A complete cycle of a traffic light takes 60 seconds. During each cycle the light is green for 25 seconds, yellow for 5 seconds, and red for 30 seconds. At a randomly chosen time, what is the probability that the light will NOT be green? $ \text{(A)}\ \frac{1}{4}\qquad\text{(B)}\ \frac{1}{3}\qquad\text{(C)}\ \frac{5}{12}\qquad\text{(D)}\ \frac{1}{2}\qquad\text{(E)}\ \frac{7}{12} $

Petya has $10, 000$ balls, among them there are no two balls of equal weight. He also has a device, which works as follows: if he puts exactly $10$ balls on it, it will report the sum of the weights of some two of them (but he doesn't know which ones). Prove that Petya can use the device a few times so that after a while he will be able to choose one of the balls and accurately tell its weight.

In convex quadrilateral $ABCD, AB \perp AD, AD = DC$. Let $ E$ be a point on side $BC$, and $F$ be a point on the extension of $DE$ such that $\angle ABF = \angle DEC>90^{\circ}$. Let $O$ be the circumcenter of $\triangle CDE$, and $P$ be a point on the side extension of $FO$ satisfying $FB =FP$. Line BP intersects AC at point Q. Prove that $\angle AQB =\angle DPF.$

Let $DEF$ be a triangle and H the foot of the altitude from $D$ to $EF$. If $DE = 60$, $DF = 35$, and $DH = 21$, what is the difference between the minimum and the maximum possible values for the area of $DEF$?

For each integer $k\geq 2$, determine all infinite sequences of positive integers $a_1$, $a_2$, $\ldots$ for which there exists a polynomial $P$ of the form \[ P(x)=x^k+c_{k-1}x^{k-1}+\dots + c_1 x+c_0, \] where $c_0$, $c_1$, \dots, $c_{k-1}$ are non-negative integers, such that \[ P(a_n)=a_{n+1}a_{n+2}\cdots a_{n+k} \] for every integer $n\geq 1$.

Let $n \geq 3$ and $k \geq 2$ be integers, and form the forward differences of the members of the sequence $1,n,n^2,...n^{k-1}$ and successive forward differences thereof, as illustrated on the right for case $(n,k) = (3,5)$. Prove that all entries of the resulting triangles of positive integers are distinct from one another. Diagram: http://www.cms.math.ca/Competitions/IMTS/imts5.html

Given three quadratic trinomials $f, g, h$ without roots. Their elder coefficients are the same, and all their coefficients for x are different. Prove that there is a number $c$ such that the equations $f (x) + cg (x) = 0$ and $f (x) + ch (x) = 0$ have a common root.

In a right triangle with the length legs $b$ and $c$, and the length hypotenuse $a$, the ratio between the length of the hypotenuse and the length of the diameter of the inscribed circle does not exceed $1 + \sqrt2$. Determine the numerical value of the expression of $E =\frac{a}{b + c}+\frac{b}{c + a}+\frac{c}{a + b}$.

Let $D$ be the interior of the circle $C$ and let $A \in C$. Show that the function $f : D \to \mathbb R, f(M)=\frac{|MA|}{|MM'|}$ where $M' = AM \cap C$, is strictly convex; i.e., $f(P) <\frac{f(M_1)+f(M_2)}{2}, \forall M_1,M_2 \in D, M_1 \neq M_2$ where $P$ is the midpoint of the segment $M_1M_2.$

A function $ f$ defined on the positive integers (and taking positive integers values) is given by: $ \begin{matrix} f(1) \equal{} 1, f(3) \equal{} 3 \\ f(2 \cdot n) \equal{} f(n) \\ f(4 \cdot n \plus{} 1) \equal{} 2 \cdot f(2 \cdot n \plus{} 1) \minus{} f(n) \\ f(4 \cdot n \plus{} 3) \equal{} 3 \cdot f(2 \cdot n \plus{} 1) \minus{} 2 \cdot f(n), \end{matrix}$ for all positive integers $ n.$ Determine with proof the number of positive integers $ \leq 1988$ for which $ f(n) \equal{} n.$

Find a real number $c$ and a positive number $L$ for which \[\lim_{r\to\infty}\frac{r^c\int_0^{\pi/2}x^r\sin x\,dx}{\int_0^{\pi/2}x^r\cos x\,dx}=L.\]

Let $f:[0,1]\rightarrow \mathbb{R}$ a differentiable function such that $f(0)=f(1)=0$ and $|f'(x)|\le 1,\ \forall x\in [0,1]$. Prove that: \[\left|\int_0 ^1f(t)dt\right|<\frac{1}{4}\]

Let $P$ and $Q$ be points on line $\ell$ with $PQ = 12$. Two circles, $\omega$ and ­$\Omega$, are both tangent to $\ell$ at $P$ and are externally tangent to each other. A line through $Q$ intersects $\omega$ at $A$ and $B$, with $A$ closer to $Q$ than $B$, such that $AB = 10$. Similarly, another line through $Q$ intersects ­ ­$\Omega$ at $C$ and $D$, with $C$ closer to $Q$ than $D$, such that $CD = 7$. Find the ratio $AD/BC$.

Find the area of the region of the points such that the total of three tangent lines can be drawn to two parabolas $y=x-x^2,\ y=a(x-x^2)\ (a\geq 2)$ in such a way that there existed the points of tangency in the first quadrant.

In a parallelogram $ABCD$ with $\angle A < 90$, the circle with diameter $AC$ intersects the lines $CB$ and $CD$ again at $E$ and $F$ , and the tangent to this circle at $A$ meets the line $BD$ at $P$ . Prove that the points $P$, $E$, $F$ are collinear.

Let $ABC$ be a triangle and $D,E$ be points on $BC,CA$ such that $AD,BE$ are angle bisectors of $\triangle ABC$. Let $F,G$ be points on the circumcircle of $\triangle ABC$ such that $AF||DE$ and $FG||BC$. Prove that $\frac{AG}{BG}= \frac{AB+AC}{AB+BC}$.

Two persons are playing the following game. They have a pile of stones and take turns removing stones from it, with the first player taking the first turn. At each turn, the first player removes either 1 or 10 stones from the pile, and the second player removes either m or n stones. The player who can not make his move loses. It is known that for any number of stones in the pile, the first player can always win (regardless of the second player's moves). What are the possible values of m and n?

Four semicircles of radius $1$ are placed in a square, as shown below. The diameters of these semicircles lie on the sides of the square and each semicircle touches a vertex of the square. Find the absolute difference between the shaded area and the "hatched" area. [asy] import patterns; add("hatch",hatch(1.2mm)); add("checker",checker(2mm)); real r = 1 + sqrt(3); filldraw((0,0)--(r,0)--(r,r)--(0,r)--cycle,gray(0.4),linewidth(1.5)); fill((1,0)--(r,1)--(r-1,r)--(0,r-1)--cycle,white); fill((1,0)--(r,1)--(r-1,r)--(0,r-1)--cycle,pattern("hatch")); filldraw(arc((1,0),1,0,180)--(0,0)--cycle,white,linewidth(1.5)); filldraw(arc((r,1),1,90,270)--(r,0)--cycle,white,linewidth(1.5)); filldraw(arc((r-1,r),1,180,360)--(r,r)--cycle,white,linewidth(1.5)); filldraw(arc((0,r-1),1,270,450)--(0,r)--cycle,white,linewidth(1.5)); [/asy] [i]Proposed by Connor Gordon[/i]

Let $AH_A, BH_B, CH_C$ be the altitudes of triangle $ABC$. Prove that if $\frac{H_BC}{AC} = \frac{H_CA}{AB}$, then the line symmetric to $BC$ with respect to line $H_BH_C$ is tangent to the circumscribed circle of triangle $H_BH_CA$. [i](Proposed by Mykhailo Bondarenko)[/i]

A $\pm 1$-[i]sequence[/i] is a sequence of $2022$ numbers $a_1, \ldots, a_{2022},$ each equal to either $+1$ or $-1$. Determine the largest $C$ so that, for any $\pm 1$-sequence, there exists an integer $k$ and indices $1 \le t_1 < \ldots < t_k \le 2022$ so that $t_{i+1} - t_i \le 2$ for all $i$, and $$\left| \sum_{i = 1}^{k} a_{t_i} \right| \ge C.$$

Let $ S$ be a set of $ 100$ points in the plane. The distance between every pair of points in $ S$ is different, with the largest distance being $ 30$. Let $ A$ be one of the points in $ S$, let $ B$ be the point in $ S$ farthest from $ A$, and let $ C$ be the point in $ S$ farthest from $ B$. Let $ d$ be the distance between $ B$ and $ C$ rounded to the nearest integer. What is the smallest possible value of $ d$?

For finite sets $A,M$ such that $A \subseteq M \subset \mathbb{Z}^+$, we define $$f_M(A)=\{x\in M \mid x\text{ is divisible by an odd number of elements of }A\}.$$ Given a positive integer $k$, we call $M$ [i]k-colorable[/i] if it is possible to color the subsets of $M$ with $k$ colors so that for any $A \subseteq M$, if $f_M(A)\neq A$ then $f_M(A)$ and $A$ have different colors. Determine the least positive integer $k$ such that every finite set $M \subset\mathbb{Z}^+$ is k-colorable.

In a kindergarten, a nurse took $n$ congruent cardboard rectangles and gave them to $n$ kids, one per each. Each kid has cut its rectangle into congruent squares(the squares of different kids could be of different sizes). It turned out that the total number of the obtained squares is a prime number. Prove that all the initial squares were in fact squares.

Circle $\odot O$ with diameter $\overline{AB}$ has chord $\overline{CD}$ drawn such that $\overline{AB}$ is perpendicular to $\overline{CD}$ at $P$. Another circle $\odot A$ is drawn, sharing chord $\overline{CD}$. A point $Q$ on minor arc $\overline{CD}$ of $\odot A$ is chosen so that $\text{m} \angle AQP + \text{m} \angle QPB = 60^\circ$. Line $l$ is tangent to $\odot A$ through $Q$ and a point $X$ on $l$ is chosen such that $PX=BX$. If $PQ = 13$ and $BQ = 35$, find $QX$. [i]Proposed by Aaron Lin[/i]

Let \( f: \mathbb{R} \to \mathbb{R} \) be a differentiable function such that \( f(0) = 0 \) and \( 0 < f'(t) \leq 1 \) for all \( t \in [0, 1] \). Show that: \[ \left( \int_0^1 f(t) \, dt \right)^2 \geq \int_0^1 f(t)^3 \, dt. \]