Found problems: 85335
2025 Harvard-MIT Mathematics Tournament, 3
Point $P$ lies inside square $ABCD$ such that the areas of $\triangle{PAB}, \triangle{PBC}, \triangle{PCD},$ and $\triangle{PDA}$ are $1, 2, 3,$ and $4,$ in some order. Compute $PA \cdot PB \cdot PC \cdot PD.$
2016 India IMO Training Camp, 1
We say a natural number $n$ is perfect if the sum of all the positive divisors of $n$ is equal to $2n$. For example, $6$ is perfect since its positive divisors $1,2,3,6$ add up to $12=2\times 6$. Show that an odd perfect number has at least $3$ distinct prime divisors.
[i]Note: It is still not known whether odd perfect numbers exist. So assume such a number is there and prove the result.[/i]
1988 IMO Longlists, 93
Given a natural number $n,$ find all polynomials $P(x)$ of degree less than $n$ satisfying the following condition \[ \sum^n_{i=0} P(i) \cdot (-1)^i \cdot \binom{n}{i} = 0. \]
2009 Canadian Mathematical Olympiad Qualification Repechage, 6
Triangle $ABC$ is right-angled at $C$. $AQ$ is drawn parallel to $BC$ with $Q$ and $B$ on opposite sides of $AC$ so that when $BQ$ is drawn, intersecting $AC$ at $P$, we have $PQ = 2AB$. Prove that $\angle ABC = 3\angle PBC$.
1991 Vietnam Team Selection Test, 2
For every natural number $n$ we define $f(n)$ by the following rule: $f(1) = 1$ and for $n>1$ then $f(n) = 1 + a_1 \cdot p_1 + \ldots + a_k \cdot p_k$, where $n = p_1^{a_1} \cdots p_k^{a_k}$ is the canonical prime factorisation of $n$ ($p_1, \ldots, p_k$ are distinct primes and $a_1, \ldots, a_k$ are positive integers). For every positive integer $s$, let $f_s(n) = f(f(\ldots f(n))\ldots)$, where on the right hand side there are exactly $s$ symbols $f$. Show that for every given natural number $a$, there is a natural number $s_0$ such that for all $s > s_0$, the sum $f_s(a) + f_{s-1}(a)$ does not depend on $s$.
2022 Dutch Mathematical Olympiad, 4
In triangle $ABC$, the point $D$ lies on segment $AB$ such that $CD$ is the angle bisector of angle $\angle C$. The perpendicular bisector of segment $CD$ intersects the line $AB$ in $E$. Suppose that $|BE| = 4$ and $|AB| = 5$.
(a) Prove that $\angle BAC = \angle BCE$.
(b) Prove that $2|AD| = |ED|$.
[asy]
unitsize(1 cm);
pair A, B, C, D, E;
A = (0,0);
B = (2,0);
C = (1.8,1.8);
D = extension(C, incenter(A,B,C), A, B);
E = extension((C + D)/2, (C + D)/2 + rotate(90)*(C - D), A, B);
draw((E + (0.5,0))--A--C--B);
draw(C--D);
draw(interp((C + D)/2,E,-0.3)--interp((C + D)/2,E,1.2));
dot("$A$", A, SW);
dot("$B$", B, S);
dot("$C$", C, N);
dot("$D$", D, S);
dot("$E$", E, S);
[/asy]
2010 Contests, 2
There are $n$ students standing in a circle, one behind the other. The students have heights $h_1<h_2<\dots <h_n$. If a student with height $h_k$ is standing directly behind a student with height $h_{k-2}$ or less, the two students are permitted to switch places. Prove that it is not possible to make more than $\binom{n}{3}$ such switches before reaching a position in which no further switches are possible.
1989 IMO Longlists, 19
Let $ a_1, \ldots, a_n$ be distinct positive integers that do not contain a $ 9$ in their decimal representations. Prove that the following inequality holds
\[ \sum^n_{i\equal{}1} \frac{1}{a_i} \leq 30.\]
2011 Saudi Arabia Pre-TST, 1.2
Find all triples $(a, b, c)$ of integers such that $a+ b + c = 2010 \cdot 2011 $ and the solutions to the equation $$2011x^3 +ax^2 +bx+c = 0$$ are all nonzero integers.
2006 China Team Selection Test, 3
Given $n$ real numbers $a_1$, $a_2$ $\ldots$ $a_n$. ($n\geq 1$). Prove that there exists real numbers $b_1$, $b_2$ $\ldots$ $b_n$ satisfying:
(a) For any $1 \leq i \leq n$, $a_i - b_i$ is a positive integer.
(b)$\sum_{1 \leq i < j \leq n} (b_i - b_j)^2 \leq \frac{n^2-1}{12}$
1987 All Soviet Union Mathematical Olympiad, 444
Prove that $1^{1987} + 2^{1987} + ... + n^{1987}$ is divisible by $n+2$.
1994 China Team Selection Test, 3
Find the smallest $n \in \mathbb{N}$ such that if any 5 vertices of a regular $n$-gon are colored red, there exists a line of symmetry $l$ of the $n$-gon such that every red point is reflected across $l$ to a non-red point.
2011 NZMOC Camp Selection Problems, 3
There are $16$ competitors in a tournament, all of whom have different playing strengths and in any match between two players the stronger player always wins. Show that it is possible to find the strongest and second strongest players in $18$ matches.
2004 Harvard-MIT Mathematics Tournament, 5
There exists a positive real number $x$ such that $ \cos (\arctan (x)) = x $. Find the value of $x^2$.
2018 Iranian Geometry Olympiad, 5
$ABCD$ is a cyclic quadrilateral. A circle passing through $A,B$ is tangent to segment $CD$ at point $E$. Another circle passing through $C,D$ is tangent to $AB$ at point $F$. Point $G$ is the intersection point of $AE,DF$, and point $H$ is the intersection point of $BE$, $CF$. Prove that the incenters of triangles $AGF$, $BHF$, $CHE$, $DGE$ lie on a circle.
Proposed by Le Viet An (Vietnam)
1963 Kurschak Competition, 1
$mn$ students all have different heights. They are arranged in $m > 1$ rows of $n > 1$. In each row select the shortest student and let $A$ be the height of the tallest such. In each column select the tallest student and let $B$ be the height of the shortest such. Which of the following are possible: $A < B$, $A = B$, $A > B$? If a relation is possible, can it always be realized by a suitable arrangement of the students?
2014 HMNT, 9
For any positive integers $a$ and $b$, define $a \oplus b$ to be the result when adding $a$ to $b$ in binary (base $2$), neglecting any carry-overs. For example, $20 \oplus 14 = 10100_2 \oplus 1110_2 = 11010_2 = 26$.
(The operation $\oplus$ is called the [i]exclusive or.[/i])
Compute the sum $$\sum^{2^{2014} -1}_{k=0} \left( k \oplus \left\lfloor \frac{k}{2} \right \rfloor \right).$$ Here $\lfloor x\rfloor$ is the greatest integer not exceeding $x$.
2014 Contests, 2
Roy's cat eats $\frac{1}{3}$ of a can of cat food every morning and $\frac{1}{4}$ of a can of cat food every evening. Before feeding his cat on Monday morning, Roy opened a box containing $6$ cans of cat food. On what day of the week did the cat finish eating all the cat food in the box?
${ \textbf{(A)}\ \text{Tuesday}\qquad\textbf{(B)}\ \text{Wednesday}\qquad\textbf{(C)}\ \text{Thursday}\qquad\textbf{(D)}}\ \text{Friday}\qquad\textbf{(E)}\ \text{Saturday}$
2018 Math Prize for Girls Problems, 7
For every positive integer $n$, let $T_n = \frac{n(n+1)}{2}$ be the $n^{\text{th}}$ triangular number. What is the $2018^{\text{th}}$ smallest positive integer $n$ such that $T_n$ is a multiple of 1000?
2002 Italy TST, 1
Given that in a triangle $ABC$, $AB=3$, $BC=4$ and the midpoints of the altitudes of the triangle are collinear, find all possible values of the length of $AC$.
2009 AMC 12/AHSME, 9
Triangle $ ABC$ has vertices $ A\equal{}(3,0)$, $ B\equal{}(0,3)$, and $ C$, where $ C$ is on the line $ x\plus{}y\equal{}7$. What is the area of $ \triangle ABC$?
$ \textbf{(A)}\ 6\qquad
\textbf{(B)}\ 8\qquad
\textbf{(C)}\ 10\qquad
\textbf{(D)}\ 12\qquad
\textbf{(E)}\ 14$
Estonia Open Junior - geometry, 2013.2.3
In an isosceles right triangle $ABC$ the right angle is at vertex $C$. On the side $AC$ points $K, L$ and on the side $BC$ points $M, N$ are chosen so that they divide the corresponding side into three equal segments. Prove that there is exactly one point $P$ inside the triangle $ABC$ such that $\angle KPL = \angle MPN = 45^o$.
2014 ISI Entrance Examination, 7
Let $f: [0,\infty)\to \mathbb{R}$ a non-decreasing function. Then show this inequality holds for all $x,y,z$ such that $0\le x<y<z$.
\begin{align*} & (z-x)\int_{y}^{z}f(u)\,\mathrm{du}\ge (z-y)\int_{x}^{z}f(u)\,\mathrm{du} \end{align*}
1966 IMO Shortlist, 17
Let $ABCD$ and $A^{\prime }B^{\prime}C^{\prime }D^{\prime }$ be two arbitrary parallelograms in the space, and let $M,$ $N,$ $P,$ $Q$ be points dividing the segments $AA^{\prime },$ $BB^{\prime },$ $CC^{\prime },$ $DD^{\prime }$ in equal ratios.
[b]a.)[/b] Prove that the quadrilateral $MNPQ$ is a parallelogram.
[b]b.)[/b] What is the locus of the center of the parallelogram $MNPQ,$ when the point $M$ moves on the segment $AA^{\prime }$ ?
(Consecutive vertices of the parallelograms are labelled in alphabetical order.
1949 Moscow Mathematical Olympiad, 157
a) Prove that if a planar polygon has several axes of symmetry, then all of them intersect at one point.
b) A finite solid body is symmetric about two distinct axes. Describe the position of the symmetry planes of the body.