SMO 2024 Open Round 1

Let $S_k=1+2+3+\cdots+k$ for any positive integer $k$. Find $S_1+S_2+S_3+\cdots+S_{20}$.

Let $S=\displaystyle\sum_{r=1}^{64}r\binom{64}{r}$, where $\binom nr=\dfrac{n!}{r!(n-r)!}$ and $0!=1$. Find $\log_2 S$.

Let $x$ be the largest number in the interval $[0,2\pi]$ such that $(\sin x)^{2024}-(\cos x)^{2024}=1$. Find $\lfloor x\rfloor$.

(Note: If you think that such a number $x$ does not exist, enter your answer “99999”.)

Find the number of real numbers $x$ that satisfies the equation $|x-2|+|x-3|=|2x-5|$.

(Note: If you think that there are no such numbers, enter “0”; if you think that there are infinitely many such numbers, enter “99999”.)

Among all the real numbers that satisfies the inequality $e^x\ge1+2e^{-x}$, find the minimum value of $\lceil e^x+e^{-x}\rceil$.

Find the smallest positive integer $C$ greater than $2024$ such that the sets $A=\{2x^2+2x+C:x\in\mathbb Z\}$ and $B=\{x^2+2024x+2:x\in\mathbb Z\}$ are disjoint.

Let $ABCD$ be a convex quadrilateral inscribed in a circle $\omega$. The bisector of $\angle BAC$ meets $\omega$ at $E$ $(\ne A)$, the bisector of $\angle ABD$ meets $\omega$ at $F$ $(\ne B)$, $AE$ intersects $BF$ at $P$ and $CF$ intersects $DE$ at $Q$. Suppose $EF=20$, $PQ=11$. Find the area of the quadrilateral $PEQF$.

Let $f(x)=\sqrt{x^2+1}+\sqrt{(4-x)^2+4}$. Find the minimum value of $f(x)$.

It is known that $a\ge0$ satisfies $\sqrt{4+\sqrt{4+\sqrt{4+\sqrt{4+a}}}}=a$. Find the value of $(2a-1)^2$.

A rectangle with sides parallel to the horizontal and vertical axes is inscribed in the region bounded by the graph of $y=60-x^2$ and the $x$-axis. If the area of the largest such rectangle has area $k\sqrt5$, find the value of $k$.

Let $x$ be a real number satisfying the equation $x^{x^5}=100$. Find the value of $\lfloor x^5\rfloor$.

Let $a,b,c,d,e$ be distinct integers with $a+b+c+d+e=9$. If $m$ is an integer such that $$(m-a)(m-b)(m-c)(m-d)(m-e)=2009,$$ determine the value of $m$.

Let $\{x\}$ be the fractional part of the number $x$, i.e., $\{x\}=x-\lfloor x\rfloor$. If $S=\int_0^9\{x\}^2\,dx$, find $\lfloor S\rfloor$.

The solution of the inequality $|(x+1)(x-6)|\gt |(x+4)(x-2)|$ can be expressed as $x\lt a$ or $b\lt x\lt c$. If $S=|a|+|b|+|c|$, find $\lfloor14S\rfloor$.

Given that $x,y\gt 0$ and $x\sqrt{2-y^2}+y\sqrt{2-x^2}=2$, find the value of $x^2+y^2$.

A convex polygon has $n$ sides such that no three diagonals are concurrent. It is known that all its diagonals divide the polygon into $2500$ regions. Determine $n$.

Find the number of integers $n$ between $-2029$ and $2029$ inclusive such that $(n+2)^2+n^2$ is divisible by $2029$.

Let $f$ be a function such that for any real number $x$, we have $f(x)+2f(2-x)=x+x^2$. Find the value of $f(1)+f(2)+f(3)+\cdots+f(34)$.

Find the largest possible positive prime integer $p$ such that $p$ divides $$S(p)=1^{p-2}+2^{p-2}+3^{p-2}+4^{p-2}+5^{p-2}+6^{p-2}+7^{p-2}+8^{p-2}.$$

Let $f$ be a function such that $f(x)+f\left(\dfrac1{1-x}\right)=1+\dfrac1x$ for all $x\notin\{0,1\}$. Find the value of $\lfloor180\cdot f(10)\rfloor$.