Let $f:[a,b]\to\mathbb R$ be a continous function with the property that there exists a constant $\lambda\in\mathbb R$ so that for every $x\in[a,b]$ there exists a $y\in[a,b]-\{x\}$ s.t. $\int_x^yf(x)dx=\lambda$. Prove that the function $f$ has at least two zeros in $(a,b)$. [i]Eugen Paltanea[/i]

Let $f:[0,1]\to\mathbb R$ be a differentiable function, while $f'$ is continuous on $[0,1]$ and $|f'(x)|\le1$, $(\forall)x\in[0,1]$. If $$2\left|\int^1_0f(x)dx\right|\le1$$ Show that: $$(n+2)\left|\int^1_0x^nf(x)dx\right|\le1,~(\forall)x\ge1$$ [i]Proposed by Florin Stănescu and Şerban Cioculescu[/i]

A circle in the first quadrant with center on the curve $y=2x^2-27$ is tangent to the $y$-axis and the line $4x=3y$. The radius of the circle is $\frac{m}{n}$ where $M$ and $n$ are relatively prime positive integers. Find $m+n$.

Find the positive integer $n$ such that $10^n$ cubic centimeters is the same as 1 cubic kilometer.

Let $\{u_{n}\}_{n \ge 0}$ be a sequence of positive integers defined by \[u_{0}= 1, \;u_{n+1}= au_{n}+b,\] where $a, b \in \mathbb{N}$. Prove that for any choice of $a$ and $b$, the sequence $\{u_{n}\}_{n \ge 0}$ contains infinitely many composite numbers.

The $12$ numbers on a clock face are rearranged. Show that we can still find three adjacent numbers whose sum is $21$ or more.

Show that if $D_1$ and $D_2$ are two skew lines, then there are infinitely many straight lines such that their points have equal distance from $D_1$ and $D_2.$

Show that there are infinitely many positive integer numbers $n$ such that $n^2+1$ has two positive divisors whose difference is $n$.

Let there be a pentagon $ABCDE$ inscribed in a circle $O$. The tangent to $O$ at $E$ is parallel to $AD$. A point $F$ lies on $O$ and it is in the opposite side of $A$ with respect to $CD$, and satisfies $AB \cdot BC \cdot DF = AE \cdot ED \cdot CF$ and $\angle CFD = 2\angle BFE$. Prove that the tangent to $O$ at $B,E$ and line $AF$ concur at one point.

If positive real numbers $x,y,z$ satisfy the following system of equations, compute $x+y+z$. $$xy+yz = 30$$ $$yz+zx = 36$$ $$zx+xy = 42$$ [i]Proposed by Nathan Xiong[/i]

The base of a triangle is of length $b$, and the latitude is of length $h$. A rectangle of height $x$ is inscribed in the triangle with the base of the rectangle in the base of the triangle. The area of the rectangle is: $ \textbf{(A)}\ \frac{bx}{h}(h-x)\qquad\textbf{(B)}\ \frac{hx}{b}(b-x)\qquad\textbf{(C)}\ \frac{bx}{h}(h-2x)\qquad$ $\textbf{(D)}\ x(b-x)\qquad\textbf{(E)}\ x(h-x) $

The quadrilateral $ABCD$ is inscribed in circle $\omega$. $F$ is the intersection point of $AC$ and $BD$. $BA$ and $CD$ meet at $E$. Let the projection of $F$ on $AB$ and $CD$ be $G$ and $H$, respectively. Let $M$ and $N$ be the midpoints of $BC$ and $EF$, respectively. If the circumcircle of $\triangle MNG$ only meets segment $BF$ at $P$, and the circumcircle of $\triangle MNH$ only meets segment $CF$ at $Q$, prove that $PQ$ is parallel to $BC$.

Given a triangle $ABC$ for which $C=90$ degrees, prove that given $n$ points inside it, we can name them $P_1, P_2 , \ldots , P_n$ in some way such that: $\sum^{n-1}_{k=1} \left( P_K P_{k+1} \right)^2 \leq AB^2$ (the sum is over the consecutive square of the segments from $1$ up to $n-1$). [i]Edited by orl.[/i]

An isosceles triangle $ABC$ with base $AC$ is given. Point $H$ is the intersection of altitudes. On the sides $AB$ and $BC$, points $M$ and $K$ are selected, respectively, so that the angle $KMH$ is right. Prove that a right-angled triangle can be constructed from the segments $AK, CM$ and $MK$.

In rectangle $ABCD$, $AB=6$, $AD=30$, and $G$ is the midpoint of $\overline{AD}$. Segment $AB$ is extended $2$ units beyond $B$ to point $E$, and $F$ is the intersection of $\overline{ED}$ and $\overline{BC}$. What is the area of $BFDG$? $ \textbf{(A)}\ \frac{133}{2}\qquad\textbf{(B)}\ 67\qquad\textbf{(C)}\ \frac{135}{2}\qquad\textbf{(D)}\ 68\qquad\textbf{(E)}\ \frac{137}{2}$

Let $a,b,c,x,y,z$ be positive real numbers, such that $a+b+c=xyz=1$ Prove that: $$ \frac{x^2}{3a+2}+\frac{y^2}{3b+2}+\frac{z^2}{3c+2} \ge 1 $$ When does equality hold?

Fold the next seven corners into a rectangle. [img]https://cdn.artofproblemsolving.com/attachments/b/b/2b8b9d6d4b72024996a66d41f865afb91bb9b7.png[/img]

Define a domino to be an ordered pair of distinct positive integers. A proper sequence of dominos is a list of distinct dominos in which the first coordinate of each pair after the first equals the second coordinate of the immediately preceding pair, and in which $(i,j)$ and $(j,i)$ do not both appear for any $i$ and $j.$ Let $D_{40}$ be the set of all dominos whose coordinates are no larger than 40. Find the length of the longest proper sequence of dominos that can be formed using the dominos of $D_{40}.$

A ball of mass $m_\text{1}$ travels along the x-axis in the positive direction with an initial speed of $v_{\text{0}}$. It collides with a ball of mass $m_\text{2}$ that is originally at rest. After the collision, the ball of mass $m_\text{1}$ has velocity $v_{\text{1x}}\hat{x}+v_{\text{1y}}\hat{y}$ and the ball of mass $m_\text{2}$ has velocity $v_{\text{2x}}\hat{x}+v_{\text{2y}}\hat{y}$. Consider the following five statements: $\text{I)} \ \ \ \ \ \ 0=m_{\text{1}}v_{\text{1x}}+m_{\text{1}}v_{\text{2x}}$ $\text{II)} \ \ \ \ \ m_{\text{1}}v_{\text{0}}=m_{\text{1}}v_{\text{1y}}+m_{\text{2}}v_{\text{2y}}$ $\text{III)} \ \ \ \ 0=m_{\text{1}}v_{\text{1y}}+m_{\text{2}}v_{\text{2y}}$ $\text{IV)} \ \ \ \ m_{\text{1}}v_{\text{0}}=m_{\text{1}}v_{\text{1x}}+m_{\text{1}}v_{\text{1y}}$ $\text{V)} \ \ \ \ \ m_{\text{1}}v_{\text{0}}=m_{\text{1}}v_{\text{1x}}+m_{\text{2}}v_{\text{2x}}$ Of these five statements, the system must satisfy (a) $\text{I and II}$ (b) $\text{III and V}$ (c) $\text{II and V}$ (d) $\text{III and IV}$ (e) $\text{I and III}$

Find the number of pairs of elements, from a group of order $ 2011, $ such that the square of the first element of the pair is equal to the cube of the second element. [i]Teodor Radu[/i]

Find all positive integers $n>2$ such that $$ n! \mid \prod_{ p<q\le n, p,q \, \text{primes}} (p+q)$$

1. Let $f(n) = n^2+6n+11$ be a function defined on positive integers. Find the sum of the first three prime values $f(n)$ takes on. [i]Proposed by winnertakeover[/i]

Let $ABC$ be an equilateral triangle whose angle bisectors of $B$ and $C$ intersect at $D$. Perpendicular bisectors of $BD$ and $CD$ intersect $BC$ at points $E$ and $Z$ respectively. a) Prove that $BE=EZ=ZC$. b) Find the ratio of the areas of the triangles $BDE$ to $ABC$

A cube was split into 8 parallelepipeds by three planes parallel to its faces. The resulting parts were painted in a chessboard pattern. The volumes of the black parallelepipeds are 1, 6, 8, 12. Find the volumes of the white parallelepipeds. [i]Oleg Smirnov[/i]

In triangle $ABC$, $\angle A=90^{\circ}$ and $M$ is the midpoint of $BC$. Point $D$ is chosen on segment $AC$ such that $AM=AD$ and $P$ is the second meet point of the circumcircles of triangles $\Delta AMC,\Delta BDC$. Prove that the line $CP$ bisects $\angle ACB$.