prove that $ ab$ is a root of $ x^6+x^4+x^3-x^2-1=0$.
1977 USAMO
Prove that if the opposite sides of a skew (non-planar) quadrilateral are congruent, then the line joining the midpoints of the two diagonals is perpendicular to these diagonals, and conversely, if the line joining the midpoints of the two diagonals of a skew quadrilateral is perpendicular to these diagonals, then the opposite sides of the quadrilateral are congruent.
1977 USAMO
If $ a,b,c,d,e$ are positive numbers bounded by $ p$ and $ q$, i.e, if they lie in $ [p,q], 0 < p$, prove that

$$ (a + b + c + d + e)\left(\frac {1}{a} + \frac {1}{b} + \frac {1}{c} + \frac {1}{d} + \frac {1}{e}\right) \le 25 + 6\left(\sqrt {\frac {p}{q}} - \sqrt {\frac {q}{p}}\right)^2$$

and determine when there is equality.
1976 USAMO
(a) Suppose that each square of a 4 x 7 chessboard is colored either black or white. Prove that with any such coloring, the board must contain a rectangle (formed by the horizontal and vertical lines of the board) whose four distinct unit corner squares are all of the same color.

(b) Exhibit a black-white coloring of a 4 x6 board in which the four corner squares of every rectangle, as described above, are not all of the same color.
1976 USAMO
If $ A$ and $ B$ are fixed points on a given circle and $ XY$ is a variable diameter of the same circle, determine the locus of the point of intersection of lines $ AX$ and $ BY$. You may assume that $ AB$ is not a diameter.
1976 USAMO
Determine all integral solutions of $$ a^2+b^2+c^2=a^2b^2.$$
1976 USAMO
If the sum of the lengths of the six edges of a trirectangular tetrahedron $ PABC$ (i.e., $ \angle APB = \angle BPC = \angle CPA = 90^\circ$) is $ S$, determine its maximum volume.
1976 USAMO
If $ P(x),Q(x),R(x)$, and $ S(x)$ are all polynomials such that $$ P(x^5)+xQ(x^5)+x^2R(x^5)=(x^4+x^3+x^2+x+1)S(x),$$ prove that $ x-1$ is a factor of $ P(x)$.
1975 USAMO
(a) Prove that $$ [5x]+[5y] \ge [3x+y] + [3y+x],$$ where $ x,y \ge 0$ and $ [u]$ denotes the greatest integer $ \le u$ (e.g., $ [\sqrt{2}]=1$).

(b) Using (a) or otherwise, prove that $$ \frac{(5m)!(5n)!}{m!n!(3m+n)!(3n+m)!}$$ is integral for all positive integral $ m$ and $ n$.
1975 USAMO
Let $ A,B,C,$ and $ D$ denote four points in space and $ AB$ the distance between $ A$ and $ B$, and so on. Show that $$ AC^2+BD^2+AD^2+BC^2 \ge AB^2+CD^2.$$
1975 USAMO
If $ P(x)$ denotes a polynomial of degree $ n$ such that $ P(k)=\frac{k}{k+1}$ for $ k=0,1,2,\ldots,n$, determine $ P(n+1)$.
1975 USAMO
Two given circles intersect in two points $ P$ and $ Q$. Show how to construct a segment $ AB$ passing through $ P$ and terminating on the circles such that $ AP \cdot PB$ is a maximum.
1975 USAMO
A deck of $ n$ playing cards, which contains three aces, is shuffled at random (it is assumed that all possible card distributions are equally likely). The cards are then turned up one by one from the top until the second ace appears. Prove that the expected (average) number of cards to be turned up is $ (n+1)/2$.
1974 USAMO
Let $ a,b,$ and $ c$ denote three distinct integers, and let $ P$ denote a polynomial having integer coefficients. Show that it is impossible that $ P(a) = b, P(b) = c,$ and $ P(c) = a$.
1974 USAMO
Prove that if $ a,b,$ and $ c$ are positive real numbers, then $$ a^ab^bc^c \ge (abc)^{(a+b+c)/3}.$$
1974 USAMO
Two boundary points of a ball of radius 1 are joined by a curve contained in the ball and having length less than 2. Prove that the curve is contained entirely within some hemisphere of the given ball.
1974 USAMO
A father, a mother and son hold a family tournament, playing a two person board game with no ties. The tournament rules are:

(i) The weakest player chooses the first two contestants.

(ii) The winner of any game plays the next game against the person left out.

(iii) The first person to win two games wins the tournament.

The father is the weakest player, the son the strongest, and it is assumed that any player's probability of winning an individual game from another player does not change during the tournament.

Prove that the father's optimal strategy for winning the tournament is to play the first game with his wife.
1974 USAMO
Consider the two triangles $ ABC$ and $ PQR$ shown below. In triangle $ ABC, \angle ADB = \angle BDC = \angle CDA = 120^\circ$. Prove that $ x=u+v+w$.

[asy]unitsize(7mm);

defaultpen(linewidth(.7pt)+fontsize(10pt));



pair C=(0,0), B=4*dir(5);

pair A=intersectionpoints(Circle(C,5), Circle(B,6))[0];



pair Oc=scale(sqrt(3)/3)*rotate(30)*(B-A)+A;

pair Ob=scale(sqrt(3)/3)*rotate(30)*(A-C)+C;



pair D=intersectionpoints(Circle(Ob,length(Ob-C)), Circle(Oc,length(Oc-B)))[1];



real s=length(A-D)+length(B-D)+length(C-D);

pair P=(6,0), Q=P+(s,0), R=rotate(60)*(s,0)+P;

pair M=intersectionpoints(Circle(P,length(B-C)), Circle(Q,length(A-C)))[0];



draw(A--B--C--A--D--B);

draw(D--C);



label("$B$",B,SE);

label("$C$",C,SW);

label("$A$",A,N);

label("$D$",D,NE);

label("$a$",midpoint(B--C),S);

label("$b$",midpoint(A--C),WNW);

label("$c$",midpoint(A--B),NE);

label("$u$",midpoint(A--D),E);

label("$v$",midpoint(B--D),N);

label("$w$",midpoint(C--D),NNW);



draw(P--Q--R--P--M--Q);

draw(M--R);



label("$P$",P,SW);

label("$Q$",Q,SE);

label("$R$",R,N);

label("$M$",M,NW);

label("$x$",midpoint(P--R),NW);

label("$x$",midpoint(P--Q),S);

label("$x$",midpoint(Q--R),NE);

label("$c$",midpoint(R--M),ESE);

label("$a$",midpoint(P--M),NW);

label("$b$",midpoint(Q--M),NE);[/asy]
1973 USAMO
Two points $ P$ and $ Q$ lie in the interior of a regular tetrahedron $ ABCD$. Prove that angle $ PAQ < 60^\circ$.
1973 USAMO
Let $ \{X_n\}$ and $ \{Y_n\}$ denote two sequences of integers defined as follows:

\begin{align*} X_0 = 1,\ X_1 = 1,\ X_{n + 1} = X_n + 2X_{n - 1} \quad (n = 1,2,3,\ldots), \\
Y_0 = 1,\ Y_1 = 7,\ Y_{n + 1} = 2Y_n + 3Y_{n - 1} \quad (n = 1,2,3,\ldots).\end{align*}Prove that, except for the "1", there is no term which occurs in both sequences.
1973 USAMO
Three distinct vertices are chosen at random from the vertices of a given regular polygon of $ (2n+1)$ sides. If all such choices are equally likely, what is the probability that the center of the given polygon lies in the interior of the triangle determined by the three chosen random points?
1973 USAMO
Determine all roots, real or complex, of the system of simultaneous equations

\begin{align*} x+y+z &= 3, \\
x^2+y^2+z^2 &= 3, \\
x^3+y^3+z^3 &= 3.\end{align*}
1973 USAMO
Show that the cube roots of three distinct prime numbers cannot be three terms (not necessarily consecutive) of an arithmetic progression.
1972 USAMO
The symbols $ (a,b,\ldots,g)$ and $ [a,b,\ldots,g]$ denote the greatest common divisor and least common multiple, respectively, of the positive integers $ a,b,\ldots,g$. For example, $ (3,6,18)=3$ and $ [6,15]=30$. Prove that $$ \frac{[a,b,c]^2}{[a,b][b,c][c,a]}=\frac{(a,b,c)^2}{(a,b)(b,c)(c,a)}.$$
1972 USAMO
A given tetrahedron $ ABCD$ is isoceles, that is, $ AB=CD$, $ AC=BD$, $ AD=BC$. Show that the faces of the tetrahedron are acute-angled triangles.
1972 USAMO
A random selector can only select one of the nine integers $ 1,2,\ldots,9$, and it makes these selections with equal probability. Determine the probability that after $ n$ selections ($ n>1$), the product of the $ n$ numbers selected will be divisible by 10.
1972 USAMO
Let $ R$ denote a non-negative rational number. Determine a fixed set of integers $ a,b,c,d,e,f$, such that for every choice of $ R$, $$ \left| \frac{aR^2+bR+c}{dR^2+eR+f}-\sqrt[3]{2}\right| < \left|R-\sqrt[3]{2}\right|.$$
1972 USAMO
A given convex pentagon $ ABCDE$ has the property that the area of each of five triangles $ ABC, BCD, CDE, DEA$, and $ EAB$ is unity (equal to 1). Show that all pentagons with the above property have the same area, and calculate that area. Show, furthermore, that there are infinitely many non-congruent pentagons having the above area property.
2014 Uzbekistan National Olympiad
Find all integers $\,a,b,c\,$ with $\,1<a<b<c\,$ such that $$ (a-1)(b-1)(c-1)  $$ is a divisor of $abc-1.$
2014 Uzbekistan National Olympiad
Find all functions $f:R\rightarrow R$ such that $$ f(x^3)+f(y^3)=(x+y)(f(x^2)+f(y^2)-f(xy)) $$ for all $x,y\in R$.
2014 Uzbekistan National Olympiad
For all $x,y,z\in \mathbb{R}\backslash \{1\}$, such that $xyz=1$, prove that $$ \frac{x^2}{(x-1)^2}+\frac{y^2}{(y-1)^2}+\frac{z^2}{(z-1)^2}\ge 1 $$
2014 Uzbekistan National Olympiad
A circle passes through the points $A,C$ of triangle $ABC$ intersects with the sides $AB,BC$ at points $D,E$ respectively. Let $ \frac{BD}{CE}=\frac{3}{2}$, $BE=4$, $AD=5$ and $AC=2\sqrt{7} $.

Find the angle $ \angle BDC$.
2014 Uzbekistan National Olympiad
Let $PA_1A_2...A_{12} $  be the regular pyramid, $ A_1A_2...A_{12} $ is regular polygon,  $S$ is area of the triangle $PA_1A_5$ and angle  between of the planes $A_1A_2...A_{12} $ and $ PA_1A_5 $ is equal to $ \alpha $.

Find the volume of the pyramid.
2013 Uzbekistan National Olympiad
Let  real numbers $a,b$ such that $a\ge b\ge 0$.  Prove that  $$ \sqrt{a^2+b^2}+\sqrt[3]{a^3+b^3}+\sqrt[4]{a^4+b^4} \le 3a+b .$$
2013 Uzbekistan National Olympiad
Let  $x$ and $y$ are  real numbers such that  $x^2y^2+2yx^2+1=0.$  If  $S=\frac{2}{x^2}+1+\frac{1}{x}+y(y+2+\frac{1}{x})$,  find

(a)max$S$   and

(b) min$S$.
2013 Uzbekistan National Olympiad
Find  all functions  $f:Q\rightarrow Q$ such that  $$ f(x+y)+f(y+z)+f(z+t)+f(t+x)+f(x+z)+f(y+t)\ge 6f(x-3y+5z+7t) $$  for all  $x,y,z,t\in Q.$
2013 Uzbekistan National Olympiad
Let  circles $ \Gamma $ and $ \omega $ are  circumcircle and incircle  of the  triangle $ABC$,   the incircle  touches  sides  $BC,CA,AB$ at the points  $A_1,B_1,C_1$.   Let $A_2$  and  $B_2$  lies the lines $A_1I$ and $B_1I$ ($A_1$ and $A_2$ lies  different sides  from $I$, $B_1$ and $B_2$ lies  different sides from $I$) such that   $IA_2=IB_2=R$.  Prove that :

(a) $AA_2=BB_2=IO$;

(b)   The lines $AA_2$  and  $BB_2$   intersect   on the circle  $ \Gamma ;$
2013 Uzbekistan National Olympiad
Let  $SABC$  is  pyramid,  such that   $SA\le 4$, $SB\ge 7$, $SC\ge 9$, $AB=5$, $BC\le 6$ and  $AC\le 8$.

Find max value  capacity(volume) of the pyramid $SABC$.
2012 Uzbekistan National Olympiad
Given a digits {$0,1,2,...,9$} . Find the number of numbers of 6 digits which cantain $7$ or $7$'s digit and they is permulated(For example 137456 and 314756 is one numbers).
2012 Uzbekistan National Olympiad
For any positive integers $n$ and $m$ satisfying the equation $n^3+(n+1)^3+(n+2)^3=m^3$, prove that $4\mid n+1$.
2012 Uzbekistan National Olympiad
Given $a,b$ and $c$ positive real numbers with $ab+bc+ca=1$. Then prove that

$\frac{a^3}{1+9b^2ac}+\frac{b^3}{1+9c^2ab}+\frac{c^3}{1+9a^2bc} \geq \frac{(a+b+c)^3}{18}$
2012 Uzbekistan National Olympiad
Given points $A,B,C$ and $D$ lie a circle. $AC\cap BD=K$. $I_1, I_2,I_3$ and $I_4$ incenters of $ABK,BCK,CDK,DKA$. $M_1,M_2,M_3,M_4$ midpoints of arcs $AB,BC,CA,DA$ . Then prove that $M_1I_1,M_2I_2,M_3I_3,M_4I_4$ are concurrent.
2011 Uzbekistan National Olympiad
Let a,b,c Postive real numbers such that $a+b+c\geq 6$. Find the minimum value $A=\sum_{cyc}{a^2}$+$\sum_{cyc}{\frac{a}{b^2+c+1}}$