If the function $f\colon [1,2]\to R$ is such that $\int_{1}^{2}f(x) dx=\frac{73}{24}$, then show that there exists a $x_{0}\in (1;2)$ such that \[x_{0}^{2}<f(x_{0})<x_{0}^{3}\] [Edit: $f$ is continuous]

Let $ABC$ be an acute-angled triangle. $M ,N$ are any two points on the sides $AB , AC$ respectively. The circles with the diameters $BN$ and $CM$ intersect at points $P$ and $Q$. Show that the points $P, Q$ and the orthocenter of the triangle $ABC$ lie on a straight line.

Joe has a collection of 23 coins, consisting of 5-cent coins, 10-cent coins, and 25-cent coins. He has 3 more 10-cent coins than 5-cent coins, and the total value of his collection is 320 cents. How many more 25-cent coins does Joe have than 5-cent coins? $\textbf{(A) } 0 \qquad \textbf{(B) } 1 \qquad \textbf{(C) } 2 \qquad \textbf{(D) } 3 \qquad \textbf{(E) } 4 $

What is the least positive integer $n$ such that $\overbrace{f(f(\dots f}^{21 \text{ times}}(n)))=2013$ where $f(x)=x+1+\lfloor \sqrt x \rfloor$? ($\lfloor a \rfloor$ denotes the greatest integer not exceeding the real number $a$.) $ \textbf{(A)}\ 1214 \qquad\textbf{(B)}\ 1202 \qquad\textbf{(C)}\ 1186 \qquad\textbf{(D)}\ 1178 \qquad\textbf{(E)}\ \text{None of above} $

In triangle $ABC$, right at $C$, let $F$ be the intersection point of the altitude $CD$ with the angle bisector $AE$ and $G$ be the intersection point of $ED$ with $BF$. Prove that the area of the quadrilateral $CEGF$ is equal to the area of the triangle $BDG$ .

Let $P(x)=x^n+a_{n-1}x^{n-1}+\cdots+a_0$ be a polynomial of degree $n\geq 3.$ Knowing that $a_{n-1}=-\binom{n}{1}$ and $a_{n-2}=\binom{n}{2},$ and that all the roots of $P$ are real, find the remaining coefficients. Note that $\binom{n}{r}=\frac{n!}{(n-r)!r!}.$

Circles $ \Gamma_1$ and $ \Gamma_2$ are internally tangent to circle $ \Gamma$ at $ P$ and $ Q$, respectively. Let $ P_1$ and $ Q_1$ are on $ \Gamma_1$ and $ \Gamma_2$ respectively such that $ P_1Q_1$ is the common tangent of $ P_1$ and $ Q_1$. Assume that $ \Gamma_1$ and $ \Gamma_2$ intersect at $ R$ and $ R_1$. Define $ O_1,O_2,O_3$ as the intersection of $ PQ$ and $ P_1Q_1$, the intersection of $ PR$ and $ P_1R_1$, and the intersection $ QR$ and $ Q_1R_1$. Prove that the points $ O_1,O_2,O_3$ are collinear. [i]Rudi Adha Prihandoko, Bandung[/i]

Suppose that $(a_1,b_1),$ $(a_2,b_2),$ $\dots,$ $(a_{100},b_{100})$ are distinct ordered pairs of nonnegative integers. Let $N$ denote the number of pairs of integers $(i,j)$ satisfying $1\leq i<j\leq 100$ and $|a_ib_j-a_jb_i|=1$. Determine the largest possible value of $N$ over all possible choices of the $100$ ordered pairs. [i]Proposed by Ankan Bhattacharya[/i]

Two non-intersecting circles, $\omega$ and $\Omega$, have centers $C_\omega$ and $C_\Omega$ respectively. It is given that the radius of $\Omega$ is strictly larger than the radius of $\omega$. The two common external tangents of $\Omega$ and $\omega$ intersect at a point $P$, and an internal tangent of the two circles intersects the common external tangents at $X$ and $Y$. Suppose that the radius of $\omega$ is $4$, the circumradius of $\triangle PXY$ is $9$, and $XY$ bisects $\overline{PC_\Omega}$. Compute $XY$.

Pairwise distinct real numbers are arranged into an $m \times n$ rectangular array. In each row the entries are arranged increasingly from left to right. Each column is then rearranged in decreasing order from top to bottom. Prove that in the reorganized array, the rows remain arranged increasingly.

For which real numbers $a$ is the set of all solutions of the inequality $$(x^2 + ax + 4)(x^2 - 5x + 6) < 0$$ an interval?

A square has been divided into $2022$ rectangles with no two of them having a common interior point. What is the maximal number of distinct lines that can be determined by the sides of these rectangles?

On the dart board shown in the figure, the outer circle has radius 6 and the inner circle has radius 3. Three radii divide each circle into the three congruent regions, with point values shown. The probability that a dart will hit a given region is proportional to to the area of the region. What two darts hit this board, the score is the sum of the point values in the regions. What is the probability that the score is odd? [asy] draw(Circle(origin, 2)); draw(Circle(origin, 1)); draw(origin--2*dir(90)); draw(origin--2*dir(210)); draw(origin--2*dir(330)); label("$1$", 0.35*dir(150), dir(150)); label("$1$", 1.3*dir(30), dir(30)); label("$1$", (0,-1.3), dir(270)); label("$2$", 1.3*dir(150), dir(150)); label("$2$", 0.35*dir(30), dir(30)); label("$2$", (0,-0.35), dir(270));[/asy] $ \textbf{(A)}\: \frac{17}{36}\qquad \textbf{(B)}\: \frac{35}{72}\qquad \textbf{(C)}\: \frac{1}{2}\qquad \textbf{(D)}\: \frac{37}{72}\qquad \textbf{(E)}\: \frac{19}{36}\qquad $

[b]p1.[/b] For what integers $n$ is $2^6 + 2^9 + 2^n$ the square of an integer? [b]p2.[/b] Two integers are chosen at random (independently, with repetition allowed) from the set $\{1,2,3,...,N\}$. Show that the probability that the sum of the two integers is even is not less than the probability that the sum is odd. [b]p3.[/b] Let $X$ be a point in the second quadrant of the plane and let $Y$ be a point in the first quadrant. Locate the point $M$ on the $x$-axis such that the angle $XM$ makes with the negative end of the $x$-axis is twice the angle $YM$ makes with the positive end of the $x$-axis. [b]p4.[/b] Let $a,b$ be positive integers such that $a \ge b \sqrt3$. Let $\alpha^n = (a + b\sqrt3)^n = a_n + b_n\sqrt3$ for $n = 1,2,3,...$. i. Prove that $\lim_{n \to + \infty} \frac{a_n}{b_n}$ exists. ii. Evaluate this limit. [b]p5.[/b] Suppose $m$ and $n$ are the hypotenuses are of Pythagorean triangles, i.e,, there are positive integers $a,b,c,d$, so that $m^2 = a^2 + b^2$ and $n^2= c^2 + d^2$. Show than $mn$ is the hypotenuse of at least two distinct Pythagorean triangles. Hint: you may not assume that the pair $(a,b)$ is different from the pair $(c,d)$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $a,b,c,d$ be four positive real numbers. If they satisfy \[a+b+\frac{1}{ab}=c+d+\frac{1}{cd}\quad\text{and}\quad\frac1a+\frac1b+ab=\frac1c+\frac1d+cd\] then prove that at least two of the values $a,b,c,d$ are equal.

The positive $n > 3$ called ‘nice’ if and only if $n +1$ and $8n + 1$ are both perfect squares. How many positive integers $k \le 15$ such that $4n + k$ are composites for all nice numbers $n$?

$100$ points are marked in the plane so that no three of marked points are collinear. One of marked points is red, and the others are blue. A triangle with vertices at blue points is called [i]good[/i] if the red point lies inside it. Determine if it is possible that the number of good triangles is not less than the half of the total number of traingles with vertices at blue points.

If $\sum_{i=1}^{n} \cos ^{-1}\left(\alpha_{i}\right)=0,$ then find $\sum_{i=1}^{n} \alpha_{i}$ [list=1] [*] $\frac{n}{2} $ [*] $n $ [*] $n\pi $ [*] $\frac{n\pi}{2} $ [/list]

In the Faculty of Cybernetics football championship, \( n \geq 3 \) teams participated. The competition was held in a round-robin format, meaning that each team played against every other team exactly once. For a win, a team earns 3 points, for a loss no points are awarded, and for a draw, both teams receive 1 point each. It turned out that the winning team scored strictly more points than any other team and had at most as many wins as losses. What is the smallest \( n \) for which this could happen? [i]Proposed by Bogdan Rublov[/i]

Determine all pairs of polynomials $f$ and $g$ with real coefficients such that \[ x^2 \cdot g(x) = f(g(x)). \]

What is the $7$th term of the sequence $\{-1, 4,-2, 3,-3, 2,...\}$? (A) $ -1$ (B) $ -2$ (C) $-3$ (D) $-4$ (E) None of the above

Let $S=2^3+3^4+5^4+7^4+\cdots+17497^4$ be the sum of the fourth powers of the first $2014$ prime numbers. Find the remainder when $S$ is divided by $240$.

In an acute-angled triangle $ABC, D$ is the foot of the altitude from $A$. Let $D_1$ and $D_2$ be the symmetric points of $D$ wrt $AB$ and $AC$, respectively. Let $E_1$ be the intersection of $BC$ and the line through $D_1$ parallel to $AB$ . Let $E_2$ be the intersection of$ BC$ and the line through $D_2$ parallel to $AC$. Prove that $D_1, D_2, E_1$ and $E_2$ on one circle whose center lies on the circumscribed circle of $\vartriangle ABC$.

Let $A_1,A_2,\ldots,A_{2n}$ are the vertices of a regular $2n$-gon and $P$ is a point from the incircle of the polygon. If $\alpha_i=\angle A_iPA_{i+n}$, $i=1,2,\ldots,n$. Prove the equality $$\sum_{i=1}^n\tan^2\alpha_i=2n\frac{\cos^2\frac\pi{2n}}{\sin^4\frac\pi{2n}}.$$

A jeweler makes necklaces with round stones, four emeralds (green) and four rubies (red), arranged at equal distances from each other. One day, they decide to give away some necklaces. How many necklaces can they give away without the risk of two friends ending up with the same necklace? (*Observation: The necklace is completely symmetrical except for the type of stone, meaning there is not a unique way to form it. Consider this while solving the problem.*)