However, according to the rule of IMO, we can not disclose the ISL problems before IMO 2011. So the campers MUST NOT share those problems (that are omitted here) with other people.
Like IMO, every exam had 3 problems and the contestants were given 4.5 hours for solving those problems.
Exam 1: Geometry 1
Problem 1:
Problem 2:
For an acute triangle $ABC$ satisfying $AB \neq AC$, denote by $H$ the feet of the perpendicular line segment drawn from the point $A$ to the side $BC$. Take points $P$ and $Q$ in such a way that the $3$ points $A,B,P$ and the $3$ points $A,C,Q$ lie on a straight line in the given order, respectively. If the $4$ points $B,C,P,Q$ lie on the circumference of a circle, and also $HP=HQ$, prove that the point $H$ must coincide with the circumcenter of the triangle $APQ$.
viewtopic.php?f=25&t=808
Problem 3:
Let $\omega$ be the circumcircle of a triangle $ABC$. Let $D$ be a variable point on the arc $AB$ that does not contain $C$, note that $D\neq A,B$. Let $E,F$ be the incenters of the triangles $CAD$ and $CBD$, respectively. Find the locus of the second intersection point of the circumcircle of triangle $DEF$ and $\omega$, as $D$ varies on the arc $AB$.
viewtopic.php?f=25&t=809
Exam 2: Geometry 2
Problem 1:
Given an arbitrary triangle $ABC$ with area $T$ and perimeter $L$. Let $P,Q,R$ be the points of tangency of the sides $BC,CA,AB$ respectively with the inscribed circle. Prove that \[ \left ( \frac{AB}{PQ} \right )^3+ \left ( \frac{BC}{QR} \right )^3+\left ( \frac{CA}{RP} \right )^3 \geq \frac{2}{\sqrt{3}}\cdot \frac{L^2}{T}\]
viewtopic.php?f=25&t=810
Problem 2:
Given trapezoid $ABCD$ with parallel sides $AB$ and $CD$, let $E$ be a point on line $BC$ outside segment $BC$, such that segment $AE$ intersects segment $CD$. Assume that there exists a point $F$ inside segment $AD$ such that $\angle EAD=\angle CBF$. Denote by $I$ the intersection point of $CD$ and $EF$. Let $K$ be the midpoint of segment $EF$, and assume that $K$ is different from $I$ and $J$.
Prove $K$ belong to the circumcircle of $\triangle ABI$ if and only if $K$ belongs to the circumcircle of $\triangle CDJ$.
viewtopic.php?f=25&t=811
Problem 3:
Exam 3: Number Theory 1
Problem 1:
Problem 2:
Consider a polygon with $n$ sides. Each side has the same length $l$. The vertices of the polygon $(x_i,y_i)$ are rational coordinates, $a_i=x_{i+1}-x_i$, $b_i=y_{i+1}-y_i$, and ${a_i}^2+{b_i}^2=l^2$ for $i=1,2,\cdots,n$. Prove that $n$ is even.
viewtopic.php?f=26&t=812
Problem 3:
The integer $x$ is at least $3$ and $n=x^6-1$. Let $p$ be a prime and $k$ be a positive integer such that $p^k$ is a factor of $n$. Show that $p^{3k}<8n$.
viewtopic.php?f=26&t=813
Exam 4: Number Theory 2
Problem 1:
Find all prime numbers $p$ for which $p^3-4p+9$ is a perfect square.
viewtopic.php?f=26&t=814
Problem 2:
Problem 3:
$N$ is a $5$ digit number, of which the first and the last digits are nonzero. $N$ is a palindromic product if
- $N$ is a palindrome (it reads the same way from the left to right or right to left such as $12321$).
- $N$ is a product of two positive integers, of which the first, when read from left to right, is equal to the second, when read from right to left. For example, $20502$ is a palindromic product, since $102\cdot 201=20502$ and $20502$ itself is a palindrome.
viewtopic.php?f=26&t=815
