This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 85335

2024 Malaysia IMONST 2, 3
Tags: number theory
Ivan claims that for all positive integers $n$, $$\left\lfloor\sqrt[2]{\frac{n}{1^3}}\right\rfloor + \left\lfloor\sqrt[2]{\frac{n}{2^3}}\right\rfloor + \left\lfloor\sqrt[2]{\frac{n}{3^3}}\right\rfloor + \cdots = \left\lfloor\sqrt[3]{\frac{n}{1^2}}\right\rfloor + \left\lfloor\sqrt[3]{\frac{n}{2^2}}\right\rfloor + \left\lfloor\sqrt[3]{\frac{n}{3^2}}\right\rfloor + \cdots$$ Why is he correct? (Note: $\lfloor x \rfloor$ denotes the floor function.)

1972 Vietnam National Olympiad, 4
Tags: geometry , tetrahedron , angles , 3-Dimensional Geometry , solid geometry , 3D geometry
Let $ABCD$ be a regular tetrahedron with side $a$. Take $E,E'$ on the edge $AB, F, F'$ on the edge $AC$ and $G,G'$ on the edge AD so that $AE =a/6,AE' = 5a/6,AF= a/4,AF'= 3a/4,AG = a/3,AG'= 2a/3$. Compute the volume of $EFGE'F'G'$ in term of $a$ and find the angles between the lines $AB,AC,AD$ and the plane $EFG$.

2017 Hong Kong TST, 2
Tags: geometry
Two circles $\omega_1$ and $\omega_2$, centered at $O_1$ and $O_2$, respectively, meet at points $A$ and $B$. A line through $B$ intersects $\omega_1$ again at $C$ and $\omega_2$ again at $D$. The tangents to $\omega_1$ and $\omega_2$ at $C$ and $D$, respectively, meet at $E$, and the line $AE$ intersects the circle $\omega$ through $AO_1O_2$ at $F$. Prove that the length of segment $EF$ is equal to the diameter of $\omega$.

2021 Saudi Arabia IMO TST, 4
Tags: IMO Shortlist , combinatorics , IMO Shortlist 2020 , quadrilateral , colorings , partition , Gerhard Woeginger
In a regular 100-gon, 41 vertices are colored black and the remaining 59 vertices are colored white. Prove that there exist 24 convex quadrilaterals $Q_{1}, \ldots, Q_{24}$ whose corners are vertices of the 100-gon, so that [list] [*] the quadrilaterals $Q_{1}, \ldots, Q_{24}$ are pairwise disjoint, and [*] every quadrilateral $Q_{i}$ has three corners of one color and one corner of the other color. [/list]

1980 AMC 12/AHSME, 15
Tags:
A store prices an item in dollars and cents so that when 4% sales tax is added, no rounding is necessary because the result is exactly $n$ dollars where $n$ is a positive integer. The smallest value of $n$ is $\text{(A)} \ 1 \qquad \text{(B)} \ 13 \qquad \text{(C)} \ 25 \qquad \text{(D)} \ 26 \qquad \text{(E)} \ 100$

2013 Purple Comet Problems, 6
Tags:
Pete's research shows that the number of nuts collected by the squirrels in any park is proportional to the square of the number of squirrels in that park. If Pete notes that four squirrels in a park collect $60$ nuts, how many nuts are collected by $20$ squirrels in a park?

2016-2017 SDML (Middle School), 6
Tags: SDML Middle School , 2016-17 SDML Round 2
What is the probability that a random arrangement of the letters in the word 'ARROW' will have both R's next to each other? $\text{(A) }\frac{1}{10}\qquad\text{(B) }\frac{2}{15}\qquad\text{(C) }\frac{1}{5}\qquad\text{(D) }\frac{3}{10}\qquad\text{(E) }\frac{2}{5}$

2020 Stanford Mathematics Tournament, 5
Tags: geometry , 3D geometry
Find the smallest possible number of edges in a convex polyhedron that has an odd number of edges in total has an even number of edges on each face.

2000 Bundeswettbewerb Mathematik, 1a
Tags:
Two natural numbers have the same decimal digits in different order and have the sum $999\cdots 999$. Is this possible when each of the numbers consists of $1999$ digits?

2004 Spain Mathematical Olympiad, Problem 5
Tags: geometry , median , Inscribed circle
Demonstrate that the condition necessary so that, in triangle ${ABC}$, the median from ${B}$ is divided into three equal parts by the inscribed circumference of a circle is: ${A/5 = B/10 = C/13}$.

2017 Novosibirsk Oral Olympiad in Geometry, 7
Tags: geometry , angles
A car is driving along a straight highway at a speed of $60$ km per hour. Not far from the highway there is a parallel to him a $100$-meter fence. Every second, the passenger of the car measures the angle at which the fence is visible. Prove that the sum of all the angles he measured is less than $1100^o$

1997 Balkan MO, 1
Tags: geometry , trigonometry , inequalities , geometry proposed
Suppose that $O$ is a point inside a convex quadrilateral $ABCD$ such that \[ OA^2 + OB^2 + OC^2 + OD^2 = 2\mathcal A[ABCD] , \] where by $\mathcal A[ABCD]$ we have denoted the area of $ABCD$. Prove that $ABCD$ is a square and $O$ is its center. [i]Yugoslavia[/i]

Durer Math Competition CD Finals - geometry, 2012.D5
Tags: geometry , area of a triangle , color
The points of a circle of unit radius are colored in two colors. Prove that $3$ points of the same color can be chosen such that the area of the triangle they define is at least $\frac{9}{10}$.

2004 Germany Team Selection Test, 3
Tags: linear algebra , matrix , vector , induction , algorithm , pigeonhole principle , combinatorics solved
We consider graphs with vertices colored black or white. "Switching" a vertex means: coloring it black if it was formerly white, and coloring it white if it was formerly black. Consider a finite graph with all vertices colored white. Now, we can do the following operation: Switch a vertex and simultaneously switch all of its neighbours (i. e. all vertices connected to this vertex by an edge). Can we, just by performing this operation several times, obtain a graph with all vertices colored black? [It is assumed that our graph has no loops (a [i]loop[/i] means an edge connecting one vertex with itself) and no multiple edges (a [i]multiple edge[/i] means a pair of vertices connected by more than one edge).]

2000 Iran MO (3rd Round), 3
Tags: function , floor function , algebra proposed , algebra
Suppose $f : \mathbb{N} \longrightarrow \mathbb{N}$ is a function that satisfies $f(1) = 1$ and $f(n + 1) =\{\begin{array}{cc} f(n)+2&\mbox{if}\ n=f(f(n)-n+1),\\f(n)+1& \mbox{Otherwise}\end {array}$ $(a)$ Prove that $f(f(n)-n+1)$ is either $n$ or $n+1$. $(b)$ Determine$f$.

2003 AMC 12-AHSME, 20
Tags: counting , distinguishability , AMC12
How many $ 15$-letter arrangements of $ 5$ A's, $ 5$ B's, and $ 5$ C's have no A's in the first $ 5$ letters, no B's in the next $ 5$ letters, and no C's in the last $ 5$ letters? $ \textbf{(A)}\ \sum_{k\equal{}0}^5\binom{5}{k}^3 \qquad \textbf{(B)}\ 3^5\cdot 2^5 \qquad \textbf{(C)}\ 2^{15} \qquad \textbf{(D)}\ \frac{15!}{(5!)^3} \qquad \textbf{(E)}\ 3^{15}$

2014 Contests, 4
Tags: geometry , circumcircle , trapezoid , geometry proposed
$ABC$ is an acute triangle with orthocenter $H$. Points $D$ and $E$ lie on segment $BC$. Circumcircle of $\triangle BHC$ instersects with segments $AD$,$AE$ at $P$ and $Q$, respectively. Prove that if $BD^2+CD^2=2DP\cdot DA$ and $BE^2+CE^2=2EQ\cdot EA$, then $BP=CQ$.

Russian TST 2021, P2
Tags: geometry , rhombus , geometric transformation , IMO Shortlist , IMO Shortlist 2020 , Hi
Let $ABCD$ be a cyclic quadrilateral. Points $K, L, M, N$ are chosen on $AB, BC, CD, DA$ such that $KLMN$ is a rhombus with $KL \parallel AC$ and $LM \parallel BD$. Let $\omega_A, \omega_B, \omega_C, \omega_D$ be the incircles of $\triangle ANK, \triangle BKL, \triangle CLM, \triangle DMN$. Prove that the common internal tangents to $\omega_A$, and $\omega_C$ and the common internal tangents to $\omega_B$ and $\omega_D$ are concurrent.

1965 Putnam, A2
Tags: Putnam , binomial coefficients
Show that, for any positive integer $n$, \[ \sum_{r=0}^{[(n-1)/2]}\left\{\frac{n-2r}n\binom nr\right\}^2 = \frac 1n\binom{2n-2}{n-1}, \] where $[x]$ means the greatest integer not exceeding $x$, and $\textstyle\binom nr$ is the binomial coefficient "$n$ choose $r$", with the convention $\textstyle\binom n0 = 1$.

2023 Irish Math Olympiad, P4
Tags: function , algebra , functional equation
Find all functions $f:\mathbb{R} \rightarrow \mathbb{R}$ with the property that $$f(x)f(y) = (xy - 1)^2f\left(\frac{x + y - 1}{xy - 1}\right)$$ for all real numbers $x, y$ with $xy \neq 1$.

Kyiv City MO Seniors 2003+ geometry, 2021.10.3
Tags: geometry , circles , equal segments , parallelogram
Circles $\omega_1$ and $\omega_2$ with centers at points $O_1$ and $O_2$ intersect at points $A$ and $B$. A point $C$ is constructed such that $AO_2CO_1$ is a parallelogram. An arbitrary line is drawn through point $A$, which intersects the circles $\omega_1$ and $\omega_2$ for the second time at points $X$ and $Y$, respectively. Prove that $CX = CY$. (Oleksii Masalitin)

2009 HMNT, 3
Tags: geometry , analytic geometry
Let $C$ be the circle of radius $12$ centered at $(0, 0)$. What is the length of the shortest path in the plane between $(8\sqrt3, 0)$ and $(0, 12 \sqrt2)$ that does not pass through the interior of $C$?

2006 National Olympiad First Round, 6
Tags:
What is the sum of $3+3^2+3^{2^2} + 3^{2^3} + \dots + 3^{2^{2006}}$ in $\mod 11$? $ \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 5 \qquad\textbf{(E)}\ 10 $

2006 Tournament of Towns, 1
Tags: combinatorics , combinatorial geometry
Prove that one can always mark $50$ points inside of any convex $100$-gon, so that each its vertix is on a straight line connecting some two marked points. (4)

2006 VJIMC, Problem 4
Tags: matrix , linear algebra
Let $A=[a_{ij}]_{n\times n}$ be a matrix with nonnegative entries such that $$\sum_{i=1}^n\sum_{j=1}^na_{ij}=n.$$ (a) Prove that $|\det A|\le1$. (b) If $|\det A|=1$ and $\lambda\in\mathbb C$ is an arbitrary eigenvalue of $A$, show that $|\lambda|=1$.