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424 Problems in Total

2020	Jan	Feb	Mar	Apr	May	Jun
48	Jul	Aug	Sep	Oct	Nov November 2020 Find all primes $p$ to make $p^3-2p+4$ is a perfect square.How many sequences of integers ${a_1, a_2, …, a_7}$ are there for which $-1 \le a_i \le 1$ for every $i$, and $a_1a_2+a_2a_3+a_3a_4+a_4a_5+a_5a_6+a_6a_7 = 4$...There are 12 students in a classroom: 6 of them are Democrats and 6 of them are Republicans. Every hour the students are...In a single-elimination tournament consisting of $2^9=512$ teams, there is a strict ordering on the skill levels of the ...A sphere is centered at a point with integer coordinates and passes through the three points $(2,0,0), (0,4,0), (0,0,6)$...Solve over the integers: $615+x^2=2^y$	Dec December 2020 For $n=1,2,3,… (\sqrt{2}-1)^n=\sqrt{m+1}-\sqrt{m}$ where $m$ is a positive integer which depends on $n$. Proof by (eas...A 4-digit number divides each of its 4 digits will get 4 integers. And the sum of these 4 digits equals to the 4-digit n...If $a,b \in \mathbb{N}$ such that $\dfrac{2019}{2020}<\dfrac{a}{b} < \dfrac{2020}{2021}$, what is the smallest pos...If $\frac{1}{6}=\frac{1}{a_1}+\frac{1}{a_2}+\frac{1}{a_3}+\frac{1}{a_4}+\frac{1}{a_5}+\frac{1}{a_6}$ and $a_i \in \mathbb{N}$...Solve for real values of $x$: $2\sqrt[3]{2x+1}=x^3-1$For what $x \in \mathbb{N}$ are $8x^3-20$ and $2x^5-2$ both perfect squares $?$Tessa has a unit cube, on which each vertex is labeled by a distinct integer between 1 and 8 inclusive. She also has a d...Given $a_1,a_2,…,a_n \in \mathbb{N}, n>4$, prove: there exist $x_1,x_2,…,x_n \in \mathbb{R}$ such that $\displaystyle\prod_{i=1}^{n}{x_i} \ne 0$...Let $a$ be the greatest root of $x^3-3x^2+1=0$. Show $\lfloor {a^{1988}} \rfloor$ is divisible by $17.$Let $a$, $b$ and $c$ be positive integers with $a \ge b \ge c$ such thatThe sum of the first $m$ positive odd integers is 212 more than the sum of the first $n$ positive even integers. What is...Adam, Benin, Chiang, Deshawn, Esther, and Fiona have internet accounts. Some, but not all, of them are internet friends ...A $3 \times 3$ square is partitioned into 9 unit squares. Each unit square is painted either white or black with each co...Real numbers $x$, $y$, and $z$ are chosen independently and at random from the interval $[0, n]$ for some positive integ...In a round-robin tournament with 6 teams, each team plays one game against each other team, and each game results in one...Suppose that one of every $500$ people in a certain population has a particular disease, which displays no symptoms. A b...Jesse cuts a circular paper disk of radius $12$ along two radii to form two sectors, the smaller having a central angle ...Prove that when $p>3$ is a prime, $p!+p$ cannot be a perfect square.

2021	Jan January 2021 Aaron and Brooke are playing a game where a fair coin is being flipped repeatedly. If 4 heads in a row are ever flipped,...An ant starts at the point $(1, 0)$. Each minute, it walks from its current position to one of the four adjacent lattice...Let $x_1, x_2, x_3, x_4$ be positive reals such that $(x_1+x_3)(x_2+x_4)=1$. Prove $\dfrac{x_1^3}{x_2+x_3+x_4} + \dfrac{x_2^3}{x_1+x_3+x_4} + \dfrac{x_3^3}{x_1+x_2+x_4} + \dfrac{x_4^3}{x_1+x_2+x_3} \ge \dfrac{1}{3}.$...Adam, Bob and Charlie, each of them is given a unique number from 1,2,3,4,5,6,7,8,9. Everyone knows his number only. And...Let $S_{n,k}=\displaystyle\sum_{a=0}^{n}{ {a \choose k}{n-a \choose k} }$. Find the remainder when $\displaystyle\sum_{n=0}^{200}{\sum_{k=0}^{200}{S_{n,k}}}$...	Feb February 2021 Let $ABC$ be a triangle with $AB=13, BC=14, AC=15$. Let $H, I, M$ be the orthocenter of $▵{ABC}$, incenter of $▵{ABC}$, ...Let $A=\frac{2}{3} \cdot \frac{5}{6} \cdot \frac{8}{9} \cdots \cdot \frac{995}{996} \cdot \frac{998}{999}$. Which is big...In how many ways can the sequence 1, 2, 3, 4, 5 be rearranged so that no three consecutive terms are increasing and no t...Suppose for all $x, y \in \mathbb{R}$ we have $f(x+y)=f(x)+f(y)+xy$ and $f(4)=10$, find $f(2001)$ (2001 Dutch Math Olymp...Two distinct regular tetrahedra have all their vertices among the vertices of the same unit cube. What is the volume of ...Suppose $n=2^{499}$. There are $2n$ points on a circle and they are numbered as $1, 2, … 2n$. Prove: there exists 100 ...	Mar March 2021 For $\triangle{ABC}$ we know $2sinA +sinB = 2sinC$, find the minimum value for $\dfrac{5}{sinA} + \dfrac{9}{sinC}.$If the sum of the first $n$ positive integers equals to the sum of the next $k$ positive integers, where $n<20$, what...Given $\triangle{ABC}$ and a point $P$ inside it, $\angle{BAP}=18^{\circ}, \angle{CAP}=30^{\circ}, \angle{ACP}=48^{\circ}, AP=BC$...In the diagram below, $ABCD$ is a rectangle with side lengths $AB=3$ and $BC=11$, and $AECF$ is a rectangle with side le...Segments $\overline{AB}, \overline{AC},$ and $\overline{AD}$ are edges of a cube and $\overline{AG}$ is a diagonal throu...Solve for real values of $x$: $2\sqrt[3]{2x+1}=x^3-1$.	Apr April 2021 Let $ABC$ be a triangle and $X$ an interior point of $\triangle{ABC}$. Show that at least one of the angles $\angle{XAB}, \angle{XBC}, \angle{XCA}$...Solve $\sqrt{x-\dfrac{1}{x}}+\sqrt{1-\dfrac{1}{x}}=x$	May May 2021	Jun June 2021 Empty. Open June 2021
89	Jul July 2021 Solve angle $x$.Solve angle $x$	Aug August 2021	Sep September 2021 Calculate the total length of segments in red color.$MN=1$, randomly pick points $P,Q$ on $MN$, what is the possibility that $MP, PQ, QN$ can be three sides of a triangle?Use only a ruler to get the perpendicular line from point $A$ on a circle with diameter $BC$.	Oct October 2021 How many distinct permutations from $1,1,2,2,3,3,4,4$ that there are no equal consecutive numbers?	Nov November 2021 Two problems with known answers:$81^{x^2+y}+81^{y^2+x}=\dfrac{2}{3}, x,y \in \mathbb{R}$, solve $x, y$.$x+y+z=1, x^2+y^2+z^2=2, x^3+y^3+z^3=3, x^5+y^5+z^5=?$ (from MindYourDecisions)100 boxes contain apples, oranges and bananas. Prove that we can choose 51 boxes in such a way that they contain at leas...AMC 10A 2022 Problem 17: An architect is building a structure that will place vertical pillars at the vertices of regula...AMC 12A Problem 24: Convex quadrilateral $ABCD$ has $AB=18, \angle{A}=60^{\circ}$, and $AB \parallel CD$. In some order,...$E$ is midpoint on side of square $ABCD$ and $F$ is a point on side $CD$ so that $\angle{EAF}=45^{\circ}$. $AF$ and $BD$...Solve $\displaystyle\int{\dfrac{1}{x^4+1}}\,dx$Solve $\displaystyle \lim_{d\to 0}{\dfrac{tan(d)-d}{d^3}}$	Dec December 2021 Two green circles are externally tangent and two purple circles are externally tangent at the same point. And all these ...Given any five integers, prove that there must be four of them as $a,b,c,d$ so that 28 divides $(a^2-b^2)(c^2-d^2)$Let $a$, $b$ and $n$ be positive integers such that $n \ge 2$ and $(ab)^{n-1}+1$ divides $a^n+b^n$. Show that $\dfrac{a^n+b^n}{ab^{n-1}+1}$...

2022	Jan Jan 2022 There are five stones $A, B, C, D, E$ in a pond. A frog can jump from one stone to any other stone except current one. I...	Feb February 2022	Mar March 2022 Let $A_1=1998^{50}$, $A_{i+1}$ is the sum of all digits of $A_i$ for $i = 1, 2, … $ until $A_n < 10$. Find $A_n$.	Apr April 2022 Solve the equation group in real numbers:	May May 2022 Solve equation in positive integers $\dfrac{a}{b+c} + \dfrac{b}{c+a} + \dfrac{c}{a+b} = 4$One hundred people line up to board a plane with 100 seats. The first person in line has lost his boarding pass, so he r...What is the possibility that a rope is randomly cut into three pieces and they can form a triangle.Choose $n$ points at random (uniformly and independently) on the circumference of a circle. Find the probability $p_n$ t...	Jun June 2022 $a,b,c > 0, a^{\frac{3}{2}} + b^{\frac{3}{2}} + c^{\frac{3}{2}}=1$, prove:
48	Jul July 2022	Aug August 2022	Sep September 2022 There are 30 ant nests located on a straight line with equal distance between two adjacent nests. And there are 100 ants...Suppose $I$ is the incenter of $\triangle{ABC}$ and $AI=d, BI=e, CI=f$. Find the perimeter and area of $\triangle{ABC}$.Compute $\frac{\sqrt{10+\sqrt{1}}+\sqrt{10+\sqrt{2}}+\sqrt{10+\sqrt{3}}+\cdots +\sqrt{10+\sqrt{99}}}{\sqrt{10-\sqrt{1}}+\sqrt{10-\sqrt{2}}+\sqrt{10-\sqrt{3}}+ \cdots +\sqrt{10-\sqrt{99}}}$...	Oct October 2022 What are the last three digits of $7^{2022}$?Simplify $\cfrac{1+\sqrt{2-\sqrt{2}}+\sqrt{2-\sqrt{3}}}{\sqrt{3}+\sqrt{2+\sqrt{2}}+\sqrt{2+\sqrt{3}}}$	Nov November 2022 In $\triangle{ABC}, AB=AC=10, BC=12$. Point $D$ lies strictly between $A$ and $B$ on $\overline{AB}$ and point $E$ lies ...There are 8 horses in a horse race. Times are rounded to the nearest second, so it is very possible for horses to tie. H...	Dec December 2022

2023	Jan January 2023 $E$ is on side $AD$ of rectangle $ABCD$ such that $AE=3, DE=6, \angle{BEC}=60^{\circ}$, find $[BCE]$	Feb February 2023 (1) How many different ways to arrange 4 couples in a row such that no one sits next to his/her spouse? (2) How many way...	Mar March 2023	Apr April 2023	May May 2023 Square $DEFG$ is inside of $\triangle{ABC}$ such that $D, E, F $ are on side $AB, BC, CA$ with $AD=7, DB=6, BE=EC, CF=2$...(1) Prove $AD^2+BD^2=AF^2+CF^2$ and find $AF$[2010 AMC 12A Problem 23] The number obtained from the last two nonzero digits of $90!$ is equal to $N$. What is $N$?	Jun June 2023 $m, n$ are real numbers. $m^3 + n^3 + 3mn - 1 = 0$. Find $m+n$.$AD$ is bisector of $\angle{BAC}$ in $\triangle{ABC}$, $D$ is on $BC$, and $\angle{BAC}=60^{\circ},AB=2, BC=\sqrt{6}$, f...
92	Jul July 2023 There are $a_1, a_2, \dots, a_m \in { 1, 2, \dots, m }$, $b_1, b_2, \dots, b_m \in { 1, 2, \dots, m }$. Prove that there...	Aug August 2023 Find the values of $a$ such that $x^2 + ax + 5a = 0$ has only integer solutions.Positive integers $x, y, z$ satisfying $x \le y \le z, xyz=2(x+y+z)$, find the sum of possible $z$.$a+\dfrac{1}{b}=3, b+\dfrac{1}{c}=17, c+\dfrac{1}{a}=\dfrac{11}{25}$, find $abc$.Prove that every odd integer that does not have 5 as its ones digit is a factor of some integer consisting of only 9 as ...	Sep September 2023	Oct October 2023	Nov November 2023 $\displaystyle\sum_{n=1}^{\infty}{\dfrac{1}{n^4}}=?$	Dec December 2023

2024	Jan January 2024	Feb February 2024 $a+\dfrac{1}{b}=x, b+\dfrac{1}{c}=y, c+\dfrac{1}{a}=z, $ find $abc$.$x+\dfrac{1}{x}=3$, find $\dfrac{x^6+1}{x^5+x}$$\dfrac{x}{y+z}+\dfrac{y}{z+x}+\dfrac{z}{x+y}=1$, find $\dfrac{x^2}{y+z}+\dfrac{y^2}{z+x}+\dfrac{z^2}{x+y}$$a+b+c=0, \dfrac{1}{a+1}+\dfrac{1}{b+2}+\dfrac{1}{c+3}=0$, find $(a+1)^2+(b+2)^2+(c+3)^2$$abc \neq 0, a+b+c=0$, find $\dfrac{a^2}{bc}+\dfrac{b^2}{ca}+\dfrac{c^2}{ab}$	Mar March 2024 $x,y,z$ are real numbers satisfying $x+y+z=2024, \dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}=\dfrac{1}{2024}$, find $(x-2024)(y-2024)(z-2024)$...$ABCD$ is a cyclic quadrilateral with $AB=7, CD=8$. Points $P, Q$ are on $AB$ such that $AP=BQ=3$ . Points $R, S$ are on...	Apr April 2024	May May 2024 Factorize $x^6-4x^3-1$Solve cubic equation: $x^3=3ax^2+6bx+2c$	Jun June 2024
73	Jul July, 2024 Richard and Vanessa are going to play a game of foosball. The first player to score 5 goals wins. Vanessa is better: at ...In the card game bridge, each of 4 players is dealt a hand of 13 of the 52 cards. What is the probability that each play...	Aug August, 2024 Two points $P$ and $Q$ are randomly chosen on a circle $C$. What is the probability that the smaller arc between $P$ and...	Sep September, 2024	Oct October 2024	Nov November 2024	Dec December 2024

2025	Jan January 2025 The radius of the circumcircle of $\triangle{ABC}$ is $14$, $AB=4, AC=7, AD\perp BC$, find $AD$.	Feb February 2025	Mar March 2025	Apr April 2025 $a+b=1, a^2+b^2=2$, find $a^{11}+b^{11}$	May May 2025 $P$ is a point inside the equilateral triangle $\triangle{ABC}$ and $PA=a,PB=b,PC=c$, find $[ABC]$.solve system $\sqrt{x}+y=11, x+\sqrt{y}=7$	Jun June 2025 Consider a regular n-gon inscribed in the unit circle. Pick one vertex. Now connect it with line segments to all the oth...The function $f$ is defined by $f(x)=\dfrac{ax+b}{cx+d}$, where $a,b,c$ and $d$ are nonzero real numbers, has the proper...
47	Jul July 2025	Aug August 2025	Sep September 2025	Oct October 2025	Nov November 2025 Let $A=\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\cdots+\frac{1}{308}+\frac{1}{309}$ and $B=\frac{308}{1}+\frac{307}{2}+\frac{306}{3}+\cdots+\frac{3}{306}+\frac{2}{307}+\frac{1}{308}$...	Dec December 2025 There are 10 multiple choice questions as following, find their answers.

2026	Jan January 2026 $P$ is a point on the incircle of equilateral triangle $\triangle{ABC}$ and $PB=1, PC=\sqrt{7}$, find the radius of the ...	Feb February 2026	Mar March 2026	Apr April 2026	May May 2026	Jun June 2026 Find the minimum value of $\sqrt{x^2+9}+\sqrt{x^2-14x+65}$If $x+y=2$, $x, y > 0$, find the minimum value of $x+\sqrt{x^2+3y^2}$Find the maximum value of $\sqrt{x^2+10x+41}-\sqrt{x^2+1}$ for $x>0$Find the minimum value of $x^2+\sqrt{x^4-3x^2+2x+5}$Find the minimum value of $\dfrac{x}{2}+\sqrt{x^2-4x+7}$Find the maximum value of $2\sqrt{x-1}+\sqrt{3-2x}$Find the maximum value of $5\sqrt{x-1}+\sqrt{10-2x}$Find the maximum value of $\sqrt{2-x}+\sqrt{8+2x}$
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